English العربية کوردی
01007022

Dr. Majeed Ahmad Yousif

Assistant Professor
College of Education / Department of Mathematics
9647504738150
majeed.yousif@uoz.edu.krd
Number of Publications: 77
2
Awards
77
Published Journal Articles
2
Thesis
0
Training Course

0
Book Chapter
2
Conference
0
Presentation
0
Workshop
2024
Doctor of Philosophy

Mathematics
University of Zakho

2020
Doctor of Philosophy

Mathematics
University of Zakho

2013
Master

Mathematics
University of Zakho

2010
Bachelor

Mathematics
University of Duhok

MATLAB Software Mathematica Software

Google scholar
Researchgate
Orcid
Publons

2024

Assistant Professor

Mathematics Department, College of Education, University of Zakho

2024-04-30,current

2018

Lecturer

Mathematics Department, College of Education, University of Zakho

2018-12-27,2024-04-29

2015

Assistant Lecturer

Mathematics Department, College of Science, University of Zakho

2015-03-10,2018-12-26

2011

Assistant Researcher

Mathematics Department, College of Science, University of Duhok

2011-04-10,2015-03-09
Assistant Professor

Date: 2024-04-30

Lecturer

Date: 2018-12-27

Assistant Lecturer

Date: 2015-03-10

Assistant Researcher

Date: 2011-04-10

2017

Best Researcher at University of Zakho

Best Researcher at University of Zakho

2017-10

2010

Top Students' Awards

Top Students' Awards at University of Duhok

2010-07

2026

Novel Sequence Spaces via Tribonacci and Lacunary Sequences: Theoretical Insights, Structural Properties, and Neural Network Comparison
Fractals (Issue : 2026) (Volume : 2026)
The Tribonacci sequence, an extension of the Fibonacci sequence, exhibits rapid growth and a distinctive recursive structure. It plays a significant role in algebra, combinatorics, number theory, dynamical systems, and fractal geometry. Moreover, its applications extend to computer science-for modeling recursive algorithms, cryptographic structures-and to the natural sciences, such as population dynamics and biological growth patterns. In this study, we construct new sequence spaces based on the fusion of Tribonacci and lacunary sequences, generalized through modulus functions under appropriate conditions. We focus particularly on ψ-Tribonacci lacunary statistical convergence of order α and ψ-strong Tribonacci lacunary summability of order α. To strengthen our theoretical contributions, we integrate Neural Networks as computational tools for analyzing the convergence behavior and structural relationships among these sequence spaces. Several illustrative examples, supported by numerical simulations and figures, demonstrate the practical interpretation and effectiveness of the proposed concepts. Furthermore, refinements in lacunary sequences allow us to establish deeper interrelations, contributing to a more comprehensive understanding of these spaces and their applications in both mathematical theory and computational frameworks.
2026-05
Applications of Lucas sequences in convergence and signal processing
Scientific Reports (Issue : 2026) (Volume : 2026)
In this paper we propose a new perspective related to Lucas sequence with statistical and summability, some considerations have been given in connection with their applications of signal processing. We define the notions of lambda-Lucas statistically convergent and strongly lambda-Lucas summability through modulus functions. The study of investigates on the Lucas transform and its embedding diagram into the representative form of Lucas numbers is studied in the context of convergence and summability problems. Furthermore, we provide inclusions as well as principles of equivalence, which lead to conditions guaranteeing the uniqueness of restrictions and generalize statistical convergence theory. Numerical simulations on a wide variety of signals such as noisy sinusoidal signals and blurred images demonstrate the flexibility of the proposed approach. These experiments show a steady decay to zero, which implies considerable noise immunity. In addition, by providing alternative perspectives of assessing signal quality, compression strategies and filtering techniques, those methods help in distinguishing the relevant signal information from background noise. Bringing together theoretical results and practical real-world signal analysis examples, the paper introduces classical and recent contributions to summability theory as well as new signal-processing procedures.
2026-04
Discrete fractional operators of delta-type with Mittag-Leffler kernels and applications
Journal of Computational and Applied Mathematics (Issue : 15) (Volume : 485)
Mittag–Leffler function is a special function, which is easily affected by the measurement kernel. This paper recall a new discrete delta Mittag–Leffler function derived as an extension of the existing nabla Mittag–Leffler function for fractional difference operators. A plenty of properties of our new special function have been derived. Next, we derive the AB fractional differences and sums with our new discrete delta Mittag–Leffler function in their kernel from the left and right sides associated with both Riemann–Liouville and Liouville–Caputo operators. These operators have been reformulated as Laplace transformation summations. Moreover, integration by parts has been stated and examined for the Riemann–Liouville fractional difference operators. The applicability of these types in the Euler-Lagrange equations has been shown in the last part of the paper.
2026-03
Stability analysis of discrete delta fractional models under summation multipoint constraints for robust engineering systems
Scientific Reports (Issue : 1) (Volume : 2026)
When real-world systems like sensor networks or controllers face small disturbances, do their discrete fractional models stay reliable? We tackle this for a critical but unstudied problem: discrete delta fractional equations with summation multipoint boundary conditions (SMBCs). First, we build the mathematical foundation—the Green’s function—to prove solutions exist and are unique under Lipschitz conditions. Crucially, we show these solutions are Ulam–Hyers–Rassias stable: even when perturbed, they stay close to the true solution, with explicit guarantees for both uniform and varying disturbances. Testing this on a nonlinear system and a thermal sensor model confirms real-world resilience. This work bridges abstract fractional calculus with engineering robustness, giving designers confidence to deploy these models in next-generation technologies.
2026-03
Extensions and Improvements of Jensen's and Majorization Type Inequalities for p-harmonic Convex Functions
Contemporary Mathematics (Issue : 2026) (Volume : 7)
In this paper some integral inequalities of Jensen's type for generalised non-convex functions defined on real-line intervals and some results on Majorization type inequalities are established. We introduce a novel category of harmonic convex functions, referred to as p-harmonic convex functions. Additionally, new findings are also presented, with generalizations of previously established results. Further validation of the obtained solutions with the existing literature is also discussed.
2026-03
Variable-Order Fractional Delay Differential Equations with Integral Boundary Values: A Study on Existence, Uniqueness, and Stability
European Journal of Mathematical Sciences (Issue : 1) (Volume : 19)
Variable-order fractional differential equations (VO-FDEs) significantly generalize classical fractional calculus by permitting the order of differentiation to vary with time or other system parameters. This flexibility offers a powerful framework for modeling complex phenomena characterized by evolving memory and dynamic heterogeneities, features that are beyond the reach of constant-order models. In this work, we conduct a rigorous analysis of a coupled system of VO-FDEs that incorporates multiple delays and nonlocal integral boundary conditions within the Caputo formalism. Our investigation first establishes sufficient criteria for the existence and uniqueness of solutions using Banach’s and Schauder’s fixed-point theorems. We then perform a comprehensive stability analysis, deriving explicit conditions for Ulam–Hyers and Ulam–Hyers–Rassias stability to guarantee the robustness of solutions against small perturbations. The practical applicability of our theoretical findings is demonstrated through detailed numerical examples, which serve to validate the efficacy of the proposed framework.
2026-02
Analytic function classes defined by Mittag–Leffler inspired Poisson-type series
Alexandria Engineering Journal (Issue : 136) (Volume : 136)
In this article, we introduce new subclass S of univalent functions based on Mittag-Leffler function related to the distribution series within the unit disc U. Further the fundamental properties such as growth, distortion, extreme points, convexity, close-to-convexity, starlike and coefficient inequalities have been estimated for the subclass. In addition, we consider an integral means of inequality for the subclass. This work bridges the gap between fractional integral operators and Poisson distribution series by incorporating both into a new subclass of univalent functions. This integrative approach provides the valuable insights into the applications of geometric function theory in signal and image processing.
2026-01
A Novel Family of Discrete Variant Inequality for Jensen and Hermite-Hadamard Types in the p-Harmonic Convex Function Setting
Contemporary Mathematics (Issue : 1) (Volume : 7)
This paper introduces a novel form of the discrete Jensen-type inequality specifically designed for p-harmonic convex functions and extend it to the broader family of (p, h)-harmonic convex functions. Moreover, we have established several majorization-type inequalities in the setting of (p, h)-harmonic convexity. In addition, certain refinements of these inequalities are provided through Hermite-Hadamard-type results for p-harmonically convex functions.
2026-01

