# Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

Authors: Neda Khaksari; Mahmoud Paripour; Nasrin Karamikabir
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.


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## Body

## 1. Introduction

The integral equations technique is broadly applied for solving several equations in physics, mathematics, and engineering [1, 2]. There are many studies about integral problems which deal with the analysis of these types of equations and solution. Many equations in physics and engineering may be modeled applying second kind two-dimensional integral equations. Fredholm integral equations have utilizations in plasma physics, electrical engineering [3], and telegraph equations [4]. Due to the limited area of closed form solutions, efforts are required to make accurate numerical methods and efficient algorithms for simulating such models. A new interpolation method for functional integral equation is introduced by Rashed [5] and then developed in more detail by other researchers. Among different approaches, the RBF method is one of the best numerical methods for approximation of the solutions. In this study, RBF is used to approximate the solution of the Fredholm linear functional two-dimensional integral equation of second kind as follows:(1)lX=uX+AXuhX+λ∫ab∫cdKX,TuTdT,where AX,hX, and lX are the analytical known continuous functions, and the kernel KX,T is a function in a,b×c,d. In the above equation, T=t1,t2, and X=x1,x2. In order to approximate function uX, RBF interpolation is used in distinct points from the domain definition. To do so, the following linear combination is considered:(2)uX≈∑i=0nαiϕiX,where ϕiX can be one of the basis functions in Table 1. One of the most important bases is the radial basis, and the corresponding method for these functions is radial basis function. The RBF method is an effective method for the interpolation of scattered data. Also, we use the radial basis functions in this study to get good results at reasonably low computational cost. The advantage of the proposed method consists mainly in the fact that it is easy to use and implement but shows good approximations of the solution at a given set of nodes. Furthermore, there is another catalog of used studies [6, 7] that study the solution of problems with radial basis functions in the interior of the domain. In the RBF set, ϕii=1n is shown as follows:(3)ϕi:ℝn⟶ℝ,ϕiα=ϕx−xi,where ⋅ is the Euclidean norm and xi is the center of RBF. One of the famous functions which produces RBF is the Gaussian function, and more functions are presented in Table 1. These functions according to having the parameter αi have exponential convergence [8].Table 1
Well-known functions generating RBF.
FunctionDefinitionGaussian (GA)ϕr=exp−σr2Multiquadric (MQ)ϕr=r2+σ2Inverse multiquadric (IMQ)ϕr=r2+σ2−1/2Inverse quadric (IQ)ϕr=r2+σ2−1RBF is a function with a quantity real positive, and its values depend on distance from center. The choice of the center point can have an arbitrary distribution; therefore, scattered data can be used for the necessary calculation. Interpolation by the RBF method has many benefits including high accuracy and applicable in scattered data and irregular domains. Also, there is appearance of existence and unique result, while there are little restrictions on dimension. Another important advantage of RBF is that it does not need triangulation from the data points, unlike other numerical methods such as multivariable polynomial interpolation or splines [9]. The concept of radial basis function was first introduced by Hardy [10, 11]. Franke [12], in his study, compared interpolation methods and made RBF more famous. Buhmann and Micchelli [13, 14] proved that the multiquadric interpolation advanced and application of RBF. Kansa [15, 16] used the multiquadrics method for estimating partial derivative and partial differential equations. Since then, a number of researchers have extensively used approaches based on RBF in many articles [17–22]. Now, to approximate the target function, ϕiX is substituted for one of the bases functions. As an example, consider Gaussian function, and then, we would have(4)ϕiX=e−σX−Xi2.Similarly,(5)uhX≈∑i=0nαiϕihX,is established when h is a known function. This study is organized as follows. Section 2 presents the RBF method to approximate Fredholm two-dimensional integral equations. Section 3 presents some examples to analyze the proposed approach and show its efficiency. Finally, we will express the conclusion.


