In this paper, several types of space-time fractional partial differential equations has been solved by using most of special double linear integral transform ”double  Sumudu ”. Also, we are going to argue the truth of these solutions by another analytically method “invariant subspace method”. All results are illustrative numerically and graphically.

In this paper, we are going to study a nonlinear fractional partial differential equations by double Sumudu transformation coupled with Adomian decomposition method or with variational iteration method. For this case, we choose an important fractional partial differential equations, such as parabolic-hyperbolic with mixed fractional derivatives (homogeneous and nonhomogeneous) types. Some examples are given here to illustrate efficiency of this method.

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using  Mathcad 15.and graphic in Matlab R2015a.

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.