On a new numerical method for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics

1982 ◽  
pp. 642-651 ◽  
Author(s):  
Robert L. Sternberg ◽  
Marvin J. Goldstein ◽  
Drew Drinkard
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Nonlinear Partial Differential Equations ◽  
Geometrical Optics ◽  
Partial Differential ◽  
New Class

scholarly journals A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alemu Senbeta Bekela ◽  
Melisew Tefera Belachew ◽  
Getinet Alemayehu Wole
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Nonlinear Partial Differential Equations ◽  
Stability And Convergence ◽  
Test Problems ◽  
Proportional Delay ◽  
Variational Theory ◽  
Partial Differential ◽  
The Stability

Abstract Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

Application of New Triangular Functions to Nonlinear Partial Differential Equations

2009 ◽  
Vol 64 (1-2) ◽  
pp. 1-7 ◽  
Author(s):  
Emad A.-B. Abdel-Salam ◽  
Dogan Kaya
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Main Idea ◽  
Traveling Wave Solutions ◽  
Mkdv Equation ◽  
Kawahara Equation ◽  
Partial Differential ◽  
New Class ◽  
New Research

The results of some new research on a new class of triangular functions that unite the characteristics of the classical triangular functions are presented. Taking into consideration the great role played by triangular functions in geometry and physics, it is possible to expect that the new theory of the triangular functions will bring new results and interpretations in mathematics, biology, physics and cosmology. New traveling wave solutions of some nonlinear partial differential equations are obtained in a unified way. The main idea of this method is to express the solutions of these equations as a polynomial in the solution of the Riccati equation that satisfy the symmetrical triangular Fibonacci functions. We apply this method to the combined Korteweg-de Vries (KdV) and modified KdV (mKdV) equations, the generalized Kawahara equation, Ito’s 5th-order mKdV equation and Ito’s 7th-order mKdV equation.

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