2025

Propagation of solitary waves for hydrodynamical nonlinear complex model in a fractional derivative setting
Scientific Reports (Issue : 2025) (Volume : 2025)
This study investigates soliton solutions and dynamic wave behaviors in the complex Ginzburg–Landau equation, a model that plays a central role in describing diverse physical phenomena such as superconductivity, nonlinear optical fibers, liquid crystals, second-order phase transitions, and field theory strings. To derive closed-form solutions, we employ two advanced analytical techniques: the new rational extended sinh-Gordon equation expansion method (ShGEEM) and the modified generalized exponential rational function method (mGERFM). These methods yield a wide range of solitonic structures, such as complex and singular solitons, oscillatory periodic waves, bright, dark, and multi-wave profiles. In this work, new families of exact solitary wave solutions with ShGEEM and several hyperbolic, trigonometric, and exponential solutions with mGERFM are presented. Further, the obtained solutions are checked for accuracy by substituting them back into Mathematica. For the dynamics of solutions, 2D plots, 3D surfaces, and contour graphs have been constructed for some values of parameters in the presence of the beta-fractional derivative to understand wave structures and their evolution. In general, the present study not only consolidates the aspects of nonlinear wave dynamics in the field of chemical and physical oceanography but also provides pathways for further research on nonlinear fractional-order models. The originality of the present study lies in the point that the complex Ginzburg–Landau equation has not been studied within the ShGEEM and mGERFM frameworks.
2025-12
On Nonlinear Monge-Ampère Equations and Their Symmetry Classifications
Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería (Issue : 4) (Volume : 41)
The Monge-Ampère equation (MAE) plays a pivotal role across a broad spectrum of theoretical and applied sciences, with its solutions being essential for advancing various fields. This study explores all forms of the fully nonlinear MAE using Lie group transformations to reduce the equation into solvable forms. Analytical solutions are derived using ansatzbased methods, yielding novel and generalized results that enhance the existing body of knowledge. In particular, solutions for cases with diverse source functions and boundary conditions are obtained, addressing gaps in the literature. Stability of the solutions is studied through both analytical and numerical approaches. Comparisons with existing solutions demonstrate the efficiency and generality of the proposed methods. The results presented in this work are poised to impact numerous applications, providing a robust framework for further research on MAEs
2025-12
New horizons in analytic function classes induced by the Erdélyi–Kober fractional integral operators
An International Journal of Optimization and Control: Theories & Applications (Issue : 2025) (Volume : 2025)
This study investigates a new subclass of analytic and univalent functions in the open unit disk, through the convolution of normalized analytic functions with a generalized Erdélyi–Kober fractional integral operator. The main objective is to define a new subclass TS(β, γ) related with the normalized form of the Erdélyi–Kober fractional integral operator Kϑδ and explore its geometrical properties. This study obtains sharp coefficient bounds and geometric characteristics such as growth, and distortion properties. Furthermore, convexity, close-to-convexity, radii of and starlike-ness, extremal functions, and inclusion relations are determined. These results contribute to the broader understanding of defined subclass within geometric function theory and provide a mathematical foundation for modelling phenomena in fractional calculus, conformal mapping, and applied engineering contexts. The study also highlights limitations associated with the operator parameters and suggests extensions to numerical and control-based models for future investigation.
2025-12
A novel cryptographic framework and mathematical modeling for secure transmission of Parkinson’s disease data using RSA and block-based secret sharing
Scientific Reports (Issue : 2025) (Volume : 2025)
The secure transmission of medical data is an essential requirement in modern telemedicine systems, particularly for chronic neurological disorders such as Parkinson’s disease. This paper proposes a novel hybrid cryptographic framework that combines RSA encryption with block-based secret sharing enhanced by a Hilbert matrix-driven mathematical model. The framework introduces dynamic block-wise key generation and adaptive sharing to strengthen data confidentiality and robustness against cryptanalytic attacks. Mathematical modeling is employed to analyze encryption stability, numerical conditioning of the Hilbert matrix, and the diffusion properties of the key space. The proposed method is validated using publicly available Parkinson’s EEG and spiral drawing datasets, with quantitative analysis including encryption/decryption time, computational overhead, and image quality metrics (PSNR, SSIM). The framework is further benchmarked against AES-Shamir and ECC-based hybrid models. Experimental results indicate that the proposed system achieves higher security entropy and lower computational cost, making it suitable for deployment in resource-constrained medical IoT environments.
2025-12
Stability Results for Stochastic Delay Differential Equations in the Framework of Conformable Fractional Derivatives
European Journal of Pure and Applied Mathematics (Issue : 4) (Volume : 18)
Well-Posedness is essential in various scientific and engineering fields, including physics, engineering, biological sciences, economics, and environmental science. Well-posedness ensures that the mathematical model corresponds to a physically meaningful situation. Without well-posedness, the solution might not make sense in the context of the problem being modeled. The concept of well-posedness refers to certain desirable properties that a differential equation must satisfy, which are existence, uniqueness, and continuous dependency. Regularization is an additional feature of the solution of the differential equation, such as the smoothness of the solution. Stability theory is one of the indispensable qualitative concepts of dynamical systems. We established results about the well-posedness, regularity, and Ulam-Hyers stability of solutions to conformable fractional stochastic delay differential equations. First, we discussed the results about the existence and uniqueness of solutions when the global and local Lipschitz conditions of the coefficients are satisfied. Second, we demonstrated the results about continuously depend on the fractional order ξ and initial values under the global Lipschitz condition of coefficients. Thirdly, we constructed results regarding regularity and Ulam-Hyers stability, and two examples that demonstrate our results are presented. The main part of the proof made use of the truncation process, the Banach fixed point theorem, It\^{o} isometry, temporally weighted norm, Gr\"{o}nwall's inequality, and H\"{o}lder's inequality.
2025-11
Integrating Bessel-type summability and artificial neural networks for sequence classification
