Nonlinear Fractional Partial Differential Equations Research Articles

New soliton solutions and modulation instability analysis of fractional Huxley equation

In this research, the new auxiliary equation method (NAEM) for higher order nonlinear fractional Huxley equation is being employed to extricate the novel soliton solutions using Beta and M-Truncated fractional derivatives. For waves of finite amplitude, the Huxley equation demonstrates a substantial transfer of spectrum energy. A comparison of the solutions of the model with both fractional derivatives is also included in this research. Various kinds of solitary traveling wave solutions, such as trigonometric, hyperbolic, exponential, rational functions, etc., are found. These types of solutions demonstrate the superiority of the novelty of the method. This method’s key advantage over others is that it provides more broad solutions with certain flexible parameters. 3D and 2D graphs are used graphically to demonstrate the dynamical structures of the solutions. The results are presented in a way that demonstrate the usefulness and competence of the approach used to handle various nonlinear fractional partial differential equations. Lastly, we investigate the comparison of the gain spectra for modulation instability and the depiction of certain noteworthy outcomes by illustratively depicting the 2D figures produced by carefully considering the parameters

Shehu Variational Iteration Method For Solve Some Fractional Differential Equations

The idea proposed in the work is to extend the Shehu transform method to resolve the nonlinear fractional partial differential equations by combining them with the variational iteration method (VIM). We apply this technique to solve nonlinear fractional equations as nonlinear time fractional Fokker Planck equation.

Analytical Solution of Some Systems of Nonlinear Fractional Differential Equations by the SBA Method

We suggest a new approach to the Some Blaise Abbo (SBA) method for solving systems of nonlinear fractional partial differential equations and we have tested it with two examples.

A novel iterative method to solve nonlinear wave-like equations of fractional order with variable coefficients

In this work, we suggest a novel iterative method to give approximate solutions of nonlinear wave-like equations of fractional order with variable coefficients. The advantage of the proposed method is the ability to combine two different methods: Shehu transform method and homotopy analysis method, in addition to providing an approximate solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. This method can be called Shehu homotopy analysis method (SHAM). Three different examples are presented to illustrate the preciseness and effectiveness of the proposed method. The numerical results show that the solutions obtained by SHAM are in good agreement with the solutions found in the literature. Furthermore, the results show that this method can be implemented in an easy way and therefore can be used to solve other nonlinear fractional partial differential equations.

MODIFIED FINITE ELEMENT NUMERICAL METHOD FOR SOLVING CONFORMABLE SPACE-TIME FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

This paper shows how to approximate the solution to a nonlinear conformable space-time fractional partial differential equations. The proposed method is based on the Cubic B-spline polynomials and Galerkin method. Two test problems show that the approach we use to approximate the proposed equation is accurate and efficient. We apply the Von Neumann approach to show that stability requires some conditions.

Nonlinear fractional diffusion model for space-time neutron dynamics

In this work, a new fractional multidimensional neutron diffusion model is introduced using Cattaneo equation instead of the commonly used Fick's law for two neutron energy groups. The model represents a coupled system of nonlinear hyperbolic fractional partial differential equations considering finite fast and finite thermal neutron velocities. The model provides new quantities that relate some of the most important nuclear parameters together producing some newly physical concepts. These quantities include fast and thermal neutron relaxation times, Riemann-Liouville fractional integrals of the neutron densities and derivatives of macroscopic absorption cross sections. This model can be useful for mathematicians and physicists since the anomalous subdiffusion process describes the neutron movement closer to reality. The effect of the anomalous subdiffusion exponent (0<γ≤1) on the neutron flux is discussed. The model is successfully applied to simple homogenous and heterogonous reactor cores with step, ramp and sinusoidal perturbations for different values ofγ to estimate the average reactor powers in the fuel and moderator regions. Also, the new fractional model is applied to LRA BWR benchmark problem including adiabatic heating and Doppler feedback. On the other hand and based on the concept of prompt jump, the proposed fractional model is simplified and then analytically solved using Mittag-Liffler function. Solving fractional models numerically consumes considerable computational time costs due to the persistent memory of fractional operators, especially for smaller values of γ. For this regard, adaptively multi-step differential transform MDTM method is implemented to solve the proposed fractional model. The MDTM is developed using step size control technique for which the length of the step integration varies according to the solution behaviour.

New exact solutions of space-time fractional Schr¨odinger-Hirota equation

In this study, improved Bernoulli sub-equation function method (IBSEFM) is presented to construct the exact solutions of the nonlinear conformable fractional Schr¨odinger-Hirota equation (FSHE). By using the traveling wave transformation FSHE turns into the ordinary differential equation (ODE) and by the aid of symbolic calculation software, new exact solutions are obtained. 2D, 3D figures and contour surfaces acquired from the values of the solutions are plotted. The results show that the proposed method is powerful, effective and straightforward for formulating new solutions to various types of nonlinear fractional partial differential equations in applied sciences.

