Solving second-order nonlinear evolution partial differential equations using deep learning

2020 ◽  
Vol 72 (10) ◽  
pp. 105005
Author(s):  
Jun Li ◽  
Yong Chen
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Partial Differential

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

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.

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