A LAPLACE VARIATIONAL ITERATION METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Fractional Integro-Differential Equations (FIDEs) arise in the mathematical modelling of physical phenomena and play an important role in various branches of science and engineering. With He's variational iteration method, it is possible to obtain exact or better approximate solutions of differential equations. This paper is concerned with the solution of FIDEs by the variational iteration method via the Laplace transform. In this approach, a correction functional is constructed by a general Lagrange multiplier, which is determined by using the Laplace transform with the variational theory. The results of applying this method to the studied FIDEs show the high accuracy, simplicity and efficiency of the approach

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations