2024-04

Optical and Quantum Electronics (Issue : 2024) (Volume : 56)

This research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study’s findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures.

2024-03

Chaos Soliton and Fractals (Issue : 2024) (Volume : 182)

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.

2024-01

Results in Physics (Issue : 2024) (Volume : 57)

This study focuses on the variant Boussinesq equation, which is used to model waves in shallow water and electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to find closed-form wave solutions using the extended direct algebraic method. By employing this method, a range of wave solutions with distinct shapes, including shock, mixed-complex solitary-shock, singular,mixed-singular, mixed trigonometric, periodic, mixed-shock singular, mixed-periodic, and mixed-hyperbolic solutions, are attained. To illustrate the propagation of selected exact solutions, graphical representations in 2D, contour, and 3D are provided with various parametric values. The equation is transformed into a planar dynamical structure through the Galilean transformation. By utilizing bifurcation theory, the potential phase portraits of nonlinear and super-nonlinear traveling wave solutions are investigated. The Hamiltonian function of the dynamical system of differential equations is established, revealing the system’s conservative nature over time. The graphical representation of energy levels offers valuable insights and demonstrates that the model has closed-form solutions.

2023-12

Optical and Quantum Electronics (Issue : 129) (Volume : 56)

This study investigate the Kudryashov equation, which is used to describe the propagation pulses in optical fibers. The aim of this study is to find closed-form wave solutions using the new modified generalized Riccati equation mapping method. By employing the proposed method, a range of new optical solutions with distinct shapes including dark, bright, combined dark-bright, singular periodic solutions are attained. These solutions are integrated with physical parameters and exhibit special waveform characteristics under specific constraints. The modulation instability analysis of the governing model is also presented. To illustrate the propagation of selected exact solutions, graphical representations in 2D, and 3D are provided with various parametric values. Additionally, the impact of the time parameter and the non-linearity degree n are also presented. It can be convincingly stated that the employed method is remarkably efficient and achieves significant success in discovering exact solutions for nonlinear partial differential equations.

2023-09

Results in Physics (Issue : 2023) (Volume : 53)

This study examines the perturbed Fokas–Lenells equation using two methods: the Bernoulli sub-equation function method and the (1∕𝐺′)-expansion method. A traveling wave transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The perturbed Fokas–Lenells equation arises in optical fibers, and the article constructs various optical soliton solutions, including periodic dark-bright, singular, and periodic singular soliton solutions. The article also presents the necessary constraint conditions for the existence of valid solitons and graphs 2D, 3D, and contour surfaces based on different parameter values.

2023-08

Physica Scripta (Issue : 2023) (Volume : 98)

This research explores the Jacobi elliptic expansion function method and a modified version of the Sardar sub-equation method to discover new exact solutions for the nonlinear Hamiltonian amplitude equation. By applying these techniques, the study seeks to uncover previously unknown solutions for this equation, contributing to the understanding of its behavior and opening up new possibilities for its applications. The solutions obtained using these methods are represented by hyperbolic, trigonometric, and exponential functions, and they include optical dark-bright, periodic, singular, and bright solutions. The dynamic behaviors of these solutions are demonstrated by selecting appropriate values for physical parameters. By assigning values to these parameters, the study aims to showcase how the solutions of the nonlinear Hamiltonian amplitude equation behave under different conditions. This analysis provides insights into the system’s response and enables a deeper comprehension of its complex dynamics in various scenarios, contributing to the overall understanding of the equation’s behavior and potential real-world implications. Overall, these methods are effective in analyzing and obtaining analytic solutions for nonlinear partial differential equations.

2023-07

Communications in Theoretical Physics (Issue : 2023) (Volume : 75)

It is commonly recognized that, despite current analytical approaches, many physical aspects of nonlinear models remain unknown. It is critical to build more efficient integration methods to design and construct numerous other unknown solutions and physical attributes for the nonlinear models, as well as for the benefit of the largest audience feasible. To achieve this goal, we propose a new extended unified auxiliary equation technique, a brand-new analytical method for solving nonlinear partial differential equations. The proposed method is applied to the nonlinear Schrödinger equation with a higher dimension in the anomalous dispersion. Many interesting solutions have been obtained. Moreover, to shed more light on the features of the obtained solutions, the figures for some obtained solutions are graphed. The propagation characteristics of the generated solutions are shown. The results show that the proper physical quantities and nonlinear wave qualities are connected to the parameter values. It is worth noting that the new method is very effective and efficient, and it may be applied in the realisation of novel solutions.

