Two-dimensional multiple non-homogeneous harmonic equation and its boundary integral equations

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Abstract

The paper discusses 2-dimensional multiple non-homogeneous harmonic equation and its boundary integral equation, and gives recurrence formulae of fundamental solution. Under the assumption that non-homogeneous term is m-degree harmonic, the integral term in domain is shifted into boundary integral. Therefore the boundary integral equations are obtained that the integral in domain is not included, and the special solution can be expressed by boundary integrals. Furthermore, the error and convergence analysis are given by Taylor’s polynomial approximating non-homogeneous term f(x), and analytic expressions of boundary integral for 2-dimensional harmonic equation are given in local right-angle coordinate system.

Introduction

Two-dimensional Laplace equation or harmonic equation and biharmonic equation have important applications in scientific and engineering, and they have been better researched theoretically. To find approximate solution or numerical solution of elliptic partial differential equations, many authors have suggested some domain decomposition method and boundary integral equation method-BIEM [1], [2], [3]. Gunzburger [4] had discussed Dirichlet problem of Possion equation by domain decomposition method, and Jeon [5], [6], [7], [8], [9] and Fuglede [10] had given indirect scalar boundary integral equation formulas for the biharmonic equation. Sobolev [11] firstly proposed and researched multiple harmonic equation. However for 2-dimensional multiple non-homogeneous harmonic equation, for instance, Poisson equation and non-homogeneous biharmonic equation are less researched. Besides, applying BIEM for the problem of non-homogeneous equation, the boundary integral equation has integral term in domain. For eliminating the integral in domain and enough utilizing characteristics of decreasing dimension and partition units on the boundary, generally, the integral in domain is transformed into the integral on the boundary used mathematics method and the problem is reduced to homogeneous by finding special solution. But these can be only done in some special cases, for example, the non-homogeneous term is constant or harmonic. The author [12] discussed 2-dimensional Poisson’s equation and applied in elastic plane problem. The aim of the paper is to propose integral equation for 2-dimensional multiple non-homogeneous harmonic equation. We also provide error and convergence analysis.

The rest of the paper is organized as follows. In Section 2 the fundamental solutions of multiple harmonic equation are given, and the integral in domain is shifted into the boundary integral under the assumption that non-homogeneous term is m-degree harmonic and the boundary integral equations are given that do not contain the integral term in domain. In Section 3, when non-homogeneous term f(x) is sufficiently smooth function, and it is approached by Taylor’s polynomial, the error estimation and convergence analysis is given respectively. In Appendix, analytic expressions of the boundary integral of 2-dimensional harmonic equation are also given in local right-angle coordinate system for constant unit and line unit.

Section snippets

Preliminary

We consider the equationΔ(k)u=f(x),xΩ,where Ω is bounded domain in R2 with smooth or piecewise smooth boundary Γ and Δ=i=122xi2 is 2-degree Laplace operator, x = (x1, x2), and f(x) is known function.

Proposition 1

Assume that g(x, y) = Arl(lnr + a),r = x  y, where A and a are real constants, l  0 is positive integer, and that functions Fm(x, y) satisfyΔ(m)Fm=g(x,y),x,yΩ,thenFi=Airl(lnr+ai)(i=0,1,,m)andli+1=li+2,Ai+1=Aili+12,ai+1=ai-2li+1,i=0,1,,m,F0 = g(x, y) as i = 0 and A = A0, a = a0, l = l0  0.

Proposition 2

If Fi are expressed as

Main results

We are looking for a weak solution. Firstly, we need introduce some conceptions concerning weak solution.

Definition 2

The u is called to be weak m-degree harmonic if u  Hm(Ω), Δ(m−1)u  H−1/2(Γ), (Δ(m−1)u)n  H−3/2(Γ), and satisfiesΩΔ(m-1)uΔνdΩ=ΓΔ(m-1)uνnds-Γ(Δ(m-1)u)nνdsfor all v  H2(Ω), where (·)n expresses normal derivative along the Γ.

Definition 3

The u is called to be a weak solution of Eq. (1) if uHm(Ω)Hm-12(Γ) and satisfiesΩuΔ(k)ϕdΩ=ΩfϕdΩ,for ϕH0m(Ω).

From Lemma 1 and Definition 1, Definition 2, we have

Lemma 2

If f(x)  H

Error and convergence analysis

Supposing f(x)C2m(Ω¯) for x  Ω, here x =  y + h as y  Ω and h = (h1, h2), and taking that y is coordinate origin, thus the Γ:r(x, y) = {r = x  yx  Γ, y  Ω}. From multi-variable Taylor’s formula, it isf(x)=f(y)+i=12hixif(y)++1(2m-1)!i=12hixi(2m-1)f(y)+1(2m)!i=12hixi(2m)f(ξ)=N2m-1+R2m-1,where ξ = y + θh, 0 < θ < 1. If non-homogeneous term f(x) of Eq. (1) is approached by Taylor’s polynomial, then Δ(m)N2m−1 = 0, and the corresponding remainder term is Rm,k(f)=ΩR2m-1FkdΩ. We have

Theorem 3

Let Ω  R2 be a finite simply

Conclusions

From discussion of the paper, we have the following conclusions:

  • (1)

    The Proposition 1 gives recurrence formulae of Fi(i = 1, 2, 3,  ,m) in Δ(m)Fm = F0. Assume that F0 is fundamental solution of Laplace equation Δu = 0, i.e., ΔF0 =   δ(x, y), then fundamental solution of Eq. (1) can be obtained by recurrence relations (3). The Proposition 2 gives relations between Fm with Fk and Δ(m)Fk = Fkm(k  m).

  • (2)

    If the function f(x) satisfies the equation Δ(m)f = 0, the integral I(y) in domain can be transformed the boundary

Acknowledgements

The work described in this paper was supported by grants from the National Natural Science Foundation of China (50573095/E030701). Thank Dr. C.D. Li for his instructive comments.

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