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.
1. Introduction
Costa [1] and Bera et al. [2] established second order linear partial differential equations (P.D.E):
Equation (1) is valid for two-dimensional steady state flow in anisotropic medium or diffusion of mass in anisotropic medium, where , , and also
Furthermore,
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 but when is continuously changing, then solution of equation (1) is really a challenging case.
Suitable fundamental solution of equation (1) for varying coefficient is difficult though not possible. If fundamental solution is used to form integral equation for the special case when . When is uniformly changing function and is constant, then the resulting formulation is not only boundary integral but domain integral containing 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 .
This paper enhanced the use of boundary element method (BEM) for steady state diffusion equation by adding source term and taking as smoothly varying function. No restriction is imposed on , and as in the work of Ang [14], Dineva et al. [15], and Rangogni [16]. Condition of , and is to satisfy the definiteness condition (2) in solution domain. This paper also employs radial basis trial function to approximate 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 iswhere is the source term.
3. Reformation
We rewrite the system of equation (3) as
Here, are functions of and and varies smoothly, and are constant terms.
Let the substitution bewhere is related to by
is related to by
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:
Here, are constant and are radial basis functions centered at .
Using equation (9) in equation (7),where is the number of interpolation points.
We consider the following substitution:
We rearrange this linear system of equations for constants and which results in
Here, the relation for , , and are given as
Collocate at where
Equation (9) becomes with the use of equation (12):
From the above equation,
From equation (14),with
Using equation (16) into equation (10),where
Here,
We remember that and is known as radial basis trial function for partial differential equation (7).
Multiquadric radial basis function is
We consider the following trial function found in the work of Zhang et al. [19].
5. Formation of Boundary Integral Equation
Partial differential equation given by equation (8) can be converted into boundary integral equation aswhere
By putting equation (6) into equation (22),where
Discretize the boundary into straight lines denoted by .The collocation point is taken as midpoint of
By taking the following approximations,
Also,
By the use of equations (25) and (26) into equation (24),
For
It 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
or
The coefficient is given as
To obtain M-set of linear algebraic equations by using equation (12) and second part of equation (25) as
with
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 of and ,
Here, the domain of the problem is . To solve the problem above, we apply the following B. Cs:
The analytic solution of the above problem is denotes the average value of on the interior nodes, and its value is
Analytical and numerical values of are compared graphically.
Figure 1 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 at some specific points like (10, 4) and (40, 19) is considered to compare with analytic solution of at selected interior points. used in equation (12) is differentiated w.r.t in order to derive partial derivatives of first order for .
PROBLEM#02: Now considering the following values of and :
Here, the domain of the problem is . To solve the problem above, we apply the following B. Cs:
Here, again denotes the average value of on the interior nodes, and its value is same as equation (37).
The analytic solution of concerned problem is
Analytical and numerical values of are compared graphically by taking and varying such that
In the graph above, analytical solution of at fixed value of is compared with approximated value of by taking different values of The graph predicts that analytical solution agrees well with the numerical value. (Figure 2)
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].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.