A Novel Method for the Analytical Solution of Partial Differential Equations Arising in Mathematical Physics

Abstract

In this article, an efficient analytical technique, called Sumudu variational iteration method (SVIM), is used to obtain the solution of fractional partial differential equations arising in mathematical physics. The fractional derivatives are described in terms of Caputo sense. This method is the combination of the Sumudu transform (ST) and variational iteration method (VIM). The solution of the suggested technique is represented in a series form, which is convergent to the exact solution of the given problems. Furthermore, the results of the present method have shown close relations with the exact approaches of the investigated problems. Illustrative examples are discussed, showing the validity of the current method. The attractive and straightforward procedure of the present method suggests that this method can easily be extended for the solutions of other nonlinear fractional-order partial differential equations.

Time-Fractional Differential Equations with an Approximate Solution

“…In general, higher performance of the fractional calculus is demonstrated by reduced error levels created during an estimating procedure. Various approximation and methodologies, like the fractional Adomian decomposition method (FADM) [8][9][10], fractional homotopy method (FHPM) [11,12,13], fractional function decomposition method [14,15], fractional variational iteration method (FVIM) [16][17][18], fractional reduce differential transform method (FRDTM) [18,19,20,21], fractional differential transform method [22,23,24], fractional Laplace variational iteration method [25][26][27][28][29][30][31][32], fractional Laplace homotopy perturbation method (FLHPM) [33], fractional Laplace decomposition method (FLDM) [34,35], fractional Sumudu homotopy analysis method [36], fractional Sumudu variational iteration method (FVIM) [37,38], fractional Sumudu decomposition method (FSDM) [39][40][41], fractional natural decomposition method (FNDM) [42,43], fractional Sumudu homotopy perturbation method (FSHPM) [44,45], energy balance method (EBM) [46], power series methods (PSM) [47], have been used in latest years to analyze partial di...…”

An efficient hybrid technique for the solution of fractional-order partial differential equations

“…In recent decades, many of the numerical and analytical techniques have been implemented to solve fractional-order PDEs, such as the fractional variational iteration method [23,34,42,44,45], fractional differential transform method [25,36,46], fractional series expansion method [9,29], fractional Sumudu variational iteration method [20,31], fractional natural decomposition method [32,38], fractional Sumudu decomposition method [17,30,33], fractional Sumudu homotopy perturbation method [28], fractional reduce differential transform method [24,26,41], fractional Adomian decomposition method [16,21,47], fractional Laplace decomposition method [27], fractional Laplace homotopy perturbation method [14], fractional Laplace variational iteration method [13,15,18,35,37], variational iteration method [4][5][6][7][8] and local mesh less УДК 517.95 2020 Mathematics Subject Classification: 34K37, 45J99, 34A08.…”