This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.
Citation: Sima Karamseraji, Shokrollah Ziari, Reza Ezzati. Approximate solution of nonlinear fuzzy Fredholm integral equations using bivariate Bernstein polynomials with error estimation[J]. AIMS Mathematics, 2022, 7(4): 7234-7256. doi: 10.3934/math.2022404
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This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.
We consider the following family of nonlinear oscillators
yzz+k(y)y3z+h(y)y2z+f(y)yz+g(y)=0, | (1.1) |
where k, h, f≠0 and g≠0 are arbitrary sufficiently smooth functions. Particular members of (1.1) are used for the description of various processes in physics, mechanics and so on and they also appear as invariant reductions of nonlinear partial differential equations [1,2,3].
Integrability of (1.1) was studied in a number of works [4,5,6,7,8,9,10,11,12,13,14,15,16]. In particular, in [15] linearization of (1.1) via the following generalized nonlocal transformations
w=F(y),dζ=(G1(y)yz+G2(y))dz. | (1.2) |
was considered. However, equivalence problems with respect to transformations (1.2) for (1.1) and its integrable nonlinear subcases have not been studied previously. Therefore, in this work we deal with the equivalence problem for (1.1) and its integrable subcase from the Painlevé-Gambier classification. Namely, we construct an equivalence criterion for (1.1) and a non-canonical form of Ince Ⅶ equation [17,18]. As a result, we obtain two new integrable subfamilies of (1.1). What is more, we demonstrate that for any equation from (1.1) that satisfy one of these equivalence criteria one can construct an autonomous first integral in the parametric form. Notice that we use Ince Ⅶ equation because it is one of the simplest integrable members of (1.1) with known general solution and known classification of invariant curves.
Moreover, we show that transformations (1.2) preserve autonomous invariant curves for equations from (1.1). Since the considered non-canonical form of Ince Ⅶ equation admits two irreducible polynomial invariant curves, we obtain that any equation from (1.1), which is equivalent to it, also admits two invariant curves. These invariant curves can be used for constructing an integrating factor for equations from (1.1) that are equivalent to Ince Ⅶ equation. If this integrating factor is Darboux one, then the corresponding equation is Liouvillian integrable [19]. This demonstrates the connection between nonlocal equivalence approach and Darboux integrability theory and its generalizations, which has been recently discussed for a less general class of nonlocal transformations in [20,21,22].
The rest of this work is organized as follows. In the next Section we present an equivalence criterion for (1.1) and a non-canonical form of the Ince Ⅶ equation. In addition, we show how to construct an autonomous first integral for an equation from (1.1) satisfying this equivalence criterion. We also demonstrate that transformations (1.2) preserve autonomous invariant curves for (1.1). In Section 3 we provide two examples of integrable equations from (1.1) and construct their parametric first integrals, invariant curves and integrating factors. In the last Section we briefly discuss and summarize our results.
We begin with the equivalence criterion between (1.1) and a non-canonical form of the Ince Ⅶ equation, that is [17,18]
wζζ+3wζ+ϵw3+2w=0. | (2.1) |
Here ϵ≠0 is an arbitrary parameter, which can be set, without loss of generality, to be equal to ±1.
The general solution of (1.1) is
w=e−(ζ−ζ0)cn{√ϵ(e−(ζ−ζ0)−C1),1√2}. | (2.2) |
Here ζ0 and C1 are arbitrary constants and cn is the Jacobian elliptic cosine. Expression (2.2) will be used below for constructing autonomous parametric first integrals for members of (1.1).