## 2. Methodology

In then-dimensional Euclidean space, take m distinct point into consideration x1,x2,…,xm∈ℝn. In this space, the approximation function known as lx1,lx2,…,lxn are the value of the given points. Let us consider the following continuous functions w:ℝn⟶R, so that wxj=lj for j=1,2,…,m. RBF is based on a continuous function, so that ϕ:R+⟶R+ and ⋅ is a norm in ℝn. Therefore, we can have w in the following form:(6)wx=∑j=1mαjϕx−xj,where αj are the scalar parameters which have to be determined. So, w approximates l at xj point for j=1,2,…,m. Now, due to interpolation conditions, a linear system is defined as(7)Δα=L,where L=l1,l2,…,lnT, α=α1,α2,…,αnT, and Δ∈ℝn×n is called the interpolation matrix and is shown as(8)Δ=Aijn×n,where Aij=ϕxi−xj. The interpolation matrix is nonsingular because it is a positive definite matrix that definition of the positive definite matrix is given in [11]; therefore, there is unique and existence for coefficients αj.


### 2.1. The Convergence of RBF Approach

To investigate the convergence of the RBF approach, suppose that the data points are on equispaced grids inℝn. The distance is indicated by h, when h⟶0. In fact, there are infinite uniform grids from h. Considering w as an approximation of function l by using RBF, the uniform difference between w and l with the same rate is same power of h toward zero. In this article, the convergence of Gaussian function is examined. Although, the same approach is applicable for other basis functions given in Table 1.Definition 1 (Native space [23]).
The native space that is conditionally positive definite onΩ with respect to the p-unisolvent subset is defined by(9)NϕΩ≔RFϕΩ+p,where R:FϕΩ⟶CΩ,Rfx=f,G⋅,xϕ, and G⋅,x is given in [23]. The space is equipped with a semiinner product via(10)f,gNϕΩ=R−1f−Tpf,R−1g−Tpgϕ,where Tp:CΩ⟶p and Tpf=∑k=1QfξKpk.
The concept of convergence for RBF interpolation in the domainΩ∈ℝn is based on functions over native spaces NϕΩ. For same radial basis functions such as Gaussian, which are strictly positive definite basis functions, we can make the native space from positive conditional basis functions as the completion of the pre-Hilbert spaces. Take the following linear space into consideration:(11)FϕΩ=∑j=1Nαjϕ⋅−xj,α∈ℝN,xj∈Ω,j=1,…,N.
Definition of inner productFϕΩ becomes a pre-Hilbert space:(12)∑j=1Nαjϕ·−xj,∑k=1Nβkϕ⋅−xkϕΩ≔∑j=1N∑k=1Nαjβkϕxj−xk.
By applying RBF, the fill distance fromh is considered as follows:(13)hN≔supminx∈Ωx−xj2,j=1,2,…,N.
So, for sufficiently smallhN and data xj, the following relations have resulted for Gaussian:(14)l−wf,NL∞Ω≤e−αloghN/hNfNϕΩ.
The concept of convergence for RBF approximation is defined for functions in the domainΩ∈ℝN that are on native spaces NϕΩ. Convergence analysis can be proved by a similar process in [24] for this method.