Modeling Earth Systems and Environment (Issue : 47) (Volume : 12)
The study of sequence convergence and summability has made significant strides, yet exploring alternative frameworks remains a promising avenue of research. This paper introduces a novel approach to the convergence and boundedness of triple sequences, utilizing Bessel functions in combination with modulus functions. This framework provides a refined analytical perspective on sequence behavior in higher dimensions. We construct new sequence spaces based on strong Bessel summability and investigate their structural properties, such as inclusion relations, completeness, and other key characteristics. Additionally, we incorporate Artificial Neural Networks (ANNs) for the classification of sequences generated by linear, quadratic, and exponential functions. Through the integration of ANN models, we demonstrate their ability to classify these sequences with high accuracy, showcasing the potential of machine learning techniques in advancing sequence space theory. Several illustrative examples are provided to highlight the effectiveness of both the theoretical and computational frameworks. The results of this study contribute to the ongoing development of sequence space theory, with implications for future research in multidimensional summability, convergence analysis, and machine learning applications in sequence classification.
2025-11
Generalized Convergence of Sequences of Fuzzy Numbers by Means of Modulus Functions
Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería (Issue : 4) (Volume : 41)
In this paper, we extend the concepts of statistical convergence and strong summability for the sequences of fuzzy numbers using modulus functions. By introducing appropriate conditions on the modulus functions, we generalize and refine existing notions of convergence within the fuzzy setting. Additionally, we establish several interrelationships between these extended concepts, thereby contributing to the deeper understanding of summability and convergence behavior in the sequences of fuzzy numbers.
2025-11
Some New Inequalities Involving Generalized Convex Functions in the Katugampola Fractional Setting
Contemporary Mathematics (Issue : 4) (Volume : 6)
In this study, we explore a new class of convex functions termed cr-log-h-convex functions within the framework of interval-valued functions and the cr-order. We introduce and analyze fundamental properties of these functions and establish several Hermite-Hadamard inequalities by employing Katugampola fractional integrals. To illustrate the theoretical results, we present numerical examples that validate the proposed inequalities. This work extends the understanding of convexity concepts and their applications, offering a broader perspective on inequalities in real analysis and fuzzy systems.
2025-10
On behavior of solution for delta fractional differences associated with special functions
Alexandria Engineering Journal (Volume : 130)
In this paper, a general idea of Mittag-Leffler function using discrete fractional of delta-type in the Riemann–Liouville sense is initiated. Asymptotic behavior of solutions associated with the Riemann–Liouville fractional difference is proposed herein to complement its corresponding Mittag-Leffler solution related to comparison theorems. Illustrative examples are presented to support the assertions of the theoretical results. Additionally, Neural Network (NN) approximation is employed to compare and validate the numerical solutions, demonstrating its effectiveness in approximating discrete fractional problems.
2025-10
Advanced fractional soliton solutions of the Joseph–Egri equation via Tanh–Coth and Jacobi function methods
scientific reports (Issue : 2025) (Volume : 15)
This study introduces new exact soliton solutions of the time-fractional Joseph–Egri equation by employing the Tanh–Coth and Jacobi Elliptic Function methods. Using Jumarie’s modified Riemann–Liouville derivative, a wide variety of soliton structures—such as periodic, bell-shaped, W-shaped, kink, and anti-bell-shaped waves—are obtained and expressed through hyperbolic, trigonometric, and Jacobi functions. The analysis reveals the significant impact of fractional-order derivatives on soliton dynamics, with graphical illustrations highlighting their physical relevance. This work expands the known solution space of the fractional Joseph–Egri equation, demonstrates the effectiveness of advanced analytical techniques, and provides fresh insights into the behavior of fractional nonlinear waves, with potential applications in physics and engineering.
2025-10
Novel Insights into the Theory of Sequence Spaces for Fuzzy Numbers via Bessel Functions
IEEE Access (Issue : 2025) (Volume : 13)
This paper introduces a novel theoretical framework for sequence spaces of fuzzy numbers based on Bessel statistical convergence and strong Bessel summability. The proposed approach integrates Bessel functions with modulus functions, forming a unified setting that generalizes classical convergence and summability concepts in fuzzy analysis. We establish key inclusion relations among the newly defined spaces and extend the classical fuzzy Korovkin-type theorem through the Bessel–modulus framework, offering broader flexibility and applicability than existing results. A new notion, termed the fuzzy rate of Bessel statistical convergence, is also formulated to quantify the speed of convergence in fuzzy approximation processes. The presented framework effectively captures the behavior of non-classically convergent or oscillatory fuzzy sequences and provides a robust foundation for further developments in fuzzy approximation, control systems, and computational mathematics. Illustrative examples and visual analyses are provided to demonstrate the novelty and practical relevance of the proposed theory.
2025-10
FRACTIONAL ANALYSIS OF COEFFICIENT ESTIMATES FOR STARLIKE FUNCTIONS IN SHELL-LIKE DOMAINS
Fractal (Issue : 2025) (Volume : 2025)
Recent studies in analytic functions have increasingly focused on shell-like domains and their connections to fractional operators. This work introduces a novel class of analytic functions, denoted as M-rho-p-q, and explores the Fekete–Szegö inequality along with new coefficient values for this class. To align with fractional analysis, we investigate the applications of the Erdelyi–Kober fractional integral operator within this framework. In addition, we examine the function zeta Phi of zeta and its inverse Phi inverse, extending the analysis to fractional-order settings. A new class, denoted as M-phi-rho-p-q, is defined and studied in relation to convolution with normalized analytic functions. Notably, Fekete–Szegö inequalities are derived for specific subclasses involving the Poisson distribution series, demonstrating how fractional calculus enriches geometric function theory and broadens its applications across mathematical and physical domains. In summary, this study presents an innovative class of analytic functions, M-rho-p-q, and investigates the Fekete–Szegö inequality and new coefficient estimates. The analysis is extended to the function zeta Phi of zeta and its inverse, while a further class, M-phi-rho-p-q, is introduced to explore applications involving convolution with normalized analytic functions. Finally, the derived Fekete–Szegö inequalities for subclasses based on the Poisson distribution series highlight the potential of fractional calculus in advancing geometric function theory.
