# Boundary Integral Technique of 2nd Order Partial Differential Equation by Using Radial Basis

Authors: A. Asghar; M. O. Ahmad; M. N. Bashir; Taseer Muhammad; Sami Ullah Khan; M. Ramzan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The solution of second order partial differential equation, with continuous change in coefficients by the formation of integral equation and then using radial basis function approximation (RBSA), has been developed in this paper. Use of boundary element method (BEM), which gives the solution of heat or mass diffusion in non-homogenous medium with function varying smoothly in space is also the part of this article. Discretization of boundary of integral domain instead of entire domain of the problem concerned is also the distinction of the recent work. The numerical solution of some problems with known value of the variable has also been included at the end.


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## Body

## 1. Introduction

Costa [1] and Bera et al. [2] established second order linear partial differential equations (P.D.E):(1)∂∂x1ℏ11∂w^∂x1+∂∂x1ℏ12∂w^∂x2=0,∂∂x2ℏ21∂w^∂x1+∂∂x2ℏ22∂w^∂x2=0.Equation (1) is valid for two-dimensional steady state flow in anisotropic medium or diffusion of mass in anisotropic medium, where w^=temperature or concentration, ,ℏij=diffusion or conduction coefficient and also ℏij=ℏji.Furthermore,(2)ℏ11Δ1Δ1+ℏ12Δ1Δ2>0ℏ11Δ1Δ2+ℏ12Δ2Δ2>0Such thatΔ12+Δ22≠0.Equation (2) presents the constraints for the integral base scheme associated with equation (1). Reutskiy [3], Al-Jawary and Wrobel [4], Rangelov et al. [5], and Ferreira [6] studied the behavior of graded material in nonhomogeneous media.Efficiency in computation and accuracy in treatment makes the numerical method based on integral equation more advantageous for the treatment of such B.V.Ps.To solve the integral equations derived from such numerical techniques, trial functions play a vital role. Trial functions are of many types such as polynomials, trigonometric functions, and radial basis functions. Radial basis functions as conical and multiquadric radial basis functions have been found useful recently by a number of researchers Lin et al. [7]. Use of radial basis functions in modern era as Lin and Reutskiy [8] has revolutionized the process of research in a number of fields.Clements [9] and Ooi et al. [10] established the solution of equation (1) with constant value of ℏij but when ℏij is continuously changing, then solution of equation (1) is really a challenging case.Suitable fundamental solution of equation (1) for varying coefficient ℏij is difficult though not possible. If fundamental solution is used to form integral equation for the special case when ℏij=ŗijgx1,x2. When gx1,x2 is uniformly changing function and ŗij is constant, then the resulting formulation is not only boundary integral but domain integral containing w^ as integrand Brebbia and Nardini [11] used dual reciprocity method to find approximate solution in terms of boundary integrals. Ang [12] and Tanaka et al. [13] purposed dual reciprocity method for ℏij=ŗijgx1,x2.This paper enhanced the use of boundary element method (BEM) for steady state diffusion equation by adding source term and takingℏij as smoothly varying function. No restriction is imposed on ℏ11,ℏ12, and ℏ22 as in the work of Ang [14], Dineva et al. [15], and Rangogni [16]. Condition of ℏ11,ℏ12, and ℏ22 is to satisfy the definiteness condition (2) in solution domain. This paper also employs radial basis trial function to approximate w^x1,x2 to convert the given diffusion equation to elliptic diffusion equation as Ang et al. [17], Fahmy [18] explain that integral formulation here does not involve any domain integral. At the end, specific problems are solved for unknown by converting the problem into a set of algebraic equations.


