Homotopy analysis SUMUDU transform method for fractional order partial differential equations

Abstract

In recent years various fields of science and engineering are benefited by the models of fractional order partial differential equations FPDEs such as fluid mechanics material science biological mathematics plasma physics finance chemistry medical science Fractional order differential equations such as fractional FokkerPlank equation fractional nonlinear Schrodinger equation fractional NavierStokes equation fractional quasigeostrophic equation fractional GinzburgLandau equation and fractional LandauLifshitz equation have clear physical background and have opened up related new research fields In fact some mathematicians such as LHopital, Leibniz and Euler began to consider how to define the fractional derivative as early as the end of the 17th century In 1870s Riemann and Liouville obtained the definition of fractional derivative for a given function by extending the Cauchy integral formula Many kinds of fractional derivatives definitions are commonly used including Riemann Liouville definition Caputo definition Grunwald Letnikov derivative and Weyl definition In last three decades many authors researchers have shown their interest in the development of arbitrary order differential and integration newlineNowadays the solution of fractional partial differential equations FPDEs with linear and nonlinear terms for the boundary value and initial value problems are playing a vital role in order to explain the natural and memory property of systems An integer order differential operator is a local operator whereas the fractional order differential operator is non local in the sense that it takes into account the fact that the future state not only depends upon the present state but also upon all of the history of its previous states Because of this realistic property the fractional order systems are becoming increasingly popular among the scientist and researchers newlineThe solution of nonlinear FPDEs is difficult to solve than the linear FPDEs especially by the means of analytic methods