scholarly journals A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations

2017 ◽  
Vol 15 (1) ◽  
pp. 1450-1462
Author(s):  
P.G. Dlamini ◽  
M. Khumalo
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Nonlinear Evolution ◽  
Numerical Technique ◽  
Nonlinear Evolution Equations ◽  
Finite Difference Schemes ◽  
Quasilinearization Method ◽  
Partial Differential ◽  
Compact Finite Difference

Abstract This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.

scholarly journals On the relations between some well-known methods and the projective Riccati equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 613-618
Author(s):  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Riccati Equations ◽  
Nonlinear Evolution Equations ◽  
Alternative Methods ◽  
Computational Techniques ◽  
Partial Differential ◽  
Traditional Methods

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

AN INTRODUCTION TO NONSTANDARD FINITE DIFFERENCE SCHEMES

1999 ◽  
Vol 07 (01) ◽  
pp. 39-58 ◽  
Author(s):  
RONALD E. MICKENS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Difference Schemes ◽  
Discrete Models ◽  
Finite Difference Schemes ◽  
Partial Differential ◽  
Numerical Instabilities ◽  
Nonstandard Finite Difference

Nonstandard finite difference schemes offer the potential for either constructing exact discrete models of differential equations or obtaining discrete models that do not have the elementary numerical instabilities. While the general laws for constructing such schemes are not precisely known at the present time, a number of important rules have been discovered. This paper provides an introduction to the nonstandard finite difference rules, explains their significance, and applies them to several model ordinary and partial differential equations. Several major unresolved issues and problems are briefly discussed.

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