Stability analysis of Hilfer fractional differential systems

In this paper, we present some remarks on the stability of fractional order systems with the Hilfer derivative. Using the Laplace transform, some sufficient conditions on the stability and asymptotic stability of autonomous and non-autonomous fractional differential systems are given. The results are obtained via the properties of Mittag-Leffler functions and the non-standard Gronwall inequality

Numerical solution of fractional differential equation by wavelets and hybrid functions

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By uses these methods we translate an FDE to an algebraic linear equations which can be solve. Methods has been tested by some numerical examples

Approximate analytical solutions of distributed order fractional Riccati differential equation

In this paper, the combined Laplace transform and new homotopy perturbation method is employed for solving a special class of the distributed order fractional Riccati equation. To illustrate the ability and reliability of the method some examples have been provided. The results obtained by the proposed method show that the approach is very efficient, simple and can be applied to other nonlinear problems. Keywords: Homotopy perturbation method, Laplace transform, Fractional, Riccati equation, Distributed orde

Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method

In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system. This approach can also be applied to other nonlinear systems of partial differential equations which can be converted to a second-order ordinary differential equation through the travelling wave transformation