# Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

Authors: H. Nouriani; R. Ezzati
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

In this study, at first, we propose a new approach based on the two-dimensional fuzzy Lagrange interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). Then, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm by two theorems. Finally, by some examples, we show the efficiency of the proposed method.


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## Body

## 1. Introduction

Recently many authors proposed various numerical methods for solving one-dimensional fuzzy integral equations [1–9]. Also, two-dimensional fuzzy integral equations have been noticed by a lot of researchers because of their broad applications in engineering sciences. Some of the most important papers in this area are trapezoidal quadrature rule and iterative method [10–12], triangular functions [13], quadrature iterative [14], Bernstein polynomials [15], collocation fuzzy wavelet like operator [16], homotopy analysis method (HAM) [17], open fuzzy cubature rule [18], kernel iterative method [19], modified homotopy pertubation [20], block-pulse functions [21], optimal fuzzy quadrature formula [22], and finally, iterative method and fuzzy bivariate block-pulse functions [23]. Also, some researchers have solved one-dimensional fuzzy Fredholm integral equations by using fuzzy interpolation via iterative method such as: iterative interpolation method [9], Lagrange interpolation based on the extension principle [5], and spline interpolation [7].As we know interpolation is one of the most substantial and the most applicable methods in numerical analysis. So, in this paper, we want to solve 2DLFFIEs by applying two-dimensional fuzzy Lagrange interpolation and iterative method. First of all an approximate solution of integral by applying Lagrange interpolation and iterative method is provided. Then, convergence analysis and numerical stability analysis of the proposed method in Theorems11 and 12 are proved.The paper is organized as follows: Some notations and theorems about the structure of fuzzy sets are reviewed in Section2. In Section 3, firstly, we introduce two-dimensional fuzzy Lagrange interpolation. Also, we propose two-dimensional fuzzy Lagrange interpolation and iterative method for solving 2DLFFIEs. In Section 4, we verify convergence analysis for proposed method. Also, in Section 5, we prove numerical stability analysis for the method. Two numerical examples are presented in Section 6.