The equivalence criterion between (1.1) and (2.1) can be formulated as follows:
Theorem 2.1. Equation (1.1) is equivalent to (2.1) if and only if either
(I)25515lgp2qy+2352980l10+(3430q−6667920p3)l5−14580qp3−10q2−76545lgqppy=0, | (2.3) |
or
(II)343l5−972p3=0, | (2.4) |
holds. Here
l=9(fgy−gfy+fgh−3kg2)−2f3,p=gly−3lgy+l(f2−3gh),q=25515gylp2−5103lgppy+686l5−8505p2(f2−3gh)l+6561p3. | (2.5) |
The expression for G2 in each case is either
(I)G2=126l2qp2470596l10−(1333584p3+1372q)l5+q2, | (2.6) |
or
(II)G22=−49l3G2+9p2189pl. | (2.7) |
In all cases the functions F and G1 are given by
F2=l81ϵG32,G1=G2(f−3G2)3g. | (2.8) |
Proof. We begin with the necessary conditions. Substituting (1.2) into (2.1) we get
yzz+k(y)y3z+h(y)y2z+f(y)yz+g(y)=0, | (2.9) |
where
k=FG31(ϵF2+2)+3G21Fy+G1Fyy−FyG1,yG2Fy,h=G2Fyy+(6G1G2−G2,y)Fy+3FG2G21(ϵF2+2)G2Fy,f=3G2(Fy+FG1(ϵF2+2))Fy,g=FG22(ϵF2+2)Fy. | (2.10) |
As a consequence, we obtain that (1.1) can be transformed into (2.1) if it is of the form (2.9) (or (1.1)).
Conversely, if the functions F, G1 and G2 satisfy (2.10) for some values of k, h, f and g, then (1.1) can be mapped into (2.1) via (1.2). Thus, we see that the compatibility conditions for (2.10) as an overdertmined system of equations for F, G1 and G2 result in the necessary and sufficient conditions for (1.1) to be equivalent to (2.1) via (1.2).
To obtain the compatibility conditions, we simplify system (2.10) as follows. Using the last two equations from (2.10) we find the expression for G1 given in (2.8). Then, with the help of this relation, from (2.10) we find that
81ϵF2G32−l=0, | (2.11) |
and
567lG32+(243lgh−81lf2−81gly+243lgy)G2−7l2=0,243lgG2,y+324lG32−81glyG2+2l2=0, | (2.12) |
Here l is given by (2.5).
As a result, we need to find compatibility conditions only for (2.12). In order to find the generic case of this compatibility conditions, we differentiate the first equation twice and find the expression for G22 and condition (2.3). Differentiating the first equation from (2.12) for the third time, we obtain (2.6). Further differentiation does not lead to any new compatibility conditions. Particular case (2.4) can be treated in the similar way.
Finally, we remark that the cases of l=0, p=0 and q=0 result in the degeneration of transformations (1.2). This completes the proof.
As an immediate corollary of Theorem 2.1 we get
Corollary 2.1. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then an autonomous first integral of this equation can be presented in the parametric form as follows:
y=F−1(w),yz=G2wζFy−G1wζ. | (2.13) |
Here w is the general solution of (2.1) given by (2.2). Notice also that, formally, (2.13) contains two arbitrary constants, namely ζ0 and C1. However, without loss of generality, one of them can be set equal to zero.
Now we demonstrate that transformations (1.2) preserve autonomous invariant curves for equations from (1.1).
First, we need to introduce the definition of an invariant curve for (1.1). We recall that Eq (1.1) can be transformed into an equivalent dynamical system
yz=P,uz=Q,P=u,Q=−ku3−hu2−fu−g. | (2.14) |
A smooth function H(y,u) is called an invariant curve of (2.14) (or, equivalently, of (1.1)), if it is a nontrivial solution of [19]
PHy+QHu=λH, | (2.15) |
for some value of the function λ, which is called the cofactor of H.
Second, we need to introduce the equation that is equivalent to (1.1) via (1.2). Substituting (1.2) into (1.1) we get
wζζ+˜kw3ζ+˜hw2ζ+˜fwζ+˜g=0, | (2.16) |
where
˜k=kG32−gG31+(G1,y−hG1)G22+(fG1−G2,y)G1G2F2yG22,˜h=(hFy−Fyy)G22−(2fG1−G2,y)G2Fy+3gG21FyF2yG22,˜f=fG2−3gG1G22,˜g=gFyG22. | (2.17) |
An invariant curve for (2.16) can be defined in the same way as that for (1.1). Notice that, further, we will denote wζ as v.
