So, the scientists have introduced many techniques like perturbation methods, approximation methods and numerical methods to overcome the problems of such DEs. In nonlinear sciences and engineering, analytical solutions to nonlinear PDEs have a great significance but sometimes, it becomes much problematic to deal with, that becomes a hurdle in finding analytical solutions to these DEs. It has been a requirement to choose a competent mechanism that adopts such mathematical models which cover physical processes as well. The concept of the physical phenomena, depends upon the nature of the solutions of PDEs, so to find the solutions of PDEs, the analytical, semi analytical and numerical methods are introduced. So, to understand the complete physical phenomena, the mathematical modeling comes into account by means of partial differential equations (PDEs), that shows an exceptional performance in science and engineering. In any branch of natural sciences, there exists a small number of problems, that can be solved in a direct way. The importance of the nonlinear phenomena of differential equations (DEs) in sciences (biology, physics and chemistry) is significant.
įunding: There is no source of funding (financial or material support) for my study.Ĭompeting interests: The authors have declared that no competing interests exist. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.ĭata Availability: All relevant data underlying this study can be found, in the form of exact solutions of seventh order KdV-type, differential equations, theorems and definitions, at the following locations. Received: AugAccepted: DecemPublished: January 22, 2021Ĭopyright: © 2021 Saleem et al. PLoS ONE 16(1):Ĭhina University of Mining and Technology, CHINA
This article incorporates text from the article Gauss-Seidel_method on CFD-Wiki that is under the GFDL license.Citation: Saleem S, Hussain MZ, Aziz I (2021) A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven. Gauss, Carl Friedrich (1903), Werke (in German), vol. 9, Göttingen: Köninglichen Gesellschaft der Wissenschaften.
"A Unified Proof for the Convergence of Jacobi and Gauss-Seidel Methods". Abhandlungen der Mathematisch-Physikalischen Klasse der Königlich Bayerischen Akademie der Wissenschaften (in German). "Über ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineäre Gleichungen überhaupt, durch successive Annäherung aufzulösen".
0 kommentar(er)