2025-09
An Analytical Algebraic Method for Solving Nonlinear Fractional Differential Equations with Conformable Fractional Derivatives
Contemporary Mathematics (Issue : 5) (Volume : 6)
This paper presents a new and enhanced algebraic method for accurately solving nonlinear Fractional Differential Equations (FDEs). Using a complex fractional transformation, we translate nonlinear FDEs with Jumarie modified Riemann-Liouville derivatives into their corresponding ordinary differential equations. By applying this method to two nonlinear FDEs, including the time-fractional Bogoyavlenskii problem, we demonstrate its strength and adaptability. The method’s effectiveness in solving various nonlinear FDEs is showcased, laying the groundwork for future developments in this rapidly evolving field. This research has significant implications for addressing complex issues in physics, engineering, and finance, providing a reliable and efficient approach to modeling and analyzing realworld systems. Furthermore, this study advances the field of fractional calculus, which has garnered attention for its ability to illustrate and explain intricate systems and processes.
2025-09
A Novel Hybrid Approach for Obtaining Approximate Series and Exact Solutions to the Caputo Fractional Black-Scholes Equations
Contemporary Mathematics (Issue : 5) (Volume : 6)
This study employs a novel approach to derive both approximate series and exact solutions to the fractional Black-Scholes model, utilizing the Elzaki transform and residual functions. This method is called the Elzaki Residual Method (ERM). By applying the basic limit principle at zero, the ERM demonstrates a superior ability to determine the coefficients of terms in the fractional power series, whereas other well-known methods such as the Adomian decomposition method, the variational iteration method, and the Homotopy perturbation method require integration, and the residual power series method relies on differentiation, both of which are challenging in fractional contexts. Moreover, the ERM outperforms series solution techniques that rely on Adomian and He’s polynomials for solving nonlinear problems, as it eliminates the need for such polynomials. To verify the reliability of our approach, we perform recurrence and absolute error analyses. Additionally, comparative assessments with the projected method, the Adomian method, and the Aboodh decomposition method demonstrate strong agreement. Finally, graphical illustrations further confirm the accuracy of our solutions. Therefore, the ERM can serve as a valuable alternative for solving both linear and nonlinear fractional systems. Maple software is used to calculate the numerical and symbolic quantities in the paper.
2025-09
Representation of Bi-Univalent Functions to Lucas Balancing Polynomials with Geometric Properties and Coefficient Bounds
uropean Journal of Pure and Applied Mathematics (Issue : 3) (Volume : 18)
In this paper, we introduce and analyze a new subclass of bi-univalent functions defined in the open unit disk, associated with Lucas and Lucas balancing polynomials. By employing the Taylor-Maclaurin series expansion, precise bounds for the second and third coefficients, a2 and a3, are obtained. These estimates lead to important geometric interpretations related to the distortion, growth, and structural behavior of the functions near the origin. Furthermore, the mapping characteristics of these functions are examined in connection with existing bi-univalent subclasses. Special emphasis is placed on deriving a Fekete-Szegö type inequality for the proposed class, thereby extending earlier contributions in the field. The findings presented here are valuable for applications in geometric function theory and areas like fluid mechanics, conformal mappings, and engineering models where the analytic structure of bi-univalent functions plays a significant role.
2025-08
Optical Soliton Dynamics in Nonlinear Evolution Equations: Modified Kawahara and Modified Benjamin Bona Mahony Models
RIMNI (Issue : 3) (Volume : 41)
This paper explores the dynamic behavior of optical soliton solutions for the modified Kawahara (mK) equation and the modified BenjaminBona-Mahony (mBBM) equation, two significant nonlinear evolution equations. Using an advanced analytical approach, a diverse set of soliton solutions is derived, including bell-shaped, anti-bell-shaped, W-shaped, M-shaped, and periodic waveforms. These solutions unveil the intricate nonlinear dynamics underlying the equations. The robustness of the method is demonstrated through comprehensive 2D, 3D, and contour visualizations, offering clear insights into the physical significance of the solitons. The study enhances the existing catalog of soliton solutions, contributing to a deeper understanding of nonlinear wave propagation and its potential applications in fields such as optical communication and fluid dynamics
2025-08
Gross-Pitaevskii systems of fractional order with respect to multicomponent solitary wave dynamics
scientific reports (Volume : 15)
This study presents a comparative analysis of the fractional Gross–Pitaevskii equation (GPE), a fundamental nonlinear Schrödinger-type model, focusing on the derivation of exact soliton solutions critical for understanding nonlinear phenomena such as superfluidity and superconductivity. We compare the effectiveness of beta-fractional and M-truncated fractional derivatives in solving the complex fractional GPE. This comparison reveals that while both fractional derivatives enable the construction of diverse optical soliton solutions—including hyperbolic, periodic, Jacobi elliptic, and exponential forms–the beta-derivative provides smoother soliton profiles with computational simplicity, whereas the M-truncated derivative captures richer oscillatory dynamics due to its enhanced memory effects. Employing advanced analytical tools—namely the generalized extended direct algebraic method (gEDAM) and the Kummar-Malik (KM) method—along with Wolfram Mathematica for verification, we extract and rigorously validate a variety of exact solutions. The generated solutions and their corresponding wave profiles under varying parameters highlight the distinct physical implications of each fractional derivative approach. Our results offer a robust framework for modeling nonlinear fractional dynamics, with applications spanning optical fibers, plasma physics, mathematical physics, and condensed matter systems. The comparative insights gained deepen the understanding of fractional-order effects in nonlinear wave evolution, paving the way for further exploration in complex physical systems.
2025-07
Modeling and Neural Network Approximation of Asymptotic Behavior for Delta Fractional Difference Equations with Mittag-Leffler Kernels
Fractal Fract. (Issue : 7) (Volume : 9)
The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order 0.5 subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method.
2025-07
Comparative Results on Stability Analysis for Three Dimensional Functional Equations
European Journal of Pure and Applied Mathematics (Issue : 2) (Volume : 18)