2023-07

Results in Physics (Issue : 2023) (Volume : 51)

In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black-Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.

2023-05

Optical and Quantum Electronics (Volume : 2023)

This study aims to investigate the (3+1)-dimensional fourth-order KdV-type hierarchy nonlinear euation. For this aim, we use Hirota’s simple method to handle one, two, three, and four soliton solutions. Furthermore, using the ansatz method in the form of the tanhp function, the symbolic computational method is used to construct kink solitary wave solutions, multi-wave solutions, X-type solitons, and Y-type solitons, as well as shock wave solutions. Furthermore, the physical structure and propagation characteristics of the obtained solutions are simulated.

2023-05

Modern Physics Letters B (Issue : 2350073) (Volume : 2023)

In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to ð3 þ 1Þ-dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota's simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, °uid mechanics and engineering sciences can benefit from this study.

2023-03

Communications in Theoretical Physics (Volume : 75)

In this paper, we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili (eBKP) equation utilizing the condensed Hirota’s approach. In accordance with a logarithmic derivative transform, we produce solutions for single, double, and triple M-lump waves. Additionally, we investigate the interaction solutions of a single M-lump with a single soliton, a single M-lump with a double soliton, and a double M-lump with a single soliton. Furthermore, we create sophisticated single, double, and triple complex soliton wave solutions. The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics, plasma, and shallow water theory. By selecting appropriate values for the related free parameterswe also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.

2023-03

Results in Physics (Issue : 2023) (Volume : 47)

In this paper, the nonlinear complex Hirota-dynamical model is explored using the unified auxiliary equation method, which is the most powerful method for studying exact solutions to nonlinear models. This model is one of the important standards of the nonlinear Schrödinger equation in which the third derivative term represents the self-interaction in the high-frequency subsystem. Typically, in plasma, this term is isomorphic to the so known self-focusing effect. The aforementioned equation is crucial to plasma physics due to the agreements between the self-interaction at high frequency and the well-known self-focusing effect in plasma. Different structures of solutions are successfully investigated for this model. Consequently, we obtain some solutions like dark–bright, bright, dark, singular, periodic, trigonometric function, Jacobi elliptic function, and exponential solutions. We simulate the 2𝐷, 3𝐷, and counter plots of the constructed solutions by choosing the suitable values of the parameters involved.

2023-02

Results in Physics (Issue : 2023) (Volume : 45)

We examine an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in a nano-optical fiber with fourth-order spatial derivatives in this paper. We develop various optical solutions to the equation under study using the Jacobi elliptic expansion method. We investigate the consequences of nonlinearity and spatial dispersion in spatial directions 𝑥 and 𝑦. Furthermore, by investigating the stability condition of this nonlinear equation, we derive the linear stability analysis. The innovative evolutionary behaviors in the 3-dimensional profile are provided under constraint circumstances to indicate the wave propagation direction.

2023-01

Optical and Quantum Electronics (Issue : 55) (Volume : 2023)

In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi elliptic function expansion method. This results in obtaining distinct solutions including dark, bright, singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The 2- and 3-dimensional graphs of the reported solutions are presented. The reported results may be useful in explaining the physical features of the studied equation.

2022-12

International Journal of Modern Physics B (Issue : 16) (Volume : 36)

In this study, through the -expansion method, we extract soliton solutions to the coupled-Higgs equation. The studied nonlinear model is known to describe Higgs mechanism. The Higgs mechanism is essential to explain the generation mechanism of the property “mass” for gauge bosons. The proposed method is one of the most powerful methods for constructing soliton solutions for nonlinear partial differential equations. The obtained wave solutions include exponential, hyperbolic, and distinct structures of complex function solutions. The presented results may be helpful in explaining the physical features of various nonlinear physical phenomena. In order to analyze the dynamic behavior of all obtained solutions, we plot three-dimensional and two-dimensional graphs for the obtained solutions.