## 2. Preliminaries

At first, we review some basic definitions and necessary results about fuzzy set theory.Definition 1 (see [24]).
A fuzzy number is a functionf~:R→[0,1] with the following properties: (1)
∃x0∈R such that f~x0=1.(2)
f~(ηx+(1-η)y)≥min⁡f~x,f~y,∀x,y∈R,∀η∈[0,1].(3)
∀x0∈R and ∀ϵ>0, ∃ neighborhood U(x0):f~(x)≤f~(x0)+ϵ,∀x∈U(x0).(4)
InR, the set supp(f~)¯ is compact.
The set of all fuzzy numbers is denoted byRF.Definition 2 (see [24]).
Forf~∈RF and 0<α≤1, define [f~]0≔x∈R:f~x>0¯ and (1)f~α≔x∈R:f~x≥α.Then it is well known that, for each α∈0,1,f~α is a bounded and closed interval of R. We define uniquely the sum f~⊕g~ and the product μ⊙f~ for f~, g~∈RF and μ∈R by (2)f~⊕g~α=f~α+g~α,μ⊙f~α=μf~α,∀α∈0,1,where [f~]α+[g~]α means the usual addition of two intervals (as subsets of R) and μ[f~]α means the usual product between a scalar and a subset of R. Notice 1⊙f~=f~ and it holds f~⊕g~=g~⊕f~, μ⊙f~=f~⊙μ. If 0≤α1≤α2≤1 then [f~]α2⊆[f~]α1. Actually [f~]α=[f~-(α),f~+(α)], where f~-(α)≤f~+(α), f~-(α), f~+(α)∈R, ∀α∈[0,1]. For μ>0 one has μf~±(α)=(μ⊙f~)±(α), respectively.Definition 3 (see [24]).
DefineD:RF×RF→R+ by (3)Df~,g~≔supα∈0,1max⁡f~-α-g~-α,f~+α-g~+α=supα∈0,1Hausdorff  distancef~α,g~α,where [g~]α=[g~-(α),g~+(α)]; f~, g~∈RF. Clearly D is a metric on RF. Also (RF,D) is a complete metric space, with the following properties [24]: (4)Df~⊕h~,g~⊕h~=Df~,g~,∀f~,g~,h~∈RF,Dk′⊙f~,k′⊙g~=k′Df~,g~,∀f~,g~∈RF,∀k′∈R,Df~⊕g~,h~⊕e~≤Df~,h~+Dg~,e~,∀f~,g~,h~,e~∈RF.Definition 4 (see [24]).
Supposef~,g~:R→RF be fuzzy number valued functions, then the distance between f and g is defined by (5)D∗f~,g~≔supx∈RDf~x,g~x.Lemma 5 (see [24]).
(1)
If we denote0~≔χ0, then ∀f~∈RF, f~⊕0~=0~⊕f~=f~. (2)
With respect to0~, none of f~∈RF, f~≠0~ has opposite in RF. (3)
Letα, β∈R:α.β≥0, and any f~∈RF, we have (α+β)⊙f~=α⊙f~⊕β⊙f~. Notice that, for general α, β∈R, the above property is false. (4)
For anyγ∈R and any f~, g~∈RF, we have γ⊙(f~⊕g~)=γ⊙f~⊕γ⊙g~. (5)
For anyγ, η∈R and any f~∈RF, we have γ⊙(η⊙f~)=(γ⊙η)⊙f~.
If we denotef~F≔D(f~,0~), ∀f~∈RF, then ·F has the properties of a usual norm on RF, i.e., (6)f~F=0ifff~=0~,μ⊙f~F=μ·f~F,f~⊕g~F≤f~F+g~F,f~F-g~F≤Df~,g~.Notice that (RF,⊕,⊙) is not a linear space over R, and consequently (RF,·F) is not a normed space. Here ∑∗ denotes the fuzzy summation.Definition 6 (see [24]).
A fuzzy valued functionf~:[a,b]→RF is said to be continuous at x0∈[a,b], if for each ϵ>0 there exists δ>0 such that D(f~(x),f~(x0))<ϵ, whenever x∈[a,b] and x-x0<δ. We say that f~ is fuzzy continuous on [a,b] if f~ is continuous at each x0∈[a,b] and denotes the space of all such functions by CF[a,b].Definition 7 (see [11]).
Suppose thatf~:[a,b]×[c,d]→Rf is a bounded mapping. The function ωa,b×c,df~,·:R+∪0→R+ defined by (7)ωa,b×c,df~,δ=supDf~x,y,f~s,t;x,s∈a,b;y,t∈c,d;x-s2+y-t2≤δ,is called modules of oscillation of f~ on [a,b]×[c,d]. Also, if f~∈CF([a,b]×[c,d]), then ω[a,b]×[c,d](f~,δ) is called uniform modules of continuity of f~.Theorem 8 (see [11]).
The following properties hold:(1)
D(f~(x,y),f~(s,t))≤ω[a,b]×[c,d](f~,(x-s)2+(y-t)2), ∀x,s∈[a,b], y,t∈[c,d];(2)
ω[a,b]×[c,d](f~,δ) is a nondecreasing mapping in δ;(3)
ω[a,b]×[c,d](f~,0)=0;(4)
ω[a,b]×[c,d](f~,δ1+δ2)≤ω[a,b]×[c,d](f~,δ1)+ω[a,b]×[c,d](f~,δ2), ∀δ1,δ2≥0;(5)
ω[a,b]×[c,d](f~,nδ)≤nω[a,b]×[c,d](f~,δ), ∀δ≥0, n∈N;(6)
ω[a,b]×[c,d](f~,μδ)≤(μ+1)ω[a,b]×[c,d](f~,δ), ∀μ,δ≥0.Theorem 9 (see [11]).
Iff~ and g~ are Henstock integrable mapping on [a,b]×[c,d] and if D(f~(s,t),g~(s,t)) is Lebesgue integrable, then (8)DFH∫cd∫abf~s,tdsdt,FH∫cd∫abg~s,tdsdt≤L∫cd∫abDf~s,t,g~s,tdsdt.


