Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves

Abstract

In this paper, an effective modification of variational iteration algorithm-II is presented for the numerical solution of the Korteweg–de Vries-Burgers equation, Burgers equation and Kortewege–de Vries equation. In this modification, an auxiliary parameter is introduced which make sure the convergence of the standard algorithm-II. In order to assess the precision of the solutions, numerical computations obtained from the time evaluation of the solutions of the Kortewege–de Vries-Burgers equation with different values for dispersion and diffusion coefficients, show that the proposed algorithm converges rapidly, yields accurate results and offers better accuracy and robustness in comparison with other previous numerical methods. Furthermore, the method can be readily implemented for illuminating viably an enormous number of nonlinear differential equations with better accuracy. Furthermore, any transformation, weak nonlinearity assumption to find an explicit solution or any discretization to find the numerical solution is not necessary for this proposed algorithm.

An efficient approach for the numerical solution of fifth-order KdV equations

“…where λ and h are two unknown parameters, the first one is known as the Lagrange multiplier [29], while the second one is an auxiliary parameter, which was used to accelerate the convergence in different methods [30][31][32][33][34][35][36][37], but we are the first to use it in VIA-II. The Lagrange multiplier can be found by taking δ on both sides of the recurrence relation (7) w.r.t.…”

Solving Partial Differential Equations Using Deep Learning and Physical Constraints

“…The KdV-Burgers equation is a nonlinear partial differential equation containing higher-order derivatives that has been of interest to many researchers [56,57]. Today, the KdV-Burgers equation is widely studied and applied in many fields, such as the study of the flow of liquids containing bubbles, the flow of liquids in elastic tubes, and other problems [58,59]. In mathematical form, the KdV-Burgers equation is defined, as follows:…”

Hyperbolic Type Solutions for the Couple Boiti-Leon-Pempinelli System

“…There are several approaches for finding solutions of nonlinear partial differential equations which have been developed and employed successfully. Some of these are a new sub equation method [1], homotopy analysis method [2,3], homotopy-Pade method [4], homotopy perturbation method [5,6], (G ′ /G)-expansion method [7,8], modified variational iteration algorithm-I [9,10,11], sub equation method [12], Variational iteration method with an auxiliary parameter [13,14,15,16], sumudu transform approach [17], (1/G ′ )-expansion method [18,19], variational iteration method [20,21], auto-Bäcklund transformation method [22], Clarkson-Kruskal direct method [23], Bernoulli sub-equation function technique [24], decomposition method [25,26,27,28], modified variational iteration algorithm-II [29,30,31], first integral method [32], homogeneous balance method [33], modified Kudryashov technique [34], residual power series approach [35], collocation method [36], extended rational SGEEM [37], sine-Gordon expansion method [38,39] and many more [40,41,...…”