Riemann-Liouville derivative and Caputo derivative for solving Extraordinary differential equation by homotopy analysis method

The purpose of this paper is to study  Riemann-Liouville derivative and  Caputo derivative by homotopy analysis method to solve an Extraordinary differential equation. The results are obtianed by the proposed  method  show  efficient (HAM) using Riemann-Liouville derivative and Caputo derivative . Keywords: homotopy analysis method, Extraordinary differential equation, Riemann-Liouville derivative , Caputo derivative

A Generalizing of the Fractional Sub-Equation Method to Solve Systems of the Space-Time Fractional Differential Equation

In the present paper, we construct the analytical solutions of some Space-Time nonlinear fractional order systems involving Jumarie's modified Riemann-Liouville derivative in mathematical physics : such that Space-Time fractional Whitham-Broer-Kaup equations, Space-Time fractional Breaking Soliton equations, Space-Time fractional Coupled Boussinesq- Burgers equations and Space-Time fractional Coupled Burgers Equations by using The fractional sub-equation method . this method  is very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. Key words: fractional sub-equation method, modified Riemann-Liouville derivative, Mittage-Leffler function

Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives

In this work we study a generalized nonlocal thermistor problem with fractional-order Riemann-Liouville derivative. Making use of fixed-point theory, we obtain existence and uniqueness of a positive solution.Comment: Submitted 17-Jul-2011; revised 09-Oct-2011; accepted 21-Oct-2011; for publication in the journal 'Differential Equations & Applications' (http://dea.ele-math.com

Fractional variational calculus for nondifferentiable functions

We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie's modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free boundary conditions is considered, as well as problems with isoperimetric and holonomic constraints.Comment: Submitted 13-Aug-2010; revised 24-Nov-2010; accepted 28-March-2011; for publication in Computers and Mathematics with Application

Unique solvability of fractional functional differential equation on the basis of Vallée-Poussin theorem

summary:We propose explicit tests of unique solvability of two-point and focal boundary value problems for fractional functional differential equations with Riemann-Liouville derivative

Analytic Solution of Linear Fractional Differential Equations with Constant Coefficient

This paper presents direct methods for obtaining the explicit general solution to a linear sequential fractional differential equation (LSFDE), involving Jumarie’s modification of Riemann–Liouville derivative, with constant coefficients. The general solution to a homogenous LSFDE with constant coefficients is obtained by using the roots of the characteristic polynomial of the corresponding homogeneous equation. For the non-homogeneous case, two methods, undetermined coefficients and variation of parameter, are investigated to find the particular solution. The method of undetermined coefficients is independent of the integral transforms while the method of variation of parameter is not. Moreover, several examples are illustrative for demonstrating the advantage of our approach. Keywords: Fractional differential equations, Riemann–Liouville derivative, Caputo derivative, undetermined coefficients, variation of parameter