This study investigates the stability of a three-dimensional cubic functional equation within several mathematical frameworks, including (n,beta)-normed spaces, non-Archimedean (n,beta)-normed spaces, and random normed spaces. The theoretical stability results are validated through experimental approaches, offering practical insight into the behavior of these functional equations. A comparative analysis is provided, highlighting differences in stability dynamics across the various spaces. Notably, the introduction of (n,beta)-normed spaces and their non-Archimedean counterparts present a novel framework for analyzing stability, while the inclusion of random normed spaces adds a stochastic dimension to the analysis. The experimental validation further strengthens the practical application of the stability results, distinguishing this study from traditional approaches.
2025-05
Generalization of Ramsey Number for Cycle with Pendant Edges
Mathematics (Issue : 7) (Volume : 13)
This paper explores various Ramsey numbers associated with cycles with pendant edges, including the classical Ramsey number, the star-critical Ramsey number, the Gallai–Ramsey number, and the star-critical Gallai–Ramsey number. These Ramsey numbers play a crucial role in combinatorial mathematics, determining the minimum number of vertices required to guarantee specific monochromatic substructures. We establish upper and lower bounds for each of these numbers, providing new insights into their behavior for cycles with pendant edges—graphs formed by attaching additional edges to one or more vertices of a cycle. The results presented contribute to the broader understanding of Ramsey theory and serve as a foundation for future research on generalized Ramsey numbers in complex graph structures.
2025-04
Solutions comparative for a novel model of fractional delta difference type
Mathematical and Computer Modelling of Dynamical Systems (Issue : 1) (Volume : 31)
We propose a novel delta fractional model with general boundary conditions to make a comparison of the functions and their fractional operators. This allows us to construct the Green’s functions (GFs), which we interpret as the delta fractional differences in the Riemann-Liouville setting. Next, positivity results for the GF together with the delta function have been derived. The appearance of two distinct functions having the same linear operator will be investigated. This further motivates to get the comparison of the functions, corresponding to the same GF. Two particular examples are provided to validate the theoretical findings.
2025-04
Finite difference beta -fractional approach for solving the time-fractional FitzHugh–Nagumo equation
Alexandria Engineering Journal (Issue : 2025) (Volume : 125)
This study presents a numerical approach for addressing the time-fractional FitzHugh–Nagumo (TFFHN) equation, a key equation in physics. The method integrates beta-fractional derivatives s with the finite difference technique. Stability analysis confirms that the proposed method is conditionally stable. Numerical experiments demonstrate its effectiveness, outperforming the cubic B-spline method regarding norm errors. The experimental order of convergence is also presented, highlighting the accuracy and efficiency of the approach, and emphasizing its potential for solving time-fractional differential equations across various physical applications.
2025-04
Solving Time-Fractional Fisher Models by Non-Polynomial Splines in Terms of Logarithmic Derivatives
Fractals (Issue : 4) (Volume : 33)
This paper introduces a novel numerical approach, the logarithmic non-polynomial spline method (LNPSM), leveraging a non-polynomial spline function with logarithmic terms to solve the conformable time-fractional Fisher (TFF) equation. The developed scheme achieves six-order convergence, derived through truncation error analysis and the Taylor series expansion. Stability is ensured under conditional constraints verified by von Neumann stability analysis. The method’s accuracy is demonstrated through two test examples, with results presented in comparison tables alongside cubic B-spline and Caputo non-polynomial spline methods, evalu-ated by norm errors. Additionally, graphical representations, including 2D and 3D plots, further illustrate the effectiveness of LNPSM. The findings indicate that LNPSM is a suitable and ro-bust tool for solving time-fractional differential equations.
2025-04
Non-Similar Analysis of Boundary Layer Flow and Heat Transfer in Non-Newtonian Hybrid Nanofluid over a Cylinder with Viscous Dissipation Effects
Energies (Issue : 7) (Volume : 18)
Highlighting the importance of artificial intelligence and machine learning approaches in engineering and fluid mechanics problems, especially in heat transfer applications is main goal of the presented article. With the advancement in Artificial Intelligence (AI) and Machine Learning (ML) techniques, the computational efficiency and accuracy of numerical results are enhanced. The theme of the study is to use machine learning techniques to examine the thermal analysis of MHD boundary layer flow of Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder embedded in a porous medium with heat source/sink and viscous dissipation effects. The considered base fluid is water (𝐻2𝑂) and hybrid nanoparticles titanium oxide (𝑇𝑖𝑂2) and Copper oxide (𝐶𝑢𝑂). The governing flow equations are nonlinear PDEs. Non-similar system of PDEs are obtained with efficient conversion variables. The dimensionless PDEs are truncated using a local non-similarity approach up to third level and numerical solution is evaluated using MATLAB built-in-function bvp4c. Artificial Neural Networks (ANNs) simulation approach is used to train the networks to predict the solution behavior. Thermal boundary layer improves with the enhancement in the value of 𝑅𝑑. The accuracy and reliability of ANNs predicted solution is addressed with computation of correlation index and residual analysis. The RMSE is evaluated [0.04892, 0.0007597,0.0007596, 0.01546, 0.008871, 0.01686] for various scenarios. It is observed that when concentration of hybrid nanoparticles increases then thermal characteristics of the Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder.
2025-03
Hyers–Ulam–Rassias Stability of Functional Equations with Integrals in B-Metric Frameworks
Symmetry (Issue : 2) (Volume : 17)
This study investigates the stability behavior of nonlinear Fredholm and Volterra integral equations, as well as nonlinear integro-differential equations with Volterra integral terms, through the lens of symmetry principles in mathematical analysis. By leveraging fixed-point methods within b-metric spaces, which generalize classical metric spaces while preserving structural symmetry, we establish sufficient conditions for Hyers–Ulam–Rassias and Hyers–Ulam stability. The symmetric framework of b-metric spaces offers a unified approach to analyzing stability across a wide range of nonlinear systems. To illustrate the theoretical results, examples are provided that underscore the practical applicability and relevance of these findings to complex nonlinear systems, emphasizing their inherent symmetrical properties.
2025-01
Exploring a Novel Approach to Deferred Nörlund Statistical Convergence
Symmetry (Issue : 2) (Volume : 17)
This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.
2025-01

