scholarly journals On the Lie symmetries of the boundary value problems for differential and difference sine-Gordon equations

2021 ◽  
Vol 102 (2) ◽  
pp. 142-153
Author(s):  
O. Yildirim ◽  
◽  
S. Caglak ◽  
Keyword(s):  
Group Theory ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Gordon Equation ◽  
Boundary Value ◽  
Lie Point Symmetries ◽  
Sine Gordon Equation ◽  
Difference Problem ◽  
Lie Group Theory ◽  
The Difference

In general, due to the nature of the Lie group theory, symmetry analysis is applied to single equations rather than boundary value problems. In this paper boundary value problems for the sine-Gordon equations under the group of Lie point symmetries are obtained in both differential and difference forms. The invariance conditions for the boundary value problems and their solutions are obtained. The invariant discretization of the difference problem corresponding to the boundary value problem for sine-Gordon equation is studied. In the differential case an unbounded domain is considered and in the difference case a lattice with points lying in the plane and stretching in all directions with no boundaries is considered.

scholarly journals Study of a Numerical Method for Solving a Boundary Value Problem for a Differential Equation with a Fractional Time Derivative

2020 ◽  
pp. 64-69
Author(s):  
N.B Alimbekova ◽  
D.R. Baigereyev ◽  
M.N. Madiyarov
Keyword(s):  
Boundary Value Problem ◽  
Difference Scheme ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Time Derivative ◽  
Smooth Functions ◽  
Differential Problem ◽  
Difference Problem ◽  
The Difference ◽  
Fractional Time Derivative

Recently, there has been an increased interest in the problem of numerical implementation of multiphase filtration models due to its enormous economic importance in the oil industry, hydrology, and nuclear waste management. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives. In the paper, a finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for solving a difference problem under the assumption that there is a solution to the problem in the class of sufficiently smooth functions that prove the uniqueness of the solution and the stability of the difference scheme. The convergence of the solution of the difference problem to the solution of the original differential problem with the second order in time and space variables is shown. The results of computational experiments confirming the reliability of theoretical analysis are presented.

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