scholarly journals Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Xinguang Zhang ◽  
Yong Hong Wu ◽  
Dragoş-Pătru Covei ◽  
Xinan Hao
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Computational Methods ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Complex Boundary

scholarly journals Complex Boundary Value Problems of Nonlinear Differential Equations 2014

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Xinguang Zhang ◽  
Yong Hong Wu ◽  
Dragoș-Pãtru Covei ◽  
Xinan Hao
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Complex Boundary

On Simplifying Complex Equations by Applying Block Elements

2020 ◽  
pp. 8-12
Author(s):  
V.A. Babeshko ◽  
O.V. Evdokimova ◽  
O.M. Babeshko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Block Element ◽  
Helmholtz Equations ◽  
Complex Boundary ◽  
Block Element Method ◽  
Element Method

There are several approaches aimed at simplifying complex partial differential equations or their systems involved in the formulation of boundary value problems by introducing simpler, but in a larger number of differential equations. Their solutions allow us to describe solutions to complex boundary value problems. However, to implement this approach, it is necessary to construct solutions of simplified boundary value problems for arbitrary boundary conditions in solvability spaces boundary value problem. In some cases, this can be done using the block element method. The block element method, which has a topological basis, reveals both global and local properties of solutions to boundary value problems for partial differential equations. At the same time, it can be used to study and solve more complex boundary value problems by applying relations that describe certain equations of the continuum by means of relatively simple equations, for example, Helmholtz. To do this, we need to construct solutions of the Helmholtz equations that satisfy boundary conditions that contain completely arbitrary values, rather than partial values, set at the boundary of functions. In relation to the Helmholtz equations, this is achieved using the block element method. Examples of constructing solutions to boundary value problems for Helmholtz equation for Dirichlet and Neumann problems and a comparative analysis of solutions are given in this article.

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