2022-11

Romanian Journal in Physics (Issue : 118) (Volume : 74)

In this paper, the Jacobi elliptic function expansion approach is used to explore the Kundu-Mukherjee-Naskar equation, which describes the fiber pulse in optics, rogue waves in the seas, and light beam bending. As a consequence, many structures of solutions based on the parameter module m, such as optical dark-bright, dark, bright, singular, Jacobi elliptic function, and exponential solutions, have been effectively developed. In the solutions produced to analyze wave propagation, specific values for free parameters are presented. We also used computer tools to display 3D surfaces, 2D images, and contour plots of some of the created solutions. To our knowledge, all the obtained solutions verify the partial differential equation.

2022-09

Optik (Issue : 169880) (Volume : 296)

This study aims to propose a new modified unified auxiliary equation method for the first time. To illustrate the validity of the new modified method, we consider the dimensionless time-dependent paraxial equation which is investigated previously in Tarla and Yilmazer (2022) using the unified auxiliary equation method. As a result, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. The obtained results are compared with the solutions obtained by the unified auxiliary equation method in Tarla and Yilmazer (2022) in the result and discussion section. We conclude that our findings are novel and distinctive, and we give the comparison in the section on results and discussion. It is demonstrated that the modified unified auxiliary equation method provides a more efficient mathematical technique for solving nonlinear partial differential equations. In addition, we draw 3D profiles and counter plots for some obtained solutions with the assistance of a computer program by giving a specific value for the involved parameters.

2022-09

Optical and Quantum Electronics (Issue : 707) (Volume : 54)

In this work, we investigate the Fokas system which expresses the nonlinear pulse propagation in mono-mode optical fibers. We use the (m+1/G')-expansion scheme and the novel modified generalized exponential rational function scheme to generate soliton solutions to the proposed model. As a result, some exact solutions are constructed including hyperbolic, exponential, dark, singular, and periodic singular solutions. In order to analyze the physical behavior of the absolute, and real parts of the obtained solutions we present the 3-D surfaces, 2-D graphs, and their contour graphs.

2022-09

Optical and Quantum Electronics (Issue : 11) (Volume : 54)

In the present study, through the generalized exponential rational function method (GERFM), we construct optical soliton solution to the coupled-Higgs equation (CHE). The proposed method is one of the most powerful method to construct abundant exact solutions for nonlinear partial differential equations. The obtained wave solutions include hyperbolic, trigonometry, and exponential functions solutions. Furthermore, we draw three-dimensional surfaces and counter plots of the obtained solutions using the appropriate value for involved parameters.

2022-09

International Journal of Modern Physics B (Volume : 37)

The coupled-Higgs equation, which depicts a system of preserved scalar nucleons cooperating with unbiased scalar mesons, is investigated using the Phi^6-model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m=0 and m=1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial differential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions.

2022-09

International Journal of Modern Physics B (Issue : 16) (Volume : 36)

The coupled-Higgs equation, which depicts a system of preserved scalar nucleons coop- erating with unbiased scalar mesons, is investigated using the 6-model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m ! 0 and m ! 1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial dier- ential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions.

2022-08

Optical and Quantum Electronics (Issue : 14) (Volume : 54)

In this paper, we use the Jacobi elliptic function expansion approach to investigate the Hirota-Maccari system which describes the dynamical behaviors of the femto-second soliton pulse in single-mode fibers. As a result, solutions with various structures are developed including dark-bright, dark, single, brilliant, exponential function, and Jacobi elliptic function solutions. In addition, a computer application is used to examine the solutions found on three-dimensional surfaces and two-dimensional visuals.