2024

Hyperbolic Non-Polynomial Spline Approach for Time-Fractional Coupled KdV Equations: A Computational Investigation
Symmetry (Issue : 12) (Volume : 16)
The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.
2024-12
High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines
Symmetry (Issue : 1) (Volume : 17)
A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2𝐷 and 3𝐷 graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.
2024-12
Bessel statistical convergence: New concepts and applications in sequence theory
PLOS ONE (Issue : 19) (Volume : 11)
This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines.
2024-11
Improved Fractional Differences with Kernels of Delta Mittag–Leffler and Exponential Functions
Symmetry (Issue : 16) (Volume : 12)
Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators.
2024-11
Weighted Statistical Convergence and Cluster Points: The Fibonacci Sequence-Based Approach Using Modulus Functions
Mathematics (Issue : 23) (Volume : 12)
In this paper, the Fibonacci sequence, renowned for its significance across various fields, its ability to illuminate numerical concepts, and its role in uncovering patterns in mathematics and nature, forms the foundation of this research. This study introduces innovative concepts of weighted density, weighted statistical summability, weighted statistical convergence, and weighted statistical Cauchy, uniquely defined via the Fibonacci sequence and modulus functions. Key theorems, relationships, examples, and properties substantiate these novel principles, advancing our comprehension of sequence behavior. Additionally, we extend the notion of statistical cluster points within a broader framework, surpassing traditional definitions and offering deeper insights into convergence in a statistical context. Our findings in this paper open avenues for new applications and further exploration in various mathematical fields.
2024-11
Positivity and uniqueness of solutions for Riemann–Liouville fractional problem of delta types
Alexandria Engineering Journal (Volume : 114)
In this paper, we explore multi positive solutions together with their existence and uniqueness, which is properly defined for delta fractional version of Riemann–Liouville difference operators. Our exploration encompasses two distinct directions. In the first direction, we construct the Green’s function formula for the proposed delta fractional boundary value problems of order $\delta \in (1,2)$, and we present some essential properties of this function. The last and main results suggest using the well-known fixed-point theorems in a Banach space for testing the existing and uniqueness of multi-positive solutions of such problems.
2024-11
Uniqueness Results Based on Delta Fractional Operators for Certain Boundary Value Problems
Fractals (Issue : 1) (Volume : 2024)
This paper primarily lies in presenting existence and uniqueness analysis of boundary fractional difference equations of a special Riemann-Liouville operators' classes. To this end, we firstly develop Green’s function to the corresponding fractional boundary value problems and provide boundary conditions to find the constants. Then we study the existence of solutions, and we examine the bounded of their solutions. Eventually, two numerical examples are given to demonstrate the efficiency and uniqueness behavior of the boundary value problem. Furthermore, such fractional problems can typically be converted to a direct fractional problem with certain boundary or initial conditions in transport in porous media.
2024-10
Efficient Study on Westervelt-Type Equations to Design Metamaterials via Symmetry Analysis
Mathematics (Issue : 12) (Volume : 18)
Metamaterials have emerged as a focal point in contemporary science and technology due to their ability to drive significant innovations. These engineered materials are specifically designed to couple the phenomena of different physical natures, thereby influencing processes through mechanical or thermal effects. While much of the recent research has concentrated on frequency conversion into electromagnetic waves, the field of acoustic frequency conversion still faces considerable technical challenges. To overcome these hurdles, researchers are developing metamaterials with customized acoustic properties. A key equation for modeling nonlinear acoustic wave phenomena is the dissipative Westervelt equation. This study investigates analytical solutions using ansatz-based methods combined with Lie symmetries. The approach presented here provides a versatile framework that is applicable to a wide range of fields in metamaterial design.
2024-09
Orthogonal Stability and Solution of a Three-Variable Functional Equation in Extended Banach Spaces
Mathematics (Issue : 12) (Volume : 18)
This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-𝛽 -Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases.
2024-09
Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting
Mathematics (Issue : 12) (Volume : 18)
This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.
2024-09
Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics
Axioms (Issue : 13) (Volume : 10)
This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems.
2024-09
Advanced Methods for Conformable Time-Fractional Differential Equations: Logarithmic Non-Polynomial Splines
Axioms (Issue : 8) (Volume : 13)
In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using 𝐿2 and 𝐿∞ norms.
2024-08
Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications
Axioms (Issue : 8) (Volume : 13)
In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size 𝑝0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method.
2024-08
Analytical and approximate monotone solutions of the mixed order fractional nabla operators subject to bounded conditions
Mathematical and Computer Modelling of Dynamical Systems (Issue : 1) (Volume : 30)
In this study, the sequential operator of mixed order is analysed on the domain (μ2,μ1)∈(0,1)×(0,1) with 1<μ2+μ1<2, Then, the positivity of the nabla operator is obtained analytically on a finite time scale under some conditions. As a consequence, our analytical results are introduced on a set, named Em,ζ, on which the monotonicity analysis is obtained. Due to the complicatedness of the set Em,ζ several numerical simulations so are applied to estimate the structure of this set and they are provided by means of heat maps.
2024-07
A computational study of time-fractional gas dynamics models by means of conformable finite difference method
AIMS Mathematics (Issue : 7) (Volume : 9)
This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $L_2$ and $L_\infty$ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.
2024-06
Efficient simulation of Time-Fractional Korteweg-de Vries equation via conformable-Caputo non-Polynomial spline method
PLOS ONE (Issue : 19) (Volume : 6)
This research presents a novel conformable-Caputo fractional non-polynomial spline method for solving the time-fractional Korteweg-de Vries (KdV) equation. Emphasizing numerical analysis and algorithm development, the method offers enhanced precision and modeling capabilities. Evaluation via the Von Neumann method demonstrates unconditional stability within defined parameters. Comparative analysis, supported by contour and 2D/3D graphs, validates the method’s accuracy and efficiency against existing approaches. Quantitative assessment using L2 and L∞ error norms confirm its superiority. In conclusion, the study proposes a robust solution for the time-fractional KdV equation.
2024-06
On Multiple-Type Wave Solutions for the Nonlinear Coupled Time-Fractional Schrödinger Model
Symmetry (Issue : 5) (Volume : 16)
Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics.
2024-05
Some New Fractional Inequalities Defined Using cr-Log-h-Convex Functions and Applications
Symmetry (Issue : 4) (Volume : 16)
There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well.
2024-04
Some Properties of a Falling Function and Related Inequalities on Green's Functions
Symmetry (Issue : 16) (Volume : 3)
Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green’s function kernel. In the deterministic case, Green’s function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type.
2024-03

2023

The fractional non-polynomial spline method: Precision and modeling improvements
Mathematics and Computers in Simulation (Issue : 2024) (Volume : 218)
This research introduces the fractional non-polynomial spline method as a novel scheme for solving the time-fractional Korteweg-de Vries (KdV) equation. The study focuses on numerical analysis and algorithm development for simulation purposes. The proposed method involves the construction of a fractional non-polynomial spline to estimate the equation's solution, offering improved precision and modelling capabilities compared to existing approaches. To assess the stability of the proposed approach, the von Neumann method is employed, demonstrating its unconditional stability within a specific parameter range. To validate the effectiveness of our numerical analysis and simulation algorithm, contour, 2D, and 3D graphs are utilized to compare the solution obtained through our method with an analytical solution. Through rigorous comparative analysis with previous works, the superiority of our approach in terms of accuracy and efficiency is demonstrated. Norm error calculations, specifically the (L_2 and L_\infinity) error norms, provide a quantitative assessment of the accuracy and reliability of our proposed scheme.
2023-11
A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation
Mathematics (Issue : 17) (Volume : 11)
Abstract This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (𝐿2 and 𝐿∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.
2023-09
Conformable non-polynomial spline method: A robust and accurate numerical technique
Ain Shams Engineering Journal (Issue : 11) (Volume : 14)
The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L_inf) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.
2023-08
A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation
A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation (Issue : 5) (Volume : 44)
In this paper, a novel numerical scheme is developed using a new construct by non-polynomial spline for solving the time fractional Generalize Fisher equation. The proposed models represent bacteria, epidemics, Brownian motion, kinetics of chemicals and fuzzy systems. The basic concept of the new approach is constructing a non-polynomial spline with different non-polynomial trigonometric and exponential functions to solve fractional differential equations. The investigated method is demonstrated theoretically to be unconditionally stable. Furthermore, the truncation error is analyzed to determine the or-der of convergence of the proposed technique. The presented method was tested in some examples and compared graphically with analytical solutions for showing the applicability and effectiveness of the developed numerical scheme. In addition, the present method is compared by norm error with the cubic B-spline method to validate the efficiency and accuracy of the presented algorithm. The outcome of the study reveals that the developed construct is suitable and reliable for solving nonlinear fractional differential equations.
2023-05