Semi-analytic solution of time-fractional Korteweg-de Vries equation using fractional residual power series method

In this paper, we have solved the non-linear Korteweg-de Vries equation by considering it in time-fraction Caputo sense and offered intrinsic properties of solitary waves. The fractional residual power series method is used to obtain the approximate solution of the aforesaid equation and compared the obtained results with Adomian Decomposition Method. Obtained results are efficient, reliable, and simple to execute on most of the non-linear fractional partial differential equations, which arise in various dynamical systems.

Diverse Precise Traveling Wave Solutions Possessing Beta Derivative of the Fractional Differential Equations Arising in Mathematical Physics

In this paper, we obtain the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions for the nonlinear time fractional generalized reaction Duffing model and density dependent fractional diffusion-reaction equation in the sense of beta-derivative by using three fertile methods, namely, Generalized tanh (GT) method, Generalized Bernoulli (GB) sub-ODE method, and Riccati-Bernoulli (RB) sub-ODE method. The derived solutions to the aforementioned equations are validated through symbolic soft computations. To promote the vital propagated features; some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the specific values to the parameters under the confine conditions. The accomplished solutions show that the presented methods are not only powerful mathematical tools for generating more solutions of nonlinear time fractional partial differential equations but also can be applied to nonlinear space-time fractional partial differential equations.

New Numerical Approach of Solving Highly Nonlinear Fractional Partial Differential Equations via Fractional Novel Analytical Method

In this work, the fractional novel analytic method (FNAM) is successfully implemented on some well-known, strongly nonlinear fractional partial differential equations (NFPDEs), and the results show the approach’s efficiency. The main purpose is to show the method’s strength on FPDEs by minimizing the calculation effort. The novel numerical approach has shown to be the simplest technique for obtaining the numerical solution to any form of the fractional partial differential equation (FPDE).

Time-Fractional Differential Equations with an Approximate Solution

This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with the Caputo fractional operator (CF) to solve time fractional linear and nonlinear partial differential equations. The Sumudu transform (ST) and the homotopy perturbation technique (HP) are combined in this approach. In the Caputo definition, the fractional derivative is defined. In general, the method is straightforward to execute and yields good results. There are some examples offered to demonstrate the technique's validity and use.

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

The solutions of nonlinear fractional partial differential equations by using a novel technique

Abstract In this article, the solutions of higher nonlinear partial differential equations (PDEs) with the Caputo operator are presented. The fractional PDEs are modern tools to model various phenomena more accurately. The residual power series method (RPSM) is used for the solution analysis of fractional partial differential equations (FPDEs), which has direct implementation for the solutions of fractional partial differential equations. In this work, the solutions to a few nonlinear FPDEs are handled by the proposed technique. The general and particular schemes of RPSM are constructed and implemented successfully. The fractional solutions of PDEs have provided many useful dynamics of the targeted problems. The RPSM results for both integer and fractional-order FPDEs are further explained and elaborated by using graphs and tables. It is observed that the higher accuracy of RPSM is achieved with fewer calculations. Graphs and tables for fractional-order solutions are presented, which confirm the convergence phenomena of fractional solutions toward integer order solutions of each problem. The suggested method can be extended to the solutions of other nonlinear fractional partial differential equations.

THE INFLUENCE OF FRACTIONAL ORDER AND MAGNETIC FIELD ON ELECTRO HYDRODYNAMIC FLOW OF JEFFREY FLUID ALONG WITH MAGNETIC NANOPARTICLES THROUGH ARTERIAL SEGMENT

The study reports the influence of Caputo–Fabrizio fractional derivative and magnetic field on blood flow as an electrically conducting non-Newtonian fluid along with magnetic nanoparticles through circular cylindrical arterial segment, by assuming blood as Jeffrey fluid. The main structure of the governing fractional nonlinear partial differential equations was obtained from non-Newtonian fluid model in which convective derivative is considered instead of the time derivative which is capable of describing the phenomena of relaxation time and retardation time. The exact solutions of the governing fractional partial differential equations were obtained by the virtue of Laplace and Hankel transforms. Numerical simulations have been performed to analyze the behavior of the Jeffrey fluid flow using Mathcad software and the results are presented graphically for an explicit and detail discussion. We noticed from the graphical representation of results that the fractional derivative, particle concentration, electro-kinetic width, Jeffrey number and the applied magnetic field have profound influence on the magnitude of blood velocity along with magnetic nanoparticles. The effect of each of the influential parameters tremendously decelerates the fluid flow and thereby showing appreciable variation in the distribution of axial and radial velocities. Practically, increasing the applied magnetic field strength and the value of the Jeffrey number on the motion of fluid flow together with magnetic nanoparticles, vehemently decreases the velocities of blood along with magnetic nanoparticles, which helps to have effective control over the fluid networks. From the study, we conclude that the fractional model, Jeffrey number and the application of the external magnetic field on the fluid flow have extensive applications in the clinical and the medical sciences.

