A natural boundary element method for the Sobolev equation in the 2D unbounded domain

Fei Teng,Zhendong Luo,Jing Yang

doi:10.1186/s13661-017-0910-x

Fei Teng, Zhendong Luo + Show 1 more

Open Access

https://doi.org/10.1186/s13661-017-0910-x

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Journal: Boundary Value Problems	Publication Date: Nov 28, 2017
Citations: 7	License type: open-access

Affiliation: North China Electric Power University

Abstract

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and analyze the errors between the exact solution and the fully discretized NBE solutions. Finally, we use some numerical experiments to verify that the NBE method is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

Highlights

Let ⊂ R be a connected and bounded region with smooth boundary := ∂, c := R \, x = (x, x ), |x| = x + x
By the principle of natural boundary reduction, we establish a fully discretized natural boundary element (NBE) format based on the natural integral equation and the Poisson integral formula of this problem and provide the error estimates between the exact solution and the fully discretized NBE solutions
We find exact solution k and numerical solutions k h at time t

Summary

Introduction

For given time upper bound T , we consider the following Sobolev equation in the D unbounded domain: Problem I Find that satisfies. It has been widely applied to many practical engineering fields (see [ , ]), for example, it is used to describe the procedure of fluid flow permeating rocks, the soils moisture migration, and the different media heat transfer It usually includes complex computing domain, initial and boundary value functions, or source term for the Sobolev equation in the D unbounded domain in the real world so that it has no analytic solution. ∂n μk and the relationship between the solution k of Problem III with its Dirichlet boundary valuek is as follows: k=.

Natural boundary reduction on the external circle area

Conclusions