## 2. Steps for Solution

Steady state anisotropic diffusion equation is(3)∂∂x1ℏ11∂w^∂x1+∂∂x1ℏ12∂w^∂x2+g1x1,x2w^+g2x1,x2=0∂∂x2ℏ21∂w^∂x1+∂∂x2ℏ22∂w^∂x2+g1x1,x2w^+g2x1,x2=0,where g1x1,x2w^+g2x1,x2 is the source term.(4)w^x1,x2=ζx1,x2forx1,x2∈C˙1ℏ11x1,x2n1x1,x2∂w^∂x1+ℏ12x1,x2n2x1,x2∂w^∂x2=ςx1,x2forx1,x2∈C˙2ℏ21x1,x2n2x1,x2∂w^∂x1+ℏ22x1,x2n2x1,x2∂w^∂x2=ςx1,x2forx1,x2∈C˙2here,C˙=C˙1UC˙2,ni=unit normal vector toC˙


## 3. Reformation

We rewrite the system of equation (3) as(5)ℏ110∂2w^∂x12+∂∂x1ℏ111∂w^∂x1+ℏ210∂2w^∂x1∂x2+∂∂x2ℏ211∂w^∂x1+ℏ120∂w^∂x1∂x2+ℏ220∂2w^∂2x2+∂∂x1ℏ121∂w^∂x2+∂∂x2ℏ221∂w^∂x2=−g1x1,x2w^−g2x1,x2.Here,ℏij1=ℏij−ℏij0 are functions of x1 and x2 and varies smoothly, and hij0 are constant terms.Let the substitution be(6)w^x1,x2=v1x1,x2+v2x1,x2,where v1 is related to w^ by(7)ℏ110∂2v1∂x12+∂∂x1ℏ111∂2w^∂x1+ℏ120∂2v1∂x1∂x2+∂∂x1ℏ121∂w^∂x2+ℏ210∂2v1∂x2∂x1+∂∂x2ℏ211∂w^∂x1+ℏ220∂2v1∂2x2+∂∂x2ℏ221∂w^∂x2=−g1x1,x2w^−g2x1,x2.v2 is related to w^ by(8)ℏ110∂2v2∂x12+ℏ120∂2v2∂x1∂x2+ℏ210∂2v2∂x2∂x1+ℏ220∂v2∂2x2=0.It is obvious that above equations satisfy equation (3).Discretize equation (8) into linear algebraic equations. Solve resulting algebraic equations using B.Cs.


## 4. Trial Function Substitution

Trial functions such as radial basis function (RBF) are used to approximate unknown and are also used to discretize the domain and numerically solve partial differential equations by considering following approximation:(9)∑j=12ℏij0∂v1∂xj+ℏij1∂w^∂xj≃∑j=1N˜αir′þr′x1,x2.Here,αir′ are constant and þr′x1,x2 are radial basis functions centered at η1r′,η2r′.Using equation (9) in equation (7),(10)∑r′=1N˜α1r′∂∂x1þr′x1,x2+α1r′∂∂x2þr′x1,x2=−g1x1,x2w^−g2x1,x2,where N˜ is the number of interpolation points.We consider the following substitution:(11)w^x1,x2≃β11þ1x1,x2+β12þ2x1,x2+β13þ3x1,x2+...+β1w^þw^x1,x2r′=1,2,..,w⌢υx1,x2≃d11þ1x1,x2+d12þ2x1,x2+d13þ3x1,x2+...+d1w^þw^x1,x2r′=1,2,..,w⌢.We rearrange this linear system of equations for constantsβr′ and dr′ which results in(12)w^x1,x2≃∑r′=1N˜∑m=1N˜Ψr′m′w^m′þr′x1,x2υx1,x2≃∑r′=1N˜∑m=1N˜Ψr′m′υm′þr′x1,x2.Here, the relation forw^m′ , υm′, and Ψr′m′ are given as(13)w^m′=w^η1m′,η2m′υm′=υη1m′,η2m′∑m′=1N˜Ψr′m′þsη1m′,η2m′=1,whenr′=S′,0,whenr′≠S′,.Collocate atx1,x2=η1n′,η2n′ where n′=1,2,3,...,N˜Equation (9) becomes with the use of equation (12):(14)αi1þ1η1ń,η2ń+αi2þ2η1ń,η2ń+...+αiN˜þw^η1ń,η2ń=∑m′=1N˜ϑin′m′υm′+ϕin′m′w^m′,where n′=1,2,3,...,N˜From the above equation,(15)ϑin′m′=∑j=12ℏij0∑r′=1N˜ϑr′m′∂∂xjþr′x1,x2atx1,x2=η1n′,η2n′ϕin′m′=∑j=12ℏij0∑r′=1N˜ϕr′m′∂∂xjþr′x1,x2atx1,x2=η1n′,η2n′.From equation (14),(16)αir′=∑m′=1N˜ŗir′m′υm′+ßiw^m′.with(17)ŗir′m′=∑n′=1N˜ϑin′m′ϕr′n′ßir′m′=∑ń=1N˜Φin′m′ϕr′n′.Using equation (16) into equation (10),(18)g1n′w^n′+∑m′=1N˜μn′m′υm′+ϖn′m′um′=−g2n′forn′=1,2,3,...,N˜,where gin′=giη1n′,η2n′Here,(19)μn′m′=∑r=1N˜∑i=12υrm′∂∂xiþrx1,x2whereη1n′,η2n′=x1,x2ϖn′m′=∑r′=1N˜∑i=12ßir′m′∂∂xiþr′x1,x2whereη1n′,η2n′=x1,x2.We remember thatw^m′ and υm′m=1,2,...,N˜ is known as radial basis trial function for partial differential equation (7).Multiquadric radial basis function is(20)þr′x1,x2=1+x1−η1r′2+x2−η2r′21/2,We consider the following trial function found in the work of Zhang et al. [19].(21)þrx1,x2=1+x1−η1r2+x2−η2r2+x1−η1r2+x2−η2r23/2.