2022-08

Modern Physics Letters B (Issue : 23) (Volume : 36)

The Jacobi elliptic function expansion method is one of the most powerful tools for exploring exact solutions of nonlinear partial differential models, which is used in this work to characterize the interaction between one long longitudinal wave and one short transverse wave propagation in a generalized elastic medium. First, we applied a wave transform to the proposed system of equations and obtained an ordinary differential equation. We obtain the values of involved free parameters after performing necessary operations, then substitute the obtained values to the ordinary differential equation by considering the constructed solutions to the ordinary and then to the partial differential equation. As a result, different structures of solutions are constructed such as solitary dark–bright, dark, bright, singular, trigonometric function, Jacobi elliptic function, and hyperbolic function solutions. In order to illustrate the tsunami and tidal oscillations, we draw 3D surfaces and 2D graphics for the obtained solutions by giving a specific value for the involved parameters under the given conditions.

2022-07

Chaos, Solitons & Fractals (Volume : 161)

In this study, we investigate a system of equations based on ion sound and Langmuir waves equations using a generalized exponential rational function, and a new technique named as a modified generalized exponential rational function method. The concept of the new approach is to modify the generalized exponential rational function method. To evaluate the efficiency of the new scheme, we applied the two methods to the given system of equations, and we chose the same values of the parameters in all families, as well as we presented the difference between the solutions in the result and the discussion section. One might easily conclude that the new approach is quite effective and successful in seeking the exact solutions of the nonlinear differential equations. As a result, we have identified a variety of new families of exact travel wave solutions. In addition, we plotted 2D, 3D and contour graphs for some reported solutions by choosing the suitable parameters values.

2022-06

Communications in Theoretical Physics

In this paper, Hirota’s simple method and long-wave method are used to study the (2+1)-dimensional KdV equation including constant and time-dependent coefficients. One-, two-, and three-M-lump solutions are constructed for both cases. Also, the interaction phenomena of M-lump solution with one-soliton, and two-soliton waves are derived for these equations. Moreover, to investigate the physical characteristics of the obtained results, the 3-dimensional figures and contour plots are graphed. All gained solutions verify the proposed equations.

2022-06

Communications in Theoretical Physics (Issue : 7) (Volume : 74)

This study investigates the perturbed Chen-Lee-Liu (CLL) model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of non-linear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications

2022-06

Modern Physics Letters B (Volume : 33)

This study investigates the Hamiltonian amplitude equation by using the generalized exponential rational function method. As a result, a variety of soliton solutions have been established with different wave structures such as optical exponential, dark, singular, periodic, periodic-singular, and hyperbolic solutions. 2D and 3D graphics of certain acquired solutions are drawn with the help of computer program. By picking appropriate values for the free parameters in the obtained solutions, the physical phenomena for these obtained solutions are graphically studied and depicted.

2022-05

Results in Physics (Volume : 36)

In this study, the Fokas system which represents the spread of irregular pulse in monomode optical fibers is investigated via the Jacobi elliptic function expansion (JEFE) method. This method is the most powerful technique to explore solutions for a wide range of various models. As a result, different solutions such as dark–bright, singular, bright, Jacobi elliptic function, and exponential solutions are obtained. In addition, 3D and 2D graphics for the obtained solutions are investigated with the assist of a computer program by assigning particular values to the parameters involved.

2022-05

Optical and Quantum Electronics (Issue : 5) (Volume : 54)

In this study, we express the Radhakrishnan–Kundu–Lakshmanan equation with an arbitrary index of. We investigated the solitary wave solutions of the Radhakrishnan–Kundu–Lakshmanan equation by mean of the Jacobi elliptic function expansion technique. As a result, we constructed several distinct solutions include dark, bright, singular, periodic, hyperbolic, trigonometric, and Jacobi elliptic function types solutions. To highlight the dynamic behavior of the generated solutions, specific values for the parameters are also assigned. The above techniques could also be employed to get a variety of exact solutions for other nonlinear models in physics, applied mathematics, and engineering.

