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.

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

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.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

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.

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.

On the partial stability of nonlinear impulsive Caputo fractional systems

In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.

Existence of positive solutions for integral boundary value problem of fractional differential equations

In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-a),where h is the space step size and α∈(1,2)is the order of Riemann-Liouville fractional derivative.Based on this scheme,we introduce a shifted finite difference method for solving space fractional partial differential equations.We obtained the error estimates with the convergence orders O(τ+h^3-a+h^β),where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation.Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Griinwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the firstorder convergence,the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition.Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Griinwald-Letnikov formula or Lubich's higher order approximation schemes.

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].

The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations

In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.

Particular Solutions of Generalized Linear Second Differential Equations by Fractional Calculus

本文依据Ｋ．Ｎｉｓｈｉｍｕｔｏ教授于１９７９年所定义的分数微积分定义Ｌｅｍｍａｓ为基础，给出一种线性阶微分方程及偏微分方程（包括均匀及非均匀）的方法，使其更一般化，概括范围更宽、更广．

Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation

In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

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.

Comparison between the Laplace Decomposition Method and Adomian Decomposition in Time-Space Fractional Nonlinear Fractional Differential Equations

The aim of this paper is to discuss application of Laplace Decomposition Method with Adomian Decomposition in time-space Fractional Nonlinear Fractional Differential Equations. The approximate solutions result from Laplace Decomposition Method and Adomian decomposition;those two accessions are comfortable to perform and firm when to PDEs. For caption and further representation of the thought, several examples are tool up.