Exact solutions for nonlinear partial differential equations via a fusion of classical methods and innovative approaches

被引:3
作者
Mhadhbi, Noureddine [1 ]
Gana, Sameh [2 ]
Alsaeedi, Mazen Fawaz [1 ]
机构
[1] King Abdulaziz Univ, Coll Sci & Arts, Dept Math, Rabigh Campus,POB 344, Jeddah 21911, Saudi Arabia
[2] Imam Abdulrahman Bin Faisal Univ, Dept Basic Sci, Deanship Preparatory Year & Supporting Studies, POB 1982, Dammam 34212, Saudi Arabia
关键词
Partial differential equations; Nonlinear partial differential equations; Variation of parameters; Method of characteristics; Mathematica; TRAVELING-WAVE SOLUTIONS; OPERATORS;
D O I
10.1038/s41598-024-57005-1
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form utt+a(x,t)uxt+b(t)ut=alpha(x,t)+G(u)(ut+a(x,t)ux)e-integral b(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e<^>{-\int b(t)dt}$$\end{document} and utm(utt+a(x,t)uxt)+b(t)utm+1=e-(m+1)integral b(t)dt(ut+a(x,t)ux)F(u,ute integral b(t)dt).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}<^>{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}<^>{m+1}=e<^>{-(m+1)\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e<^>{\int b(t)dt}).$$\end{document} We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.
引用
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页数:14
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