### 2.2. Application of RBF with Fredholm Linear Functional Two-Dimensional Integral Equation

In this section, we interpolate a continuous functionu:ℝn⟶ℝ on a set X=x1,…,xn with choosing the radial basis function for ϕ:ℝn⟶ℝ that is radial in the sense. Now, we use the RBF interpolation for approximation uX. In the introduction, we remember that Fredholm two-dimensional integral equation of second kind was defined as follows:(15)lX=uX+AXuhX+λ∫ab∫cdKX,TuTdT,where T=t1,t2,X=x1,x2. The RBF interpolation is used to approximate the above integral equation. In order, the linear component of function ϕi is substituted in uX as the following form:(16)uX≈∑i=0nαiϕiX,and(17)uhX≈∑i=0nαiϕihX,where h:a,b×c,d⟶a,b×c,d is a known function. We would also have(18)∫ab∫cdKX,TuTdT≈∫ab∫cdKX,T∑i=0nαiϕiTdT=∑i=0nαiγiX,where(19)γiX=∫ab∫cdKX,TϕiTdT.If we replace equations (16)–(18) into equation (1), the following equation will result:(20)lX=∑i=0nαiϕiX+AX∑i=0nαiϕihX+λ∑i=0nαiγiX.Now, putX=xj for j=1,2,…,n. Then,(21)lj=∑i=0nαiϕij+Ajαiϕ¯ij+λ∑i=0nαiγij,where lj=lxj,Aj=Axj.(22)ϕ¯ij=ϕihxj,ϕ¯ij=ϕixj,γij=γixj=∫ab∫cdKxj,TϕiTdT.Matrix form for equation (21) is Δα=L, in which L=l1,l2,…,lnT and α=α1,α2,…,αnT.So, we have the following relations:(23)Δ1,1Δ1,2…Δ1,nΔ2,1Δ2,2…Δ2,n⋮⋮⋱⋮Δn,1Δn,2…Δn,nα1α2⋮αn=l1l2⋮ln,where(24)Δij=ϕij+Ajϕ¯ij+λγij.Finally, Fredholm two-dimensional linear integral equation is approximated by the linear equation system of then variable; also, this method for functional equation (in mode λ=0 at equation (1)) is as follows:(25)uX+AXuhX=lX.It should be mentioned that the choice of the appropriate shape parameter in the RBF is critical to the numerical accuracy. Various studies [25, 26] have been performed to deal with this problem-dependent parameter in the RBF community. In the next section, the effectiveness of the present method along with its performance will be tested. In the next section, to show the efficiency and capability of the proposed approach, we examine it by providing examples.


## 2.1. The Convergence of RBF Approach

To investigate the convergence of the RBF approach, suppose that the data points are on equispaced grids inℝn. The distance is indicated by h, when h⟶0. In fact, there are infinite uniform grids from h. Considering w as an approximation of function l by using RBF, the uniform difference between w and l with the same rate is same power of h toward zero. In this article, the convergence of Gaussian function is examined. Although, the same approach is applicable for other basis functions given in Table 1.Definition 1 (Native space [23]).
The native space that is conditionally positive definite onΩ with respect to the p-unisolvent subset is defined by(9)NϕΩ≔RFϕΩ+p,where R:FϕΩ⟶CΩ,Rfx=f,G⋅,xϕ, and G⋅,x is given in [23]. The space is equipped with a semiinner product via(10)f,gNϕΩ=R−1f−Tpf,R−1g−Tpgϕ,where Tp:CΩ⟶p and Tpf=∑k=1QfξKpk.
The concept of convergence for RBF interpolation in the domainΩ∈ℝn is based on functions over native spaces NϕΩ. For same radial basis functions such as Gaussian, which are strictly positive definite basis functions, we can make the native space from positive conditional basis functions as the completion of the pre-Hilbert spaces. Take the following linear space into consideration:(11)FϕΩ=∑j=1Nαjϕ⋅−xj,α∈ℝN,xj∈Ω,j=1,…,N.
Definition of inner productFϕΩ becomes a pre-Hilbert space:(12)∑j=1Nαjϕ·−xj,∑k=1Nβkϕ⋅−xkϕΩ≔∑j=1N∑k=1Nαjβkϕxj−xk.
By applying RBF, the fill distance fromh is considered as follows:(13)hN≔supminx∈Ωx−xj2,j=1,2,…,N.
So, for sufficiently smallhN and data xj, the following relations have resulted for Gaussian:(14)l−wf,NL∞Ω≤e−αloghN/hNfNϕΩ.
The concept of convergence for RBF approximation is defined for functions in the domainΩ∈ℝN that are on native spaces NϕΩ. Convergence analysis can be proved by a similar process in [24] for this method.