Theorem 2.2. Suppose that either (1.1) possess an invariant curve H(y,u) with the cofactor λ(y,u) or (2.16) possess an invariant curve ˜H(w,v) with the cofactor ˜λ(w,v). Then, the other equation also has an invariant curve and the corresponding invariant curves and cofactors are connected via
H(y,u)=˜H(F,FyuG1u+G2),λ(y,u)=(G1u+G2)˜λ(F,FyuG1u+G2). | (2.18) |
Proof. Suppose that ˜H(w,v) is an invariant curve for (2.16) with the cofactor ˜λ(w,v). Then it satisfies
v˜Hw+(−˜kv3−˜hv2−˜fv−˜g)˜Hv=˜λ˜H. | (2.19) |
Substituting (1.2) into (2.19) we get
uHy+(−ku3−hu2−fu−g)H=(G1u+G2)˜λ(F,FyuG1u+G2)H. | (2.20) |
This completes the proof.
As an immediate consequence of Theorem 2.2 we have that transformations (1.2) preserve autonomous first integrals admitted by members of (1.1), since they are invariant curves with zero cofactors.
Another corollary of Theorem 2.2 is that any equation from (1.1) that is connected to (2.1) admits two invariant curves that correspond to irreducible polynomial invariant curves of (2.1). This invariant curves of (2.1) and the corresponding cofactors are the following (see, [23] formulas (3.18) and (3.19) taking into account scaling transformations)
˜H=±i√−2ϵ(v+w)+w2,˜λ=±√−2ϵw−2. | (2.21) |
Therefore, we have that the following statement holds:
Corollary 2.2. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then is admits the following invariant curves with the corresponding cofactors
H=±i√−2ϵ(FyuG1u+G2+F)+F2,λ=(G1u+G2)(±√−2ϵF−2). | (2.22) |
Let us remark that connections between (2.1) and non-autonomous variants of (1.1) can be considered via a non-autonomous generalization of transformations (1.2). However, one of two nonlocally related equations should be autonomous since otherwise nonlocal transformations do not map a differential equation into a differential equation [5].
In this Section we have obtained the equivalence criterion between (1.1) and (2.1), that defines two new completely integrable subfamilies of (1.1). We have also demonstrated that members of these subfamilies posses an autonomous parametric first integral and two autonomous invariant curves.
In this Section we provide two examples of integrable equations from (1.1) satisfying integrability conditions from Theorem 2.1.
Example 1. One can show that the coefficients of the following cubic oscillator
yzz−12ϵμy(ϵμ2y4+2)2y3z−6μyyz+2μ2y3(ϵμ2y4+2)=0, | (3.1) |
satisfy condition (2.3) from Theorem 2.1. Consequently, Eq (3.1) is completely integrable and its general solution can be obtained from (2.2) by inverting transformations (1.2). However, it is more convenient to use Corollary 2.1 and present the autonomous first integral of (3.1) in the parametric form as follows:
y=±√wμ,yz=w(ϵw2+2)wζ2wζ+w(ϵw2+2), | (3.2) |
where w is given by (2.2), ζ is considered as a parameter and ζ0, without loss of generality, can be set equal to zero. As a result, we see that (3.1) is integrable since it has an autonomous first integral.
Moreover, using Corollary 2.2 one can find invariant curves admitted by (3.1)
H1,2=y4[(√2±√−ϵμy2)2(√2∓√−ϵμy2)+2(ϵμy2∓√−2ϵ)u]2μ2y2(ϵμ2y4+2)−4u,λ1,2=±2(μy2(ϵμ2y4+2)−2u)(√−2ϵμy2∓2)y(ϵμ2y4+2) | (3.3) |
With the help of the standard technique of the Darboux integrability theory [19], it is easy to find the corresponding Darboux integrating factor of (3.1)
M=(ϵμ2y4+2)94(2ϵu2+(ϵμ2y4+2)2)34(μy2(ϵμ2y4+2)−2u)32. | (3.4) |
Consequently, equation is (3.1) Liouvillian integrable.
