2024-04

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.

2024-03

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.

2023-11

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.

2023-09

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.

2023-08

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.

2023-05

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.

2022-11

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.

2021-10

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.

2019-12

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.

2019-03

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.

2018-12

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.

2018-04

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

2017-12

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.

2017-04

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.

2017-03

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.

2017-02

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

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.

2017-01

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

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.

2016-04

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.

2015-10

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.

2015-05

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.

2015-01

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.

2014-06

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.

2014-01

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.

2013-12

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.