## 5. Formation of Boundary Integral Equation

Partial differential equation given by equation (8) can be converted into boundary integral equation as(22)δη1,η2w^η1,η2=C˙ϒx1,x2,η1,η2ϖx1,x2−ϕx1,x2,η1,η2∑i=12∑j=12ℏij0nix1,x2∂∂xjϖx1,x2dsx1,x2,where(23)δη1,η2=1,whenη1,η2lies in interior of domainD,12,whenη1,η2lies on smoooth part ofC˙.By putting equation (6) into equation (22),(24)δη1,η2w^η1,η2−v1η1,η2=C˙ϒx1,x2,η1,η2w^x1,x2−v1x1,x2−ϕx1,x2,η1,η2þx1,x2−ţx1,x2dsx1,x2.where(25)þx1,x2=∑i=12∑j=12ℏij0nix1,x2∂∂xjw^x1,x2ţx1,x2=∑i=12∑j=12ℏij0nix1,x2∂∂xjv1x1,x2.Discretize the boundaryC˙ into M straight lines denoted by C˙1,C˙2,...,C˙M .The collocation point η1m′,η2m′ is taken as midpoint of C˙MBy taking the following approximations,(26)C˙≃C˙1∪C˙2∪...∪C˙M−1∪C˙M.Also,(27)w^x1,x2≃w^mv1x1,x2≃v1Px1,x2≃Pmţx1,x2≃ţmwhenx1,x2∈C˙m′.By the use of equations (25) and (26) into equation (24),(28)δη1n′,η2n′w^n′−v1n′=∑m′=1Mw^m′−v1m′C˙m′ϒx1,x2,η1n′,η2n′dsx1,x2−∑m′=1MPm′−ţm′C˙m′ϕx1,x2,η1n′,η2n;dsx1,x2.Forn′=M+i,where1≤i≤NIt is the boundary integral approximation of partial differential equation (8).


## 6. Mathematical Methodology

Boundary conditions given in equation (4) can be expressed in terms of algebraic equations as(29)w^m′=ŗη1m′,η2m′whenw^isrestrictedtoC˙m′.or(30)Pm′+∑g=1M+N˜y¨m′gw^g=Şη1m′,η2m′where∑i=12∑j=12ℏijx1,x2nix1,x2∂w^∂xjisrestrictedtoC˙m′.The coefficienty¨m′g is given as(31)y¨m′g=∑i=12∑j=12ℏij1η1m′,η2m′nim′∑r′=1M+Nϕr′g∂∂xjþr′x1,x2atx1,x2=η1m′,η2m′.To obtain M-set of linear algebraic equations by using equation (12) and second part of equation (25) as(32)ţm′−∑g=1M+Nzm′gυg=0,m′=1,2,...,M,with(33)zm′g=∑i=12∑j=12ℏij0nim′∑r′=1M+Nϕrg∂∂xjþr′x1,x2atx1,x2=η1m′,η2m′.So, for the solution of boundary value problem, we solve system of 4M + 2N linear algebraic equations given by in equations (18), (28), (29), and (32).