Example 2. Consider the Liénard (1, 9) equation
yzz+(biyi)yz+ajyj=0,i=0,…4,j=0,…,9. | (3.5) |
Here summation over repeated indices is assumed. One can show that this equation is equivalent to (2.1) if it is of the form
yzz−9(y+μ)(y+3μ)3yz+2y(2y+3μ)(y+3μ)7=0, | (3.6) |
where μ is an arbitrary constant.
With the help of Corollary 2.1 one can present the first integral of (3.6) in the parametric form as follows:
y=3√−2ϵμw2−√−2ϵw,yz=7776√2ϵμ5wwζ(√−2ϵw−2)5(2√−ϵwζ+(√2ϵw+2√−ϵ)w), | (3.7) |
where w is given by (2.2). Thus, one can see that (3.5) is completely integrable due to the existence of this parametric autonomous first integral.
Using Corollary 2.2 we find two invariant curves of (3.6):
H1=y2[(2y+3μ)(y+3μ)4−2u)](y+3μ)2[(y+3μ)4y−u],λ1=6μ(u−y(y+3μ)4)y(y+3μ), | (3.8) |
and
H2=y2(y+3μ)2y(y+3μ)4−u,λ2=2(2y+3μ)(u−2y(y+3μ)4)y(y+3μ). | (3.9) |
The corresponding Darboux integrating factor is
M=[y(y+3μ)4−u]−32[(2y+3μ)(y+3μ)4−2u]−34. | (3.10) |
As a consequence, we see that Eq (3.6) is Liouvillian integrable.
Therefore, we see that equations considered in Examples 1 and 2 are completely integrable from two points of view. First, they possess autonomous parametric first integrals. Second, they have Darboux integrating factors.
In this work we have considered the equivalence problem between family of Eqs (1.1) and its integrable member (2.1), with equivalence transformations given by generalized nonlocal transformations (1.2). We construct the corresponding equivalence criterion in the explicit form, which leads to two new integrable subfamilies of (1.1). We have demonstrated that one can explicitly construct a parametric autonomous first integral for each equation that is equivalent to (2.1) via (1.2). We have also shown that transformations (1.2) preserve autonomous invariant curves for (1.1). As a consequence, we have obtained that equations from the obtained integrable subfamilies posses two autonomous invariant curves, which corresponds to the irreducible polynomial invariant curves of (2.1). This fact demonstrate a connection between nonlocal equivalence approach and Darboux and Liouvillian integrability approach. We have illustrate our results by two examples of integrable equations from (1.1).
The author was partially supported by Russian Science Foundation grant 19-71-10003.
The author declares no conflict of interest in this paper.