2022

Novel simulation of the time fractional Burgers–Fisher equations using a non-polynomial spline fractional continuity method
AIP Advances (Issue : 11) (Volume : 12)
In a recent study, we investigate the Burgers–Fisher equation through a developed scheme, namely, the non-polynomial spline fractional continuity method. The proposed models represent nonlinear optics, chemical physics, gas dynamics, and heat conduction. The basic concept of the new approach is constructing a non-polynomial spline with a fractional continuity equation instead of a natural derivative. Furthermore, the truncation error is analyzed to determine the order of convergence for the proposed scheme, and we presented theoretically the stability of the developed scheme using the von Neumann method. One might easily conclude that the new scheme is quite successful and effective in obtaining the numerical solutions of the time partial/fractional partial differential equations. In addition, we plotted contour, 2D, and 3D graphs for some reported solutions to compare the presented solution with an exact solution. The investigated method was tested in some examples and compared to previous solutions for showing the applicability and effectiveness of the developed numerical scheme. The absolute and norm errors L2 and L∞ has calculated to validate the accuracy and efficiency of the presented scheme. To our knowledge, all obtained solutions in this research paper are novel and not published beforehand.
2022-11

2021

Construction of analytical solution for Hirota–Satsuma coupled KdV equation according to time via new approach: Residual power series
AIP Advances (Issue : 11) (Volume : 10)
In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota-Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.
2021-10

2019

Simultaneous influence of thermo-diffusion and diffusion-thermo on non-Newtonian hyperbolic tangent magnetised nanofluid with Hall current through a nonlinear stretching surface
Pramana (Issue : 6) (Volume : 93)
In this article, the effect of thermo-diffusion and diffusion-thermo on hyperbolic tangent magnetised nanofluid with Hall current past a nonlinear porous stretching surface has been analysed numerically. The impact of thermal slip and chemical reaction are also examined in our current analysis. Runge–Kutta–Merson method and shooting method have been successfully employed to obtain numerical results for the governing nonlinear differential equations. The impact of Hartmann number, Hall parameter, porosity parameter, fluid parameter, Weissenberg number, Richardson number, concentration buoyancy parameter, Schmidt number, Dufour parameter, Soret number, Prandtl number, chemical reaction parameter, and power-law exponent are discussed and demonstrated graphically for the flow phenomena. Furthermore, the description for Sherwood number, rate of shear stress, and Nusselt number are displayed using tables against all the pertinent parameters. A detailed numerical comparison for the power-law exponent and Prandtl number has been elaborated via tables.
2019-12
NUMERICAL STUDY OF MOMENTUM AND HEAT TRANSFER OF MHD CARREAU NANOFLUID OVER EXPONENTIALLY STRETCHED PLATE WITH INTERNAL HEAT SOURCE/SINK AND RADIATION
Heat Transfer Research (Issue : 50) (Volume : 7)
In this article, the magnetohydrodynamic (MHD) thermal boundary layer of a Carreau fl ow of Cu–water nanofluids over an exponentially permeable stretching thin plate is investigated numerically. Internal heat source/sink is also taken into account. After gaining the system of leading equations, the appropriate transformations have been first employed to come across the fitting parallel conversions to alter the central governing equations into a suit of ODEs and then the renovated system of ODE along with appropriate boundary conditions is numerically solved by the shooting method with fourth-order Runge-Kutta technique. The consequences of the relevant factors of physical parameters on velocity and temperature of merging water (H2O) and nanoparticles (Cu) have been exemplified through graphs.
2019-03

2018

Simultaneous Effects of Slip and Wall Stretching/Shrinking on Radiative Flow of Magneto Nanofluid Through Porous Medium
Journal of Magnetics (Issue : 23) (Volume : 4)
Effects of the uniform magnetic field on aqueous magneto-Nanofluid confined in a porous domain with wall stretching/shrinking non-linearly is analyzed via this communication. The problem is modeled using continuity, momentum and energy equation along with linear thermal radiation. The effects of physical quantities are observed for Cu, Al2O3, TiO2 and Ag particles in water. The coupled boundary layer PDE’s are reduced into the system of ODEs by utilizing similarity transformation and solved using shooting and Runga-Kutta fourth order technique. Stability of the obtained results are also analyzed. The results are displayed through graphs. It is observed that the momentum boundary layer is thicker when silver particles are introduced in water. Whereas, temperature profile has the minimum value for silver nanoparticles and maximum for Titanium dioxide. Also, in case of shrinking sheet dual solutions are obtained along with smallest Eigen values.
2018-12
Three-dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods
Arab Journal of Basic and Applied Sciences (Issue : 1) (Volume : 25)
Three-dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods
2018-04