## 3. The Main Result

In this section, first, we introduce two-dimensional fuzzy Lagrange interpolation. Then, we propose two-dimensional fuzzy Lagrange interpolation and iterative method for solving (9).Consider 2DLFFIE as follows:(9)G~s,t=g~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙G~x,ydxdy,where μ>0, K(x,y,s,t) is an arbitrary positive function on [a,b]×[c,d]×[a,b]×[c,d] and g~:[a,b]×[c,d]→RF. We assume that K is continuous, and therefore it is uniformly continuous with respect to (s,t). So, there exists M>0 such that M=maxs,x∈a,b,y,t∈[c,d]K(x,y,s,t).Two-dimensional Lagrange polynomials,lijs,t, are defined as follows:(10)lijs,t=lis⊗ljt,0≤i,j≤n,where li(s) is the Lagrange polynomial and is defined as follows:(11)lis=∏r=0,r≠ins-srsi-sr,therefore, (12)lijsr,tr′=1,ifi=r,j=r′,0,otherwise.So, the two-dimensional interpolation in the Lagrange form is (see [25])(13)px,y=∑j=0n∑i=0nf~si,tj⊙lijx,y,where the coefficients f~(si,tj) are the fuzzy numbers.Here, we consider the two-dimensional fuzzy Lagrange interpolation in the given pointsa=s0<s1<⋯<sn=b and c=t0<t1<⋯<tn=d such that(14)Kx,y,s,t⊙G~x,y≈∑j=0n∑i=0nlijx,y⊙Ksi,tj,s,t⊙G~si,tj.Now, we propose a numerical method to solve (9). To do this, we suppose the following iterative procedure to approximate the solution of (9) in point (s,t)(15)u~0s,t=g~s,t,u~ks,t=g~s,t⊕μ⊙∑j=0n∑i=0nCijKsi,tj,s,t⊙u~k-1si,tj,where(16)Cij=∫cd∫ablijx,ydxdy.In Theorem 10, authors of [11] proved the existence and uniqueness solution of (9) by using the Banach fixed point theorem.Theorem 10 (see [11]).
Let the functionK(x,y,s,t) be continuous and positive for x,s∈[a,b], and y,t∈[c,d], and let g~:[a,b]×[c,d]→RF be continuous on [a,b]×[c,d]. If B=μM(d-c)(b-a)<1 then the fuzzy integral equation (9) has a unique solution G~∗∈X, where (17)X=g~:a,b×c,d→RF;g~is  continous,be the space of two-dimensional fuzzy continuous functions with the metric D∗ and it can be obtained by the following successive approximations method:(18)G~0s,t=g~s,t,G~ks,t=g~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙G~k-1x,ydxdy,∀k≥1.Moreover, the sequence of successive approximations (G~k)k≥1 converges to the solution G~∗. Furthermore, the following error bound holds:(19)D∗G~∗,G~k≤Bk+11-BM1,∀k≥1,where M1=sups∈[a,b],t∈[c,d]G~(s,t).


