Exact Solution of Two-Dimensional Fractional Partial Differential Equations

Abstract

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

“…The advantage of the double Sumudu transform is it gives a rapid convergence of the exact solution without any restrictive assumption of the solution compared to other known methods (see [27]). Unfortunately, this transform fails to solve nonlinear partial differential equations like other integral transforms; to solve this problem, this transform is often combined with other methods like the variational method [28], reduced differential transform method [29], double integral transform (Laplace-Sumudu transform) method [30], Adomian decomposition method [25,31], and homotopy perturbation method [20,32].…”

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.…”