The Solution Comparison of Time-Fractional Non-Linear Dynamical Systems by Using Different Techniques

Abstract

This comparative study of fractional nonlinear fractional Burger’s equations and their systems has been done using two efficient analytical techniques. The generalized schemes of the proposed techniques for the suggested problems are obtained in a very sophisticated manner. The numerical examples of Burger’s equations and their systems have been solved using Laplace residual power series method and Elzaki transform decomposition method. The obtained results are compared through graphs and tables. The error tables have been constructed to show the associated accuracy of each method. The procedures of both techniques are simple and attractive and, therefore, can be extended to solve other important fractional order problems.

Analytical Solutions of the Nonlinear Time-Fractional Coupled Boussinesq-Burger Equations Using Laplace Residual Power Series Technique

“…This section reviews some definitions and theorems for the fractional operators and the LT [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] which are essential in constructing the L-RPS solutions for the nonlinear T-FCB-BEs as in Equations ( 1) and ( 2). Definition 1.…”

“…Most important and useful properties of fractional operators can be summarized as below [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]:…”

“…In the past twenty years, partial fractional differential equations (P-FDEs) have been motivated due to their various applications in several fields of science such as fluid and layer flows, multi-energy groups of neutron diffusion processes, neutral and multi pantograph systems, dynamic and hyperbolic systems, statistical mechanics model, material sciences and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. These important phenomena and applications are well described by P-FDEs.…”

“…The main advantage of using fractional derivatives with an arbitrary order is that they are flexible more than classical derivatives and also they are not-local. The two famous and important fractional deriavtives in applications are: The Riemann-Liouville FD (R-L-FD) and C-FD [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The relationship between the R-L and the C-FDs are very closed since the R-L-FD can be converted to the C-FD under some regularity assumptions of the function.…”

“…In most cases, exact solutions do not exist for many Partial differential equations (PDEs), therefore, several numerical methods are created and applied to get the approximate series solutions for such P-FDEs such as the homotopy analysis, asymptotic and perturbation methods [1][2][3]5], variational iteration and Adomian decomposition methods [2,4,8], LT and differential transform techniques, RPS method [9][10][11][12][13][14] and L-RPS method [15][16][17][18][19].…”