## 4. Convergence Analysis

In this section, we obtain an error estimate between the exact solution and the approximate solution of 2DLFFIE (9).Theorem 11.
Under the hypotheses of Theorem10 and μ>0, the iterative procedure (15) converges to the unique solution of (9), G~∗, and its error estimate is as follows: (20)D∗G~∗,u~k≤Bk+11-BM1+11-B1+2C~d-cb-aml.Proof.
Clearly, we have(21)DG~∗s,t,u~ks,t≤DG~∗s,t,G~ks,t+DG~ks,t,u~ks,t.From (18) and (15), we conclude that (22)DG~ks,t,u~ks,t=Dg~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙G~k-1x,ydxdy,g~s,t⊕μ⊙∑j=0n∑i=0nCijKsi,tj,s,t⊙u~k-1si,tj=μDFR∫cdFR∫abKx,y,s,t⊙G~k-1x,ydxdy,∑j=0n∑i=0nCijKsi,tj,s,t⊙u~k-1si,tj=μD∑j=1n∑i=1nFR∫tj-1tjFR∫si-1siKx,y,s,t⊙G~k-1x,ydxdy,∑j=0n∑i=0nCijKsi,tj,s,t⊙u~k-1si,tj=μD∑j=1n∑i=1nFR∫tj-1tjFR∫si-1siKx,y,s,t⊙G~k-1x,ydxdy,∑j=1n∑i=1ntj-tj-1si-si-1Cijtj-tj-1si-si-1Ksi,tj,s,t⊙u~k-1si,tj⊕C00Ks0,t0,s,t⊙u~k-1s0,t0≤μ∑j=1n∑i=1nDFR∫tj-1tjFR∫si-1siKx,y,s,t⊙G~k-1x,ydxdy,FR∫tj-1tjFR∫si-1siCijtj-tj-1si-si-1Ksi,tj,s,t⊙u~k-1si,tjdxdy⊕μDC00Ks0,t0,s,t⊙u~k-1s0,t0,0~≤μ∑j=1n∑i=1n∫tj-1tj∫si-1siDKx,y,s,t⊙G~k-1x,y,Kx,y,s,t⊙u~k-1si,tj+DKx,y,s,t⊙u~k-1si,tj,Cijtj-tj-1si-si-1Ksi,tj,s,t⊙u~k-1si,tjdxdy+μDC00Ks0,t0,s,t⊙u~k-1s0,t0,0~≤μ∑j=1n∑i=1n∫tj-1tj∫si-1siKx,y,s,tDG~k-1x,y,u~k-1si,tj+Kx,y,s,tDu~k-1si,tj,0~+Cijtj-tj-1si-si-1Ksi,tj,s,tDu~k-1si,tj,0~dxdy+μDC00Ks0,t0,s,t⊙u~k-1s0,t0,0~.By supposing mk-1=sup(s,t)∈[a,b]×[c,d]u~k-1(s,t), we get (23)DG~ks,t,u~ks,t≤μ∑j=1n∑i=1n∫tj-1tj∫si-1siMDG~k-1x,y,u~k-1si,tj+Mmk-1+Cijtj-tj-1si-si-1Ksi,tj,s,tmk-1dxdy+μC00Ks0,t0,s,tmk-1.Also, we have (24)DG~ks,t,u~ks,t≤μ∑j=1n∑i=1n∫tj-1tj∫si-1siMD∗G~k-1,u~k-1dxdy+μ∑j=1n∑i=1n∫tj-1tj∫si-1siMmk-1dxdy+μ∑j=1n∑i=1n∫tj-1tj∫si-1siCijtj-tj-1si-si-1Ksi,tj,s,tmk-1dxdy+μC00Ks0,t0,s,tmk-1.Therefore, (25)DG~ks,t,u~ks,t≤μMd-cb-aD∗G~k-1,u~k-1+μMd-cb-amk-1+μMd-cb-amk-1C~d-cb-a+μMC~mk-1≤BD∗G~k-1,u~k-1+Bmk-1+2Bmk-1C~d-cb-a=BD∗G~k-1,u~k-1+Bmk-11+2C~d-cb-a,where C~=maxCij, i,j=0,⋯,n. Hence, we conclude that (26)D∗G~k-1,u~k-1≤BD∗G~k-2,u~k-2+Bmk-21+2C~d-cb-a⋮D∗G~1,u~1≤BD∗G~0,u~0+Bm01+2C~d-cb-a.So, (27)D∗G~k,u~k≤1+2C~d-cb-aBmk-1+B2mk-2+⋯+Bkm0.If ml=maxm0,…,mk-1, then we obtain (28)D∗G~k,u~k<11-B1+2C~d-cb-aml,therefore (29)D∗G~∗,u~≤Bk+11-BM1+11-B1+2C~d-cb-aml.


