The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Adomian's Method Applied to Solve Ordinary and Partial Fractional Differential Equations

This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.

Crank-Nicolson ADI Galerkin Finite Element Methods for Two Classes of Riesz Space Fractional Partial Differential Equations

In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations

The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.

Adaptive Single Piecewise Interpolation Reproducing Kernel Method for Solving Fractional Partial Differential Equation

It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolation reproducing kernel method(ASPIRKM) is determined to solve the FPDE. This improved method not only improves the calculation accuracy, but also reduces the waste of time. Two numerical examples show that the ASPIRKM is a more time-saving numerical method than the TRKM.

On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays

The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness properties with respect to the parameters of these delay fractional differential equations are considered.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

This paper deals with the existence,uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions.The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

(G′/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.

Moving finite element methods for time fractional partial differential equations

With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations （FPDEs）. The unconditional stability and convergence rates of 2 -a for time and r for space are proved when the method is used for the linear time FPDEs with a-th order time derivatives. Numerical exam-ples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.

An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

consistent Riccati expansion fractional partial differential equation Riccati equation modified Riemann–Liouville fractional derivative exact solution

In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.

The Averaging Principle for Stochastic Fractional Partial Differential Equations with Fractional Noises

The purpose of this paper is to establish an averaging principle for stochastic fractional partial differential equation of order a.>1 driven by a fractional noise.We prove the existence and uniqueness of the global mild solution for the considered equation by the fixed point principle.The solutions for SPDEs with fractional noises can be approximated by the solution for the averaged stochastic systems in the sense of p-moment under some suitable assumptions.

A High Order Formula to Approximate the Caputo Fractional Derivative

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

Exact solutions of some fractional differential equations arising in mathematical biology

In the last decades Exp-function method has been used for solving fractional differential equations. In this paper, we obtain exact solutions of fractional generalized reaction Duff- ing model and nonlinear fractional diffusion-reaction equation. The fractional derivatives are described in the modified Riemann-Liouville sense. The fractional complex trans- form has been suggested to convert fractional-order differential equations with modified Riemann-Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation.

CAUCHY PROBLEM FOR FRACTIONAL BEAM-LIKE EQUATIONS

In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the α- fractional time derivative, 1 〈 α ≤ 2. The solution is given in closed form in terms of the Mttag-Leffler functions in two parameters.

Analytical and approximate solutions of(2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation

In this paper,we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation,namely Burgers-Kadomtsev-Petviashvili equation(Burgers-K-P)that arises in shallow water waves.Furthermore,using the residual power series method(RPSM),approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package.We also presented a few graphical illustrations for some surfaces.The fractional derivatives were considered in the conformable sense.All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method.The numerical outcomes confirmed that both methods are simple,robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.

TWO-VARIABLE JACOBI POLYNOMIALS FOR SOLVING SOME FR ACTIONAL PA RTIAL DIFFERENTIAL EQUATIONS

Tw o variable Jacobi polynomials,as a two-dimensional basis,are applied to solve a class of temporal fractional partial differential equations.The fractional derivative operators are in the Caputo sense.The operational matrices of the integration of integer and fractional orders are presented.Using these matrices together with the Tau Jacobi method converts the main problem into the corresponding system of algebraic equations.An error bound is obtained in a two-dimensional Jacobi-weighted Sobolev space.Finally,the efficiency of the proposed method is demonstrated by implementing the algorithm to several illustrative examples.Results will be compared witli those obtained from some existing methods.