In this paper, the ( / ) G G  -expansion method is extended to solve fractional differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order. For illustrating the validity of this method, we apply it to fi...

The present work introduces an effective modification of homotopy perturbation method for the solution of nonlinear time-fractional biological population model and a system of three nonlinear time-fractional partial differential equations. In this approach, the solution is considered a series expansion that converges to the nonlinear problem. The new approximate analytical procedure depends onl...

Fractional Partial Differential Equations (FPDEs) are emerging as a powerful tool for modeling challenging multiscale phenomena including overlapping microscopic and macroscopic scales, anomalous transport, and long-range temporal or spatial interactions. The fractional order may be a function of space–time or even be a distribution, opening up tremendous opportunities for modeling and simulati...

The homotopy analysis method HAM is applied to solve linear and nonlinear fractional partial differential equations fPDEs . The fractional derivatives are described by Caputo’s sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demons...

We adopt the integral definition of fractional Laplace operator and analyze solution techniques for fractional, semilinear, elliptic optimal control problems posed on Lipschitz polytopes. consider two strategies discretization: a semidiscrete scheme where admissible set is not discretized fully discrete such with piecewise constant functions. As an instrumental step, we derive error estimates f...

This work investigates how we can extend the invariant subspace method to $$(2+1)$$ -dimensional time-fractional non-linear PDEs. More precisely, systematic study has been provided for constructing various dimensions of subspaces generalized convection–reaction–diffusion wave equation along with initial conditions first time. Additionally, special types above-mentioned are discussed through thi...

One of the efficient and powerful schemes to solve linear and nonlinear equations is homotopy analysis method (HAM). In this work, we obtain the approximate solution of a system of partial differential equations (PDEs) by means of HAM. For this purpose, we develop the concept of HAM for a system of PDEs as a matrix form. Then, we prove the convergence theorem and apply the proposed method to fi...