Analytical approximate solutions for nonplanar Burgers equations by weighted residual method

Abstract

In this work, analytical approximate progressive wave solutions for the generalized form of the nonplanar KdV-Burgers (KdV-B) and mKdV-Burgers (mKdV-B) equations are presented and the results are discussed. Motivated with the exact solutions of the planar KdV-B and mKdV-B equations, the weighted residual method is applied to propose analytical approximate solutions for the generalized form of the nonplanar KdV-B and mKdV-B equations. The structure of the KdV-B equation assumes a solitary wave type of solution, whereas the mKdV-B equation assumes a shock wave type of solution. The analytical approximate progressive wave solutions for the cylindrical(spherical) KdV-B and mKdV-B equations are obtained as some special cases and compared with numerical solutions and the results are depicted on 2D and 3D figures. The results revealed that both solutions are in good agreement. The advantage of the present method is that it is rather simple as compared to the inverse scattering method and gives the same results with the perturbative inverse scattering technique. Moreover, the present analytical solutions allow readers to carry out physical parametric studies on the behavior of the solution. In addition to the present solutions are defined overall the problem domain not only over the grid points, as well as the solution calculation has less CPU time-consuming and round-off error.