2022-03

Appl. Math (Issue : 2) (Volume : 16)

In this article, we applied two different methods namely as the (1/G′)-expansion method and the Bernoulli sub-equation method to investigate the generalized Kadomtsev-Petviashvili modified equal width-Burgers equation, which is designated the propagation of long-wave with dissipation and dispersion in nonlinear media. To transform the given equation into a nonlinear ordinary differential equation, a traveling wave transformation has been carried out. As a result, we constructed distinct exact solutions like complex solutions, singular solutions, and complex singular solutions. Besides, 2D, 3D, and contour surfaces are illustrated to demonstrate the physical properties of the obtained solutions.

2022-03

Optical and Quantum Electronics (Issue : 3) (Volume : 54)

In this article, the (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, which describes the interactions of long wave propagations and has diverse applications in physics, mathematics and engineering, is investigated. Lump and multi-lump wave solutions, one-, and two-soliton solutions, exploding and periodic wave solutions, localized and breather wave solutions, and interaction of lump waves with solitary waves are constructed through the symbolic computational method. In addition, multi-soliton solutions in complex form via using Hirota’s simple method and long-wave method are obtained. The dynamic behaviors of all obtained solutions are analyzed and illustrated in figures by choosing appropriate values of the parameters all of the obtained solutions are verified by a direct substitution in the original equation.

2022-02

Optical and Quantum Electronics (Issue : 2) (Volume : 54)

In this study, we investigate the perturbed Chen-Lee-Liu equation that represents the propagation of an optical pulse in plasma and optical fiber. The Jacobi elliptic function technique is used for this purpose. As a result, we obtain some new solitary wave solutions such as the Jacobi elliptic function, dark-bright, trigonometric, exponential, hyperbolic, periodic, and singular soliton solutions. To express the pulse propagation of the generated solutions, specific values for the free parameters under conditions are also given.

2022-01

Journal of Applied Analysis

The intention of this study is to investigate the fractal version of both one-term and three-term fractal differential equations. The fractal Laplace transform of the local derivative and the non-local fractal Caputo derivative is applied to investigate the given models. The analogues of both the Wright function with its related definitions in fractal calculus and the convolution theorem in fractal calculus are proposed. All results in this paper have been obtained by applying certain tools such as the general Wright and Mittag-Leffler functions of three parameters and the convolution theorem in the sense of the fractal calculus. Moreover, a comparative analysis is conducted by solving the governing equation in the senses of the standard version and fractal calculus. It is obvious that when , we obtain the same results as in the standard version.

2021-11

Physica Scripta (Issue : 125249) (Volume : 96)

In this paper, we consider an extended Calogero-Bogoyavlenskii-Schiff (eCBS) equation. Based on a logarithmic derivative transform and with the aid of symbolic computation, we construct complex multiple solitons for this nonlinear model. Also, by using a symbolic computational method, onelump solution, two-soliton solution, localized and breather wave solution, as well as a periodic wave solution and multiple wave solutions are obtained. The constraint conditions which ensure the validity of the wave structures are also reported. Besides, the graphs of the solution attained are recorded in 3D graphs by fixing parameters to discuss their dynamical properties. The achieved outcomes show that the applied computational strategy is direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations.

2021-09

Journal of Mathematical Modeling (Issue : 3) (Volume : 9)

In this article, we suggest a fractal Kronig-Penny model which includes a fractal lattice, a fractal potential energy comb, and a fractal Bloch's theorem on thin Cantor sets. We solve the fractal Schrodinger equation for a given potential, using an exact analytical method. We observe that the allowed band energies and forbidden bands in the fractal lattice are bigger than in the standard lattice. These results show the effect of fractal space-time or their fractal geometry on energy levels.

2021-09

Journal of Mathematics and Modeling in Finance (Issue : 1) (Volume : 1)

In this article, the price adjustment equation has been proposed and studied in the frame of fractal calculus which plays an important role in market equilibrium. Fractal time has been recently suggested by researchers in physics due to the self-similar properties and fractional dimension. We investigate the economic models from the viewpoint of local and non-local fractal Caputo derivatives. We derive some novel analytical solutions via the fractal Laplace transform. In fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard computational sense, and the non-local fractal Caputo fractal derivative is a generalization of the non-local fractional Caputo derivative. The economic models involving fractal time provide a new framework that depends on the dimension of fractal time. The suggested fractal models are considered as a generalization of standard models that present new models to economists for fitting the economic data. In addition, we carry out a comparative analysis to understand the advantages of the fractal calculus operator on the basis of the additional fractal dimension of time parameter, denoted by $alpha$, which is related to the local derivative, and we also indicate that when this dimension is equal to $1$, we obtain the same results in the standard fractional calculus as well as when $alpha$ and the nonlocal memory effect parameter, denoted by $gamma$, of the nonlocal fractal derivative are both equal to $1$, we obtain the same results in the standard calculus.