## 5. Numerical Stability Analysis

To show the numerical stability analysis of the proposed method in previous section, we consider another starting approximationf~(s,t)=Y~0(s,t) such that ∃ϵ>0 for which D(G~0(s,t),Y~0(s,t))<ϵ,∀s,t∈[a,b]×[c,d]. The obtained sequence of successive approximations is (30)Y~ks,t=f~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙Y~k-1x,ydxdy,and using the same iterative method, the terms of produced sequence are (31)v~0s,t=Y~0s,t=f~s,t,v~ks,t=f~s,t⊕μ⊙∑j=0n∑i=0nCijKsi,tj,s,t⊙v~k-1si,tj.Theorem 12.
The proposed method (15), under the assumptions of Theorem 11, is numerically stable with respect to the choice of the first iteration.Proof.
At first, we obtain that(32)Du~ks,t,v~ks,t≤Du~ks,t,G~ks,t+DG~ks,t,Y~ks,t+DY~ks,t,v~ks,t≤11-B1+2C~d-cb-aml+DG~ks,t,Y~ks,t+11-B1+2C~d-cb-aml.However, (33)DG~ks,t,Y~ks,t=Dg~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙G~k-1x,ydxdy,f~s,t⊕μ⊙FR∫cdFR∫abKx,y,s,t⊙Y~k-1x,ydxdy≤Dg~s,t,f~s,t+μDFR∫cdFR∫abKx,y,s,t⊙G~k-1x,ydxdy,FR∫cdFR∫abKx,y,s,t⊙Y~k-1x,ydxdy,≤ϵ+μ∫cd∫abKx,y,s,tDG~k-1x,y,Y~k-1x,ydxdy.We conclude that(34)D∗G~k,Y~k≤ϵ+μ∫cd∫abMD∗G~k-1,Y~k-1dxdy=ϵ+BD∗G~k-1,Y~k-1,and thus (35)D∗G~k,Y~k≤ϵ+BD∗G~k-1,Y~k-1D∗G~k-1,Y~k-1≤ϵ+BD∗G~k-2,Y~k-2⋮⋮D∗G~1,Y~1≤ϵ+BD∗G~0,Y~0.So, (36)D∗G~k,Y~k≤ϵ+Bϵ+BD∗G~k-2,Y~k-2≤ϵ+Bϵ+B2ϵ+BD∗G~k-3,Y~k-3⋮≤ϵ+Bϵ+B2ϵ+B3ϵ+⋯+BkD∗G~0,Y~0≤ϵ1+B+B2+B3+⋯+Bk≤ϵ1-B.Therefore, (37)D∗u~k,v~k≤11-B2ml1+2C~d-cb-a+ϵ.