Solving Nonlinear Fractional Partial Differential Equations Using the Elzaki Transform Method and the Homotopy Perturbation Method

In this paper, we combine the Elzaki transform method (ETM) with the new homotopy perturbation method (NHPM) for the first time. This hybrid approach can solve initial value problems numerically and analytically, such as nonlinear fractional differential equations of various normal orders. The Elzaki transform method (ETM) is used to solve nonlinear fractional differential equations, and then the homotopy is applied to the transformed equation, which includes the beginning conditions. To obtain the solution to an equation, we use the inverse transforms of the Elzaki transform method (ETM). The initial conditions have a big impact on the equation’s result. We give three beginning value issues that were solved as precise or approximation solutions with high rigor to demonstrate the method’s power and correctness. It is clear that solving nonlinear partial differential equations with the crossbred approach is the best alternative.

Stability Analysis and Computational Interpretation of an Effective Semi Analytical Scheme for Fractional Order Non-Linear Partial Differential Equations

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional order partial differential equations. Since the Laplace transform cannot be applied to non-linear problems, the combination of the variational iteration technique with it will give a better and rapidly convergent sequence. Exact solutions may also exist, but we will show that the coupled technique is much better to approximate the exact solutions. The Caputo–Fabrizio fractional derivative will be used throughout the study. In addition, some possible implications of the results given here are connected with fixed point theory.

On the oscillation of nonlinear delay fractional partial differential equations with damping term

In this article we will consider the oscillation of nonlinear delay fractional partial differential equations with damping subject to Neumann boundary conditions.Sufficient conditions are obtained for the oscillation of solutions by using the method of generalized Riccati transformation.

On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems

The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.

An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices

Purpose:In the last decades, many researchers have performed their best in order to find the solution of the nonlinear evolution equations, nonlinear partial differential equations (NLPDEs), fractional nonlinear partial differential equations and nonlinear optical models by using Kudryashov methods, which have various forms. However, most researchers have used different methods under the same name or applied the same method under the different names. This situation has created confusion for researchers and readers, such as choosing the appropriate method, algorithm and the right article title. In order to eliminate this problem and to clarify the methods, we have conducted rigorous research and wrote a certain standard in this field by bringing together the Kudryashov methods, which are widely used under different names in the literature. Besides, it has been aimed to give the general algorithm of the methods, to make some addendums on the methods and to present new approaches to these methods for the researchers. We strongly believe that the study will be of great benefit not only for those who are interested in nonlinear evolution problems, but also for studies in the field optics. Methodology:Regarding the methods, first of all, the articles of the researchers who suggested the methods and the studies of the respected researchers in this field have been examined and tried to gather under one framework and give a general name with appropriate algorithm for each Kudryashov method. The algorithms and solution functions of each method have been checked on the related Riccati and Bernoulli equations with different softwares (Mathematica, Maple, Matlab, etc.), and confirmed every results carefully. Afterwards, all methods have been applied on (2+1)-dimensional Zoomeron equation for the global test purpose. With the usual standard approach, the nonlinear partial differential form of Zoomeron equation has been transformed into the nonlinear ordinary differential (NODE) form by applying the wave transformation, then each method applied to the NODE form of Zoomeron equation. After these operations, we have obtained a system of linear algebraic equations and their appropriate solutions. By obtaining the appropriate solution sets, the soliton solutions of the Zoomeron equation have been constructed and some selected graphical presentations given. Findings:The results have been obtained from the applications that applied on the Zoomeron equation via Kudryashov methods shows us they can be used widely and effectively in the solution of not only nonlinear evolution problems, but also highly dispersive and higher order nonlinear partial differential equations. It has been seen that all the applied Kudryashov methods give effective results. Besides, controlling and using the parameters in the Kudryashov methods play important role, especially obtaining the effective and desired solution sets. Originality:Based on our detailed research, we could not find any summative study in this area like this one. In addition to all these, some of the approaches and addendums have been presented for the first time in this paper. Besides, (2+1)-dimensional Zoomeron equation has also not been previously solved by some of these methods. It is thought that this study will be used as a reference by many researchers in the future and it will be of great benefit to researchers.

Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.