[1] |
S. Abbasbandy, T. Allahviranloo, The Adomian decomposition method applied to the fuzzy system of Fredholm integral equations of the second kind, Int. J. Uncertain. Fuzz., 14 (2006), 101–110. https://doi.org/10.1142/S0218488506003868 doi: 10.1142/S0218488506003868
![]() |
[2] |
S. Abbasbandy, E. Babolian, M. Alavi, Numerical method for solving linear Fredholm fuzzy integral equations of the second kind, Chaos Soliton. Fract., 31 (2007), 138–146. https://doi.org/10.1016/j.chaos.2005.09.036 doi: 10.1016/j.chaos.2005.09.036
![]() |
[3] | K. Akhavan Zakeri, S. Ziari, M. A. Fariborzi Araghi, I. Perfilieva, Efficient numerical solution to a bivariate nonlinear fuzzy fredholm integral equation, IEEE T. Fuzzy Syst., 2019. https://doi.org/10.1109/TFUZZ.2019.2957100 |
[4] | G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory. Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-11220-1 |
[5] |
H. Attari, Y. Yazdani, A computational method for for fuzzy Volterra-Fredholm integral equations, Fuzzy Inf. Eng., 2 (2011), 147–156. https://doi.org/10.1007/s12543-011-0073-x doi: 10.1007/s12543-011-0073-x
![]() |
[6] |
E. Babolian, H. Sadeghi Goghary, S. Abbasbandy, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput., 161 (2005), 733–744. https://doi.org/10.1016/j.amc.2003.12.071 doi: 10.1016/j.amc.2003.12.071
![]() |
[7] | M. Baghmisheh, R. Ezzati, Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series, Adv. Differ. Equ-Ny, 51 (2015). |
[8] |
M. Baghmisheh, R. Ezzati, Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions, J. Intell. Fuzzy Syst., 30 (2016), 639–649. https://doi.org/10.3233/IFS-151783 doi: 10.3233/IFS-151783
![]() |
[9] | K. Balachandran, P. Prakash, Existence of solutions of nonlinear fuzzy Volterra-Fredholm integral equations, Indian J. Pure Ap. Mat., 33 (2002), 329–343. |
[10] |
K. Balachandran, K. Kanagarajan, Existence of solutions of general nonlinear fuzzy Volterra-Fredholm integral equations, J. Appl. Math. Stochastic Anal., 3 (2005), 333–343. https://doi.org/10.1155/JAMSA.2005.333 doi: 10.1155/JAMSA.2005.333
![]() |
[11] |
B. Bede, S. G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Set. Syst., 145 (2004), 359–380. https://doi.org/10.1016/S0165-0114(03)00182-9 doi: 10.1016/S0165-0114(03)00182-9
![]() |
[12] |
A. M. Bica, Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations, Inf. Sci., 178 (2008), 1279–1292. https://doi.org/10.1016/j.ins.2007.10.021 doi: 10.1016/j.ins.2007.10.021
![]() |
[13] |
A. M. Bica, C. Popescu, Approximating the solution of nonlinear Hammerstein fuzzy integral equations, Fuzzy Set. Syst., 245 (2014), 1–17. https://doi.org/10.1016/j.fss.2013.08.005 doi: 10.1016/j.fss.2013.08.005
![]() |
[14] |
A. M. Bica, Algebraic structures for fuzzy numbers from categorial point of view, Soft Comput., 11 (2007), 1099–1105. https://doi.org/10.1007/s00500-007-0167-x doi: 10.1007/s00500-007-0167-x
![]() |
[15] | A. M. Bica, S. Ziari, Iterative numerical method for solving fuzzy Volterra linear integral equations in two dimensions, Soft Comput. 21 (2017), 1097–1108. https://doi.org/10.1007/s00500-016-2085-2 |
[16] | A. M. Bica, C. Popescu, Fuzzy trapezoidal cubature rule and application to two-dimensional fuzzy Fredholm integral equations, Soft Comput., 21 (2017), 1229–1243. |
[17] |
A. M. Bica, S. Ziari, Open fuzzy cubature rule with application to nonlinear fuzzy Volterra integral equations in two dimensions, Fuzzy Set. Syst., 358 (2019), 108–131. https://doi.org/10.1016/j.fss.2018.04.010 doi: 10.1016/j.fss.2018.04.010