## 6. Numerical Examples

In this section, we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations for solving two examples. By using the proposed method forn=2, k=5,8,16, and r∈0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0 in (s,t)=(0.5,0.5), we present the absolute errors in Tables 1–4.Table 1
The absolute errors on the level sets withn=2, k=5 for Example 13 by using the proposed method (s,t)=(0.5,0.5).
r-level e _ r = G _ ~ ∗ ( s , t , r ) - u _ ~ k ( s , t , r ) e ¯ r = G ~ ¯ ∗ ( s , t , r ) - u ~ ¯ k ( s , t , r ) 0.0 0 . 1.23485e-5 0.1 6.66819 e - 7 1.141e-5 0.2 1.3501   e - 6 1.05703e-5 0.3 2.04985 e - 6 9.82941e-6 0.4 2.76606 e - 6 9.18729e-6 0.5 3.49874 e - 6 8.64395e-6 0.6 4.24788 e - 6 8.19941e-6 0.7 5.01349 e - 6 7.85365e-6 0.8 5.79556 e - 6 7.60668e-6 0.9 6.5941   e - 6 7.45849e-6 1.0 7.4091   e - 6 7.4091e-6Table 2
The absolute errors on the level sets withn=2, k=16 for Example 13 by using the proposed method in (s,t)=(0.5,0.5).
r-level e _ r = G _ ~ ∗ ( s , t , r ) - u _ ~ k ( s , t , r ) e ¯ r = G ~ ¯ ∗ ( s , t , r ) - u ~ ¯ k ( s , t , r ) 0.0 0 . 1.77636e-15 0.1 1.11022 e - 16 1.77636e-15 0.2 0 . 1.77636e-15 0.3 2.22045 e - 16 8.88178e-16 0.4 4.44089 e - 16 8.88178e-16 0.5 4.44089 e - 16 8.88178e-16 0.6 4.44089 e - 16 0. 0.7 8.88178 e - 16 0. 0.8 0 . 0. 0.9 0 . 0. 1.0 0 . 0.Table 3
The absolute errors on the level sets withn=2, k=5 for Example 14 by using the proposed method in (s,t)=(0.5,0.5).
r-level e _ r = G _ ~ ∗ ( s , t , r ) - u _ ~ k ( s , t , r ) e ¯ r = G ~ ¯ ∗ ( s , t , r ) - u ~ ¯ k ( s , t , r ) 0.0 0 . 9.60565e-13 0.1 4.80276 e - 14 9.12548e-13 0.2 9.60551 e - 14 8.64531e-13 0.3 1.44079 e - 13 8.16458e-13 0.4 1.9211   e - 13 7.68441e-13 0.5 2.40141 e - 13 7.20479e-13 0.6 2.88158 e - 13 6.72407e-13 0.7 3.36203 e - 13 6.24334e-13 0.8 3.8422   e - 13 5.76317e-13 0.9 4.32265 e - 13 5.283e-13 1.0 4.80282 e - 13 4.80282e-13Table 4
The absolute errors on the level sets withn=2, k=8 for Example 14 by using the proposed method in (s,t)=(0.5,0.5).
r-level e _ r = G _ ~ ∗ ( s , t , r ) - u _ ~ k ( s , t , r ) e ¯ r = G ~ ¯ ∗ ( s , t , r ) - u ~ ¯ k ( s , t , r ) 0.0 0 . 0. 0.1 0 . 0. 0.2 0 . 5.55112e-17 0.3 0 . 0. 0.4 0 . 0. 0.5 0 . 5.55112e-17 0.6 0 . 0. 0.7 0 . 0. 0.8 0 . 0. 0.9 2.77556 e - 17 0. 1.0 0 . 0.Example 13 (see [11]).
Consider the linear integral equation(38)G~s,t=g~s,t⊕FR∫01FR∫01Kx,y,s,t⊙G~x,ydxdy,with (39)Kx,y,s,t=xyst,g_~s,t,r=r13r+831+s+t-712st,g~¯s,t,r=2r2-4r+51+s+t-712st.The exact solution for 2DLFFIE (38) is (40)G_~∗s,t,r=r13r+83s+t+1,G~¯∗s,t,r=2r2-4r+5s+t+1.Example 14 (see [13]).
Consider the following fuzzy Fredholm integral equation (38) with (41)g_~s,t,r=rst+1676s2+t2-2,g~¯s,t,r=2-rst+1676s2+t2-2,and kernel (42)Kx,y,s,t=1169x2+y2-2s2+t2-2,0≤s,t,x,y≤1.The exact solution is (43)G_~∗s,t,r=rst,G~¯∗s,t,r=2-rst.


## 7. Conclusion

The 2DLFFIE is solved by utilizing two-dimensional fuzzy Lagrange interpolation and iterative method. As it was expected the method used to approximate the integral in this equation is a suitable one since convergence analysis and stability analysis have been proved and also absolute error in examples is nearly zero. As a result, considering the fact that the proposed method does not lead to solve fuzzy linear system, it can be utilized as an efficient method to solve this type of equations. As future researches, we can use finite and divided differences methods, fuzzy spline interpolation, and fuzzy quasi-interpolation for solving two-dimensional linear or nonlinear fuzzy Fredholm integral equations.



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