## 2.2. Application of RBF with Fredholm Linear Functional Two-Dimensional Integral Equation

In this section, we interpolate a continuous functionu:ℝn⟶ℝ on a set X=x1,…,xn with choosing the radial basis function for ϕ:ℝn⟶ℝ that is radial in the sense. Now, we use the RBF interpolation for approximation uX. In the introduction, we remember that Fredholm two-dimensional integral equation of second kind was defined as follows:(15)lX=uX+AXuhX+λ∫ab∫cdKX,TuTdT,where T=t1,t2,X=x1,x2. The RBF interpolation is used to approximate the above integral equation. In order, the linear component of function ϕi is substituted in uX as the following form:(16)uX≈∑i=0nαiϕiX,and(17)uhX≈∑i=0nαiϕihX,where h:a,b×c,d⟶a,b×c,d is a known function. We would also have(18)∫ab∫cdKX,TuTdT≈∫ab∫cdKX,T∑i=0nαiϕiTdT=∑i=0nαiγiX,where(19)γiX=∫ab∫cdKX,TϕiTdT.If we replace equations (16)–(18) into equation (1), the following equation will result:(20)lX=∑i=0nαiϕiX+AX∑i=0nαiϕihX+λ∑i=0nαiγiX.Now, putX=xj for j=1,2,…,n. Then,(21)lj=∑i=0nαiϕij+Ajαiϕ¯ij+λ∑i=0nαiγij,where lj=lxj,Aj=Axj.(22)ϕ¯ij=ϕihxj,ϕ¯ij=ϕixj,γij=γixj=∫ab∫cdKxj,TϕiTdT.Matrix form for equation (21) is Δα=L, in which L=l1,l2,…,lnT and α=α1,α2,…,αnT.So, we have the following relations:(23)Δ1,1Δ1,2…Δ1,nΔ2,1Δ2,2…Δ2,n⋮⋮⋱⋮Δn,1Δn,2…Δn,nα1α2⋮αn=l1l2⋮ln,where(24)Δij=ϕij+Ajϕ¯ij+λγij.Finally, Fredholm two-dimensional linear integral equation is approximated by the linear equation system of then variable; also, this method for functional equation (in mode λ=0 at equation (1)) is as follows:(25)uX+AXuhX=lX.It should be mentioned that the choice of the appropriate shape parameter in the RBF is critical to the numerical accuracy. Various studies [25, 26] have been performed to deal with this problem-dependent parameter in the RBF community. In the next section, the effectiveness of the present method along with its performance will be tested. In the next section, to show the efficiency and capability of the proposed approach, we examine it by providing examples.


