Abstract
The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the L_{2} error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).
Highlights
The soliton theory is an important tool in describing various phenomena of wave propagation
The Korteweg–de Vries (KdV) equation is mainly used to investigate the propagation of shallow water waves in one dimension, whereas the bad Boussinesq (BSQ) equation describes the wave propagation in two dimensions
2 Traveling wave solution In this part, we focus on extracting the exact solution of the good Boussinesq (GB) equation using the He semiinverse process [47, 48]
Summary
The soliton theory is an important tool in describing various phenomena of wave propagation. Several nonlinear equations have been successfully developed to investigate wave propagation. The BSQ equation is used in studying different processes appearing in electromagnetic waves in dielectrics [1], magnetosound waves in plasma [2], and the magnetoelastic waves in antiferromagnetic [3]. These equations and others have attracted the attention of a massive number of mathematicians and physicists since the 1970s due to their use in revealing the internal mechanisms of some sophisticated natural phenomena. Some developed techniques and principles include the inverse scattering transform [4], the trial function process [5], the sine–cosine principle [6], the Weierstrass elliptic function approach [7], the tanh–sech technique [8], the Fexpansion technique [9], Hirota’s bilinear principle [10], the modified tanh-function tech-
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