![]() |
[18] |
Y. Chalco-Cano, H. Roman-Flores, On new solutions of fuzzy differential equations, Chaos Soliton. Fract., 38 (2008), 112–119. https://doi.org/10.1016/j.chaos.2006.10.043 doi: 10.1016/j.chaos.2006.10.043
![]() |
[19] |
W. Congxin, W. Cong, The supremum and infimum of the set of fuzzy numbers and its applications, J. Math. Anal. Appl., 210 (1997), 499–511. https://doi.org/10.1006/jmaa.1997.5406 doi: 10.1006/jmaa.1997.5406
![]() |
[20] | D. Dubois, H. Prade, Fuzzy numbers: An overview. In: Analysis of Fuzzy Information, CRC Press, BocaRaton, 1 (1987), 3–39. |
[21] |
E. H. Dohaa, A. H. Bhrawyb, M. A. Saker, Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations, Appl. Math. Lett., 24 (2011), 559–565. https://doi.org/10.1016/j.aml.2010.11.013 doi: 10.1016/j.aml.2010.11.013
![]() |
[22] | R. Ezzati, S. Ziari, Numerical solution and error estimation of fuzzy Fredholm integral equation using fuzzy Bernstein polynomials, Aust. J. Basic Appl. Sci., 5 (2011), 2072–2082. |
[23] |
R. Ezzati, S. Ziari, Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method, Appl. Math. Comput., 225 (2013), 33–42. https://doi.org/10.1016/j.amc.2013.09.020 doi: 10.1016/j.amc.2013.09.020
![]() |
[24] |
R. Ezzati, S. Ziari, Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using fuzzy bivariate Bernstein polynomials, Int. J. Fuzzy Syst., 15 (2013), 84–89. https://doi.org/10.1017/S1466046612000579 doi: 10.1017/S1466046612000579
![]() |
[25] |
J. X. Fang, Q. Y. Xue, Some properties of the space fuzzy-valued continuous functions on a compact set, Fuzzy Set. Syst., 160 (2009), 1620–1631. https://doi.org/10.1016/j.fss.2008.07.014 doi: 10.1016/j.fss.2008.07.014
![]() |
[26] |
M. A. Fariborzi Araghi, N. Parandin, Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle, Soft Comput., 15 (2011), 2449–2456. https://doi.org/10.1007/s00500-011-0706-3 doi: 10.1007/s00500-011-0706-3
![]() |
[27] |
M. Friedman, M. Ma, A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Set. Syst., 106 (1999), 35–48. https://doi.org/10.1016/S0165-0114(98)00355-8 doi: 10.1016/S0165-0114(98)00355-8
![]() |
[28] |
M. Friedman, M. Ma, A. Kandel, Solutions to fuzzy integral equations with arbitrary kernels, Int. J. Approximating Reasoning, 20 (1999), 249–262. https://doi.org/10.1016/S0888-613X(99)00005-5 doi: 10.1016/S0888-613X(99)00005-5
![]() |
[29] | S. G. Gal, Approximation theory in fuzzy setting, In: Anastassiou, GA (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics, Chapman & Hall/CRC Press, Boca Raton, 2000,617–666. |
[30] | R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Set. Syst., (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6 |
[31] |
Z. Gong, C. Wu, Bounded variation absolute continuity and absolute integrability for fuzzy-number-valued functions, Fuzzy Set. Syst., 129 (2002), 83–94. https://doi.org/10.1016/S0165-0114(01)00132-4 doi: 10.1016/S0165-0114(01)00132-4
![]() |
[32] | O. Kaleva, Fuzzy differential equations, Fuzzy Set. Syst., 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7 |
[33] |
S. Karamseraji, R. Ezzati, S. Ziari, Fuzzy bivariate triangular functions with application to nonlinear fuzzy Fredholm–Volterra integral equations in two dimensions, Soft Comput., 24 (2020), 9091–9103. https://doi.org/10.1007/s00500-019-04439-9 doi: 10.1007/s00500-019-04439-9
![]() |
[34] |
A. Molabahrami, A. Shidfar, A. Ghyasi, An analytical method for solving linear Fredholm fuzzy integral equations of the second kind, Comput. Math. Appl., 61 (2011), 2754–2761. https://doi.org/10.1016/j.camwa.2011.03.034 doi: 10.1016/j.camwa.2011.03.034