## 3. Numerical Examples

The following examples have been presented to test efficiency of the suggested technique. Some numerical examples in this work are calculated by applying Mathematica software.Example 1.
Consider equation (1) which is defined in the Introduction section. We consider the points in both mesh and meshless case:(26)uX+AXuhX+λ∫ab∫cdKX,TuTdT=lX,where X=x1,x2,T=t1,t2.
Also,AX=1,hX=X, KX,T=ex1+x2−t1+t2, lX=3.21ex1+x2, and a=0, b=1.1, c=0, d=1.1, λ=1, where uX=ex1+x2 is the exact solution.
So, integral equation is given in the following form:(27)2uX+∫01.1∫01.1ex1+x2−t1+t2uTdT=3.21ex1+x2.
According to equation (24), we have the following relations:(28)Δij=ϕij+Ajϕ¯ij+λγij,where ϕij=ϕiXj and ϕ¯ij=ϕihXj.
Now, can write(29)γij=γiXj=∫01.1∫01.1ex1+x2−t1+t2ϕiTdT.
As mentioned in the previous section,α can be obtained from the Δα=L relation.
To do so,uX≈∑i=0100αiϕiX has been defined. In all forms in the following, the mesh points on the left and the meshless points are located in the right side (Figures 1–4). Also, Table 2 provides the applications of RBF in solving Fredholm linear two-dimensional integral equation for 100 points in both mesh and meshless cases.Figure 1
(a) The mesh points. (b) The meshless points. There are 100 points for interpolation.
(a)(b)Figure 2
The plot 3D errors for both mesh and meshless points.Figure 3
Contour plot errors in both mesh and meshless cases.Figure 4
Contour plot errors for the same points.Table 2
Accuracy of RBF to Fredholm linear two-dimensional integral equation for mesh and meshless points.
NEmesh pointsEmeshless points99.25711×10−27.33724×10−2251.84293×10−32.41075×10−3493.22909×10−51.41057×10−51008.14192×10−69.45367×10−6Example 2.
In the introduction, we had equation (1):(30)lX=uX+AXuhX+λ∫ab∫cdKX,TuTdT,where X=x1,x2,T=t1,t2.
Let(31)AX=e−x1+x2,hX=X2,KX,T=ex1+x2−t1+t2,lX=e−x1/2+x2/2+2.21ex1+x2,a=0,b=1.1,c=0,d=1.1,λ=1,where uX=ex1+x2 is the exact solution.
We consider the points in both mesh and meshless cases, so we will have(32)uX+e−x1+x2uX2+∫01.1∫01.1ex1+x2−t1+t2uTdT=e−x1/2+x2/2+2.21ex1+x2.
According to equation (24), the following relation is yielded:(33)Δij=ϕij+Aijϕ¯ij+λγij,where ϕij=ϕiXj, ϕ¯ij=ϕihXj.
So, we would have(34)γij=γiXj=∫01.1∫01.1ex1+x2−t1+t2ϕiTdT.
Based on what we had before,Δα=L, we will calculate the coefficient α.
To do so,uX=∑i=0100αiϕiX is defined (Figures 5–8 and Table 3).Figure 5
(a) The mesh points. (b) The meshless points. There are 100 points for interpolation.
(a)(b)Figure 6
The plot 3D errors for both mesh and meshless points.Figure 7
The Contour plot errors in both mesh and meshless cases.Figure 8
Contour plot errors for the same points.Table 3
Accuracy of RBF to Fredholm linear two-dimensional integral equation for mesh and meshless points.
NEmesh pointsEmeshless points95.85133×10−23.85156×10−2252.18032×10−33.82658×10−3496.55856×10−57.16108×10−51006.61838×10−69.45367×10−6Example 3.
Consider the following integral equation:(35)lX=uX+AXuhX+λ∫ab∫cdKX,TuTdT,where X=x1,x2,T=t1,t2.
Let(36)AX=1,hX=X,a=0,b=1.1,c=0,d=1.1,KX,T=cosx1+x2−t1+t2,lX=−0.125sin−x1−x2+0.25sin2.2−x1−x2−0.125sin4.4−x1−x2+2.605sinx1+x2,where uX=sinx1+x2 is the exact solution. We consider the points in both mesh and meshless cases, so we will have(37)2uX+∫01.1∫01.1cosx1+x2−t1+t2uTdT=−0.125sin−x1−x2+0.25sin2.2−x1−x2−0.125sin4.4−x1−x2+2.605sinx1+x2.
According to equation (24) we have the following relation:(38)Δij=ϕij+Aijϕ¯ij+λγij,where lj=lxj, Aj=Axj, ϕij=ϕixj, and ϕ¯ij=ϕihxj. So, we would have(39)γij=γixj=∫01.1∫01.1cosx1+x2−t1+t2ϕiTdT.
According to what we had before,Δα=L, we will calculate the coefficient α.
To this aim,uX=∑i=0100αiϕix is defined (Figures 9–12 and Table 4).Figure 9
(a) The mesh points. (b) The meshless points. There are 100 points for interpolation.
(a)(b)Figure 10
The plot 3D errors for both mesh and meshless points.Figure 11
Contour plot errors in both mesh and meshless cases.Figure 12
Contour plot errors for the same points.Table 4
Accuracy of RBF to Fredholm linear two-dimensional integral equation for mesh and meshless points.
NEmesh pointsEmeshless points99.44726×10−34.163663×10−2252.65143×10−43.17089×10−3498.02597×10−45.13247×10−31004.50859×10−41.78766×10−4


## 4. Conclusion

The RBF method was proposed for solving the Fredholm functional two-dimensional linear integral equations. These types of equations are very significant, and sometimes, detecting their analytic solutions is so difficult. Thus, using a useful numerical method for these types of equations is a topic of interest for researchers. At the end, the accuracy and applicability of the proposed method are shown through some numerical applications. The method is performed on three test problems, and simulation results show that the numerical method is appropriate for obtaining the approximate solutions of our models. Also, this method has higher accuracy and fewer calculations. The RBF method can also be used for higher dimensional and both linear and nonlinear integral equations.



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