Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A ?is a square matrix. At the same time, power-type estimate for them has been given.

THE NONEMPTINESS AND COMPACTNESS OF MILD SOLUTION SETS FOR RIEMANN-LIOUVILLE FRACTIONAL DELAY DIFFERENTIAL VARIATIONAL INEQUALITIES

This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities,which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality.Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder's fixed point theorem.

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.

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

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.

Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.

ON A COUPLED INTEGRO-DIFFERENTIAL SYSTEM INVOLVING MIXED FRACTIONAL DERIVATIVES AND INTEGRALS OF DIFFERENT ORDERS

By applying the standard fixed point theorems,we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives and mixed fractional integrals,supplemented with nonlocal coupled fractional integral boundary conditions.An example is also constructed for the illustration of the obtained results.

Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications

By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.

A Novel Analytical Technique of the Fractional Bagley-Torvik Equations for Motion of a Rigid Plate in Newtonian Fluids

The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.

Positive Solutions for Systems of Coupled Fractional Boundary Value Problems

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.

The Exact Solution of the Space-Time Fractional Modified Kdv-Zakharov-Kuznetsov Equation

In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.

ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction-Subdiffusion Equation

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL SYSTEM

In this paper we prove the existence and uniqueness of the solution for a class of nonlinear fractional differential system, and investigate the dependence of the solution on the orderαi.

EXISTENCE OF POSITIVE MULTIPLE SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN OPERATOR

In this paper, we consider a two-point fractional boundary value problem. We provide sufficient conditions for the existence of multiple positive solutions to the boundary value problem by Krasnosel＇skii fixed point theorem on the cone.

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.

Symmetries, Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation(FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations(FODEs) in terms of the Erd′elyi-Kober(E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.