Analytical Solution for the Time Fractional BBM‐Burger Equation by Using Modified Residual Power Series Method

Abstract

In this study, a generalized Taylor series formula together with residual error function, which is named the residual power series method (RPSM), is used for finding the series solution of the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The BBM-Burger equation is useful in describing approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems. The numerical solution of the BBM-Burger equation is calculated by Maple. The numerical results show that the RPSM is reliable and powerful in solving the numerical solutions of the BBM-Burger equation compared with the exact solutions as well as the solutions obtained by homotopy analysis transform method through different graphical representations and tables.

A fractional Temimi‐Ansari method (FTAM) with convergence analysis for solving physical equations

“…Also the power series solution in subfigure (F) and Equation (10) when n → ∞ converge to the exact solution. In Table 4, we find that the numerical results resulting from the presented method were compared with the numerical results from homotopy analysis method (HAM), 41 q-homotopy analysis method (q-HAM), 42 modified residual power series method (RPSM), 19 variational iteration method (VIM) 40 and homotopy perturbation method (HPM), 40 which show the superiority of the proposed method over other methods in obtaining a lower error rate and thus a better approximation to the exact solution. Table 2 shows a comparison of the convergence analysis resulting from the proposed method with HAM, q-HAM, modified RPSM, VIM, and HPM, which shows the advantage of the proposed method.…”

“…However, for further specification, several researchers have argued about analytical and approximate solutions to the nonlinear KdVBurgers equation utilizing a portion of the notable techniques. [15][16][17][18][19] The advantage of FDEs is that they have a nonlocal property that shows the new properties of these problems. [20][21][22][23][24][25] However, it is very difficult to come up with an exact solution for these types of problems.…”

Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma

“…Nevertheless, solving FPDEs is generally more complex than the clas-sical type since their operators are defined through integrals. There are many techniques proposed by many researchers to handle analytical and approximate solutions of nonlinear FPDEs such as the residual power series method [25][26][27][28], iterative Shehu transform method [29], Laplace decomposition method [30], q-homotopy analysis method [31][32][33][34], Adomian decomposition method [35], fractional reduced differential transform method [36,37], variational iteration method [38,39], homotopy analysis method [40], and other methods [41][42][43][44].…”

Numerical solutions of time and space fractional coupled Burgers equations using time–space Chebyshev pseudospectral method

“…Yang et al 11 first time investigated local fractional Riccati differential method for 2D Burgers equation and shown nondifferentiable exact traveling wave solutions, which are useful for mathematical physics. Further, some significant developments of fractional ordinary and partial differential equations have been discussed in monographs of Kilbas et al 12 Moreover, this equation was solved by several other numerical methods, such as finite difference method, 13,14 finite element method, 15,16 variational iteration method, 17 Adomian decomposition method, 18 B-spline method, [19][20][21] finite volume method, 22 residual power series method, 23 and homotopy perturbation method. 24 Besides, the pseudospectral method is also an emphatic and alternate numerical scheme for solving a wide range of linear and nonlinear fractional partial differential equations.…”