2017

Heat Transfer Analysis of MHD Three Dimensional Casson Fluid Flow Over a Porous Stretching Sheet by DTM-Padé
International Journal of Applied and Computational Mathematics (Issue : 3) (Volume : 7)
In this paper, three dimensional incompressible Casson fluid flow past a linear stretching porous plate under the magnetic field effect is analyzed using differential transformation method (DTM) and numerical method. For increasing the accuracy of DTM, Padé approximation is applied. Comparison betweenDTM-Padé and numerical method shows that Padé with order [15,15] can be an exact and high efficiency procedure for solving these kinds of problems. The influence of the Casson fluid parameter (β), Prandtl number (Pr), magnetic parameter (M), stretching parameter (c) and porous parameter (λ) on non-dimensional temperature and velocity profiles are investigated. The results indicated for the Casson fluid flow that increasing the Hartmann numbermake a decrease in velocity boundary layer thicknesses.
2017-12
A novel analytical solution for the modified Kawahara equation using the residual power series method
Nonlinear Dynamics (Volume : 89)
In this paper, strongly nonlinear partial differential equations termed the modified Kawahara equations are investigated analytically using residual power series method, a modern and effective method. The method supplies good accuracy for analytical solutions when compared to exact solutions. By means of an illustrative example we show that the present technique performs better than other methods for solving nonlinear equations. The action time and influence of term parameters of terms are shown graphically.
2017-04
Using Differential Transform Method and Padé Approximant for Solving MHD Three-Dimensional Casson Fluid Flow Past A Porous Linearly Stretching Sheet
Journal of Mathematics and Computer Science-JMCS (Issue : 1) (Volume : 17)
The problem of MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet is investigated analytically. Governing equations are reduced to a set of nonlinear ordinary differential equations using the similarity transformations, and solved via an efficient and suitable mathematical technique, named the differential transform method (DTM), in the form of convergent series, by applying Pad´e approximation. The results are compared with the results obtained by the shooting method of MATHEMATICA and with the fourth-order Runge-Kutta-Fehlberg results. The results of DTM-Padé are closer to numerical solutions than the results of DTM are. A comparison of our results with existing published results shows good agreement between them. Suitability end effectiveness of our method are illustrated graphically for various parameters. Moreover, it is also observed that the Casson fluid parameter, stretching parameter, Hartmann number and porosity parameter increase with increment in the velocity profiles.
2017-03
Approximate solutions for solving the Klein–Gordon and sine-Gordon equations
Journal of the Association of Arab Universities for Basic and Applied Sciences (Volume : 10)
In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving Klein–Gordon and sine-Gordon equations. To present the present method’s effectiveness many examples are given. In this study, we compare numerical results with the exact solutions, the Adomian decomposition method (ADM), the variational iteration method (VIM), homotopy perturbation method (HPM), modified Adomian decomposition method (MADM), and differential transform method (DTM). The results reveal that the VHPM is very effective.
2017-02
Thermal boundary layer analysis of nanofluid flow past over a stretching flat plate in different transpiration conditions by using DTM-Padé method
Journal of Mathematics and Computer Science-JMCS (Issue : 1) (Volume : 17)
In this paper, Differential Transformation Method (DTM) is applied on governing equations of heat and fluid flow for a nanofluid over a horizontal flat plate. After obtaining the governing equations and solving them by DTM, the accuracy of results is examined by fourth order Runge-kutta numerical method. Due to infinite boundary condition for the stretching plate, outcomes need to an improvement method to be converged. For this aim, Padé approximation is applied on the obtained results which [10,10] Padé order had the best accuracy compared to numerical method. The influence of relevant parameters such as the transpiration parameter on temperature and nanoparticle concentration profile is discussed and it is concluded that by increasing this parameter, nanoparticles concentration over the plate decrease due to more fluid penetration from pores and this is the main reason of lower thermal boundary layer caused by fewer nanoparticles over the plate.
2017-02
MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate
International Journal of ADVANCED AND APPLIED SCIENCES (Issue : 4) (Volume : 1)
In this research, we analyze heat transfer of MHD boundary layer flow of Casson fluid. Strong nonlinear ordinary differential equations are solved numerical using Shooting method with fourth order Runge-Kutta (RK4) integration technique. Variations of interesting different parameters on the velocity are showed graphically.
2017-01
A residual power series technique for solving Boussinesq–Burgers equations
Cogent Mathematics (Issue : 4) (Volume : 1)
In this paper, a residual power series method (RPSM) is combining Taylor’s formula series with residual error function, and is investigated to find a novel analytical solution of the coupled strong system nonlinear Boussinesq-Burgers equations according to the time. Analytical solution was purposed to find approximate solutions by RPSM and compared with the exact solutions and approximate solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time and concluded that the present results are more accurate and efficient than analytical methods studied. Then, analytical simulations of the results are studied graphically through representations for action of time and accuracy of method.
2017-01

2016

Numerical simulation using the homotopy perturbation method for a thin liquid film over an unsteady stretching sheet
International Journal of Pure and Applied Mathematics (Issue : 107) (Volume : 2)
In this article, the flow of liquid film over an unsteady elastic stretching surface is analyzed. Similarity transformations are used to transform the governing equations to a nonlinear ordinary differential equation. The differential equation reformulated to system of Voltera integral equations and solved analytically using the new technique of numerical solution called homotopy perturbation method. The results of the proposed method are compared with previously published work and the results are found to be in an excellent agreement. Also, discussed and presented graphically the effects of various parameters Darcy number and unsteadiness parameter.
2016-04

2015

A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations
American Journal of Computational Mathematics (Volume : 5)
In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.
2015-10
Variational Homotopy Perturbation Method for Solving Benjamin-Bona-Mahony Equation
Applied Mathematics (Issue : 6) (Volume : 4)
In this article, the application of variational homotopy perturbation method is applied to solve Benjamin-Bona-Mahony equation. Then, we obtain the numerical solution of BBM equation using the initial condition. Comparison with Adomian's decomposition method, homotopy perturbation method, and with the exact solution shows that VHPM is more effective and accurate than ADM and HPM, and is reliable and manageable for this type of equation.
2015-05
Approximate Solutions for a Couple of Reaction-diffusion Equations with Self-diffusion
British Journal of Mathematics & Computer Science (Issue : 11) (Volume : 2)
In this paper, a competition model of a reaction diffusion system with self-diffusion has been studied using homotopy perturbation method, variational iteration method and Finite element method FEM (COMSOL package). The traveling wave solutions for this system are found and compared numerically. It was shown that the competition will lead at the end of the winning of one species. The effect of self diffusion is shown in the dispersing of traveling wave solution. Also, it was shown that the solution of finite element method and homotopy perturbation method are convergent to each other compared to the variation iteration method.
2015-01

2014

Solving the Kuramoto-Sivashinsky equation via Variational Iteration Method
International Journal of Applied Mathematical Research (Issue : 3) (Volume : 3)
In this study, the approximate solutions for the Kuramoto-Sivashinsky equation by using the Variational Iteration Method (VIM) are obtained. Comparisons with the exact solutions and the solutions obtained by the Homotopy Perturbation Method (HPM), the numerical example show that the Variational Iteration Method (VIM) is accurate and effective and suitable for this kind of problem.
2014-06
Adomian Decomposition Method for Solving the Kuramoto – Sivashinsky Equation
IOSR Journal of Engineering (Issue : 10) (Volume : 1)
The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the Adomian Decomposition method (ADM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.
2014-01

2013

The Homotopy Perturbation Method for Solving the Kuramoto–Sivashinsky Equation
IOSR Journal of Engineering (Issue : 3) (Volume : 12)
The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the homotopy perturbation method (HPM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.
2013-12

2024

Numerical Solutions of Fractional Differential Equations by Non-Polynomial Spline Functions
PhD Numerical Analysis
2024-03-07

2013

Numerical Solution for Solving the Kuramoto–Sivashinsky Equation
Master Science
2013-09-15

2018

A Numerical Approach of Fornberg-Whitham Equation Using Residual Power Series Method
2018 International Conference on Advanced Science and Engineering (ICOASE)
In this paper, a modern and approach method called residual power series technique has been applied to find numerical solution for an important equation in optical communication systems and optical fibers called Fornberg-Whitham equation. In addition to that, the technique has been used in regard of the effectiveness of time. The present results have been used to compare it with the exact solution and have been shown graphically to demonstrate the effectiveness and suitability of RPS method. Furthermore, the steadiness and the action of time of the Fornberg-Whitham equation and the modified Fornberg-Whitham equation have been discussed and concluded that the RPSM method is an influential and beneficial one to solve strong nonlinear partial differential equations.
2018-10
Three-Dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods
Conference: International Conference on Chemical and Biochemical EngineeringAt: Shanghai, China
In this study, Least Square Method (LSM) and Differential Transform Method (DTM) are applied to solve the three dimensional problem of steady nanofluid deposition on an inclined rotating disk is illustrated. The governing non-linear partial differential equations are reduced to the nonlinear ordinary differential equations system by similarity transform. There is a good agreement between the present analytical and numerical results. Results indicate that increasing nanofluid volume fraction leads to increase in temperature profile and highest temperature obtained for Al2O3-water case.
2018-09