## 7. Numerical Application of Boundary Integral Technique (B.I.T)

Algorithm for the solution of specific problems by B.I.T using trial basis function introduced in equation (21) has been used in this numerical computation.PROBLEM#01: Considering the following specific values ofℏij and gi,(34)12ℏ11=ℏ22=ℏ12=ℏ21=x12−2x1x2+22,g1=g2=0.Here, the domain of the problem is0<xi<1i=1,2. To solve the problem above, we apply the following B. Cs:(35)w^x1,0=1+x1x12+2w^x1,1=x1x12−2x1+2w^0,x2=1−x22w^1,x2=2−x23−2x2whenx1,x2∈0,1.The analytic solution of the above problem is(36)w^x1,x2=1+x1−x2x12−2x1x2+2,ℏij0 denotes the average value of ℏij on the interior nodes, and its value is(37)ℏij0=1N02∑i=1N0∑j=1N0ℏij0iN0+1,jN0+1.Analytical and numerical values ofw^ are compared graphically.Figure1 shows comparison of number of boundary elements, and interior collocation points has been drawn graphically. This is worth observing here that the accuracy of solution has improved with more element discretization of boundary curve. Value of w^ at some specific points like (10, 4) and (40, 19) is considered to compare with analytic solution of w^ at selected interior points. w^ used in equation (12) is differentiated w.r.t xii=1,2 in order to derive partial derivatives of first order for w^.Figure 1
Analysis between computed and exact results for 16 values.PROBLEM#02: Now considering the following values ofℏij and gi:(38)ℏ11=ex1+1ex1,ℏ22=e−x2,ℏ12=ℏ21=0g1=−e2x1+x2−6e−x1−2e−x2−4,g2=1.Here, the domain of the problem is0<xi<1i=1,2 . To solve the problem above, we apply the following B. Cs:(39)∑j=12ℏ2j∂w^∂xj|x2=0=−e−2x1∑j=12ℏ2j∂w^∂xj|x2=1=−e−21+x1∑j=12ℏ1j∂w^∂xj|x1=0=−4e−x2w^1,x2=e−2−x2Forx1∈0,1x2∈0,1,Here, againℏij0 denotes the average value of ℏij on the interior nodes, and its value is same as equation (37).The analytic solution of concerned problem is(40)w^x1,x2=e−2x1−x2.Analytical and numerical values ofw^ are compared graphically by taking x2=0.50 and varying x1 such that x1=0.1,0.2,...,0.9In the graph above, analytical solution ofw^ at fixed value of x2=0.50 is compared with approximated value of w^ by taking different values of x1. The graph predicts that analytical solution agrees well with the numerical value. (Figure 2)Figure 2
Analysis between computed and exact results for 16 nodes.


## 8. Conclusion

Numerical technique used in the article requires only the boundary to be discretized for the solution of 2-D steady state mass diffusion or heat conduction using trial function approximation. Specialty is that it does not include collocation points only but also interior points distributed in a mannered way. The accuracy and validity of the method are verified by applying to a problem with known solutions. The solution obtained numerically agrees well with the known results.It is also noted here that in this paper, the boundary integral equation, obtained and used in this method, is discretized using elements with constant value and this makes the error as minimum as desired. Also, the reduction in error is observed by increasing the number of boundary elements and related interior collocation points. The selected method based on trial function and boundary integral approximation provides effective and reliable alternatives to all the existing mathematical techniques for the solution of heat and mass conduction in the anisotropic medium. The possibility of further improvement in the work to solve problems related to anisotropic media is also the part of this paper as was performed earlier by Fahmy [20], Marin and Lesnic [21], Baron [22] and Aksoy and Senocak [23], and Dobroskok and Linkov [24].



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