2021-08

Modern Physics Letters B (Issue : 26) (Volume : 35)

In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.

2021-06

International Journal of Nonlinear Sciences and Numerical Simulation

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

2021-03

Chaos: An Interdisciplinary Journal of Nonlinear Science (Issue : 3) (Volume : 31)

In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when α=1, we get the same results as in the standard version. Fractal calculus is a new branch of calculus that includes ordinary calculus. In the future, it will find many applications in various branches of science. Fractal calculus development is in progress.

2020-10

International Journal of Modern Physics B (Issue : 25) (Volume : 34)

In the current study, we instigate the four-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation. The NLmZK equation guides the attitude of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Two different methods are used, namely the sine-Gordon expansion method (SGEM) and the (1/G') -expansion method to the proposed model. We have successfully constructed some topological, non-topological, and wave solutions. In addition, the 2D, 3D, and contour graphs of the solutions are also plotted under the choice of appropriate values of the parameters.

2020-10

Int. J. Industrial Mathematics (Issue : 4) (Volume : 12)

An epidemiological model of smoking habit is studied by using one of flexible and accurate semi-analytical methods. For this reason, the homotopy analysis transform method (HATM) is applied. Convergence theorem is studied and several h-curves are demonstrated to show the convergence regions. Also, the optimal convergence regions are obtained by demonstrating the residual error functions versus h. The numerical tables are presented to show the precision of method.

2020-07

Indian Journal of Physics (Volume : 94)

This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions.

2020-05

Physica Scripta (Issue : 6) (Volume : 95)

The core of this research is to investigate the system of equations for the ion sound and Langmuir waves through the Bernoulli sub-equation method. It is one of the most effective methods for solving nonlinear ordinary differential equations. To transform a given system to a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. Novel wave behaviors of the guiding system are acquired, such as kink-type singular solutions, wave solutions, periodic wave solutions, and periodic singular solutions. Besides, the modulation instability analysis of the given system is investigated based on the standard linear stability analysis and the modulation instability gain spectrum analysis. Furthermore, 2D, 3D and contour graphs of all acquired solutions are also plotted under the selection of appropriate parameter values.

2020-04

Mathematical Methods in the Applied Sciences

In this study, the discrete fractional nabla calculus operator is used to investigate the k‐hypergeometric differential equation for both homogeneous and nonhomogeneous states. To solve the guiding equation, we implement certain classical transformations and also constrain the parameters needed to determine them valued. In order to achieve these results, some equipment like the Leibniz rule, the index law, the shift operator, and the power rule are set out in the frame of the discreet fractional calculus. We use all of these tools to the governing equation for homogeneous and nonhomogeneous situations. The major benefit of the fractional nabla operator is that it implements singular differential equations and converts them into fractional order equations. As a result, several new exact fractional solutions of the given equation are constructed.

2020-03

Front. Phys. (Volume : 8)

In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson-Pickering equation. The (1/G′)-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, and topological and non-topological solutions. Besides, the 2D, 3D, and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.

2020-03

Physica A: Statistical Mechanics and its Applications (Volume : 548)

In the current study, we investigate the (3+ 1)-dimensional extended quantum Zakharov–Kuznetsov equation through two different methods namely the sine-Gordon expansion method and the (1∕ G′)-expansion method. The given model represent the nonlinear propagation of the quantum ion-acoustic wave. Traveling wave transformation has been used, in order to convert the governing equation into a non-linear ordinary differential equation. As a result, we successfully construct various exact wave solutions, such as shock, topological, non-topological solutions, periodic wave solutions, and singular solutions. In addition, 2D, 3D and contour surfaces are plotted according to the appropriate parameters values.

2020-03

Front. Phys. (Volume : 8)

The research paper aims to investigate the space-time fractional cubic-quartic nonlinear Schrödinger equation in the appearance of the third, and fourth-order dispersion impacts without both group velocity dispersion, and disturbance with parabolic law media by utilizing the extended sinh-Gordon expansion method. This method is one of the strongest methods to find the exact solutions to the nonlinear partial differential equations. In order to confirm the existing solutions, the constraint conditions are used. We successfully construct various exact solitary wave solutions to the governing equation, for example, singular, and dark-bright solutions. Moreover, the 2D, 3D, and contour surfaces of all obtained solutions are also plotted. The finding solutions have justified the efficiency of the proposed method.

2020-02

East Asian Journal on Applied Mathematics (Issue : 3) (Volume : 10)

Periodic wave solutions of (3 + 1)-dimensional potential-Yu-Toda-Sasa-Fukuyama (YTSF) equation are constructed. Using the bilinear form of this equation, we chose ansatz as a combination of rational, trigonometric and hyperbolic functions. Density graphs of certain solutions in 3D and 2D situations show different cross-kink waveforms and new multi wave and cross-kink wave solutions. Moreover, we employ the semi-inverse variational principle (SIVP) in order to study the solitary, bright and dark soliton wave solutions of the YTSF equation.

2020-01

AIMS Mathematics (Issue : 2) (Volume : 5)

In the current article, we investigate the second order singular differential equation namely the effective mass Schrödinger equation by means of the fractional nabla operator. We apply some classical transformations in order to reduce the governing equation, and also restrict the difference parameters involved in order to find them values. In order to achieve these important results, certain tools such as the Leibniz rule, the index law, the shift operator, and the power rule are provided in view of the discrete fractional calculus. We use all these mentioned data for two representations of the given model for homogeneous and non-homogeneous instances. The main advantage of the fractional nabla operator is to apply the singular differential equations and transform them into a fractional order model. As a result, we produce some new exact fractional solutions to the present model for a given potential.

2019-12

Thermal Science (Issue : 6) (Volume : 23)

Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions of hydrogen atom type equations by using discrete fractional nabla operator ∇α(0 < α < 1). This operator is applied homogeneous and non-homogeneous hydrogen atom type equations. So, we obtain many particular solutions of these equations.

2019-11

Thermal Science (Issue : 6) (Volume : 23)

Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions (dfs) of hydrogen atom type equations (HAEs) by using discrete fractional nabla operator α(0 < α < 1). This operator is applied homogeneous and nonhomogeneous HAEs. So, we obtain many particular solutions of these equations.

2019-07

JP Journal of Heat and Mass Transfer (Issue : 1) (Volume : 18)

This survey investigates the characteristic flow, heat transfer and chemical reaction of Weissenberg and Williamson over a stretching sheet with thermal radiation and viscous dissipation. The nonlinear differential equation of the thermal boundary layer, which covers the physical problem, is derived and used to transform its equations into a system of higher nonlinear ordinary differential equations with similarity. The resulting non-monthly equation system is numerically solved using the explicit Runge-Kutta technique with the shooting technique. The behavior of the favorite physical parameters with different values in the speed profile, heat transfer, concentration of nanoparticles, skin friction, Nusselt number and Sherwood number has been discussed and graphically analyzed.

2017-10

Advances and Applications in Fluid Mechanics (Issue : 4) (Volume : 20)

This research is presented to analyze and investigate the magnetohydrodynamic Casson flow of liquid film over the stretching surface. Continuity and Navier-Stokes equation that covers problem is transformed to the nonlinear ordinary differential equation, next arising governing equations are solved numerically using ADM-Pade in order to investigate variations of interesting parameters on the velocity profile, finally all physical parameters are presented graphically .

2017-01

International Journal of Advanced and Applied Sciences (Issue : 1) (Volume : 4)

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.

2016-04

International Journal of Pure and Applied Mathematics (Issue : 2) (Volume : 107)

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.