![]() |
[35] |
N. Parandin, M. A. Fariborzi Araghi, The numerical solution of linear fuzzy Fredholm integral equations of the second kind by using finite and divided differences methods, Soft Comput., 15 (2010), 729–741. https://doi.org/10.1007/s00500-010-0606-y doi: 10.1007/s00500-010-0606-y
![]() |
[36] |
J. Y. Park, H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations, Fuzzy Set. Syst., 105 (1999), 481–488. https://doi.org/10.1016/S0165-0114(97)00238-8 doi: 10.1016/S0165-0114(97)00238-8
![]() |
[37] |
J. Y. Park, J. U. Jeong, On the existence and uniqueness of solutions of fuzzy Volttera-Fredholm integral equations, Fuzzy Set. Syst., 115 (2000), 425–431. https://doi.org/10.1016/S0165-0114(98)00341-8 doi: 10.1016/S0165-0114(98)00341-8
![]() |
[38] |
J. Y. Park, S. Y. Lee, J. U. Jeong, The approximate solution of fuzzy functional integral equations, Fuzzy Set. Syst., 110 (2000), 79–90. https://doi.org/10.1016/S0165-0114(98)00008-6 doi: 10.1016/S0165-0114(98)00008-6
![]() |
[39] |
A. Rivaz, F. Yousefi, H. Salehinejad, Using block pulse functions for solving two-dimensional fuzzy Fredholm integral equations of the second kind, Int. J. Appl. Math., 25 (2012), 571–582. https://doi.org/10.1590/S0103-51502012000300013 doi: 10.1590/S0103-51502012000300013
![]() |
[40] | S. M. Sadatrasoul, R. Ezzati, Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations, Abstract Appl. Anal., 2014, https://doi.org/10.1155/2014/413570 |
[41] |
S. M. Sadatrasoul, R. Ezzati, Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations, Fuzzy Set. Syst., 280 (2015), 91–106. https://doi.org/10.1016/j.fss.2014.12.008 doi: 10.1016/j.fss.2014.12.008
![]() |
[42] |
S. M. Sadatrasoul, R. Ezzati, Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula, J. Comput. Appl. Math., 292 (2016), 430–446. https://doi.org/10.1016/j.cam.2015.07.023 doi: 10.1016/j.cam.2015.07.023
![]() |
[43] |
V. Samadpour Khalifeh Mahaleh, R. Ezzati, Numerical solution of linear fuzzy Fredholm integral equations of second kind using iterative method and midpoint quadrature formula, J. Int. Fuzzy Syst., 33 (2017), 1293 –1302. https://doi.org/10.3233/JIFS-162044 doi: 10.3233/JIFS-162044
![]() |
[44] |
P. V. Subrahmanyam, S. K. Sudarsanam, A note on fuzzy Volterra integral equations, Fuzzy Set. Syst., 81 (1996), 237–240. https://doi.org/10.1016/0165-0114(95)00180-8 doi: 10.1016/0165-0114(95)00180-8
![]() |
[45] | C. Wu, S. Song, H. Wang, On the basic solutions to the generalized fuzzy integral equation, Fuzzy Set. Syst., (1998), 255–260. |
[46] |
C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Set. Syst., 102 (2001), 523–532. https://doi.org/10.1016/S0165-0114(99)00057-3 doi: 10.1016/S0165-0114(99)00057-3
![]() |
[47] |
H. Yang, Z. Gong, Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods, Fuzzy Set. Syst., 358 (2019), 132–149. https://doi.org/10.1016/j.fss.2018.05.010 doi: 10.1016/j.fss.2018.05.010
![]() |
[48] | S. Ziari, R. Ezzati, S. Abbasbandy, Numerical solution of linear fuzzy Fredholm integral equations of the second kind using fuzzy Haar wavelet. In: Advances in Computational Intelligence. Communications in Computer and Information Science, 299 (2012), 79–89. |
[49] |
S. Ziari, A. M. Bica, New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein-Fredholm integral equations, Fuzzy Set. Syst., 295 (2016), 136–152. https://doi.org/10.1016/j.fss.2015.09.021 doi: 10.1016/j.fss.2015.09.021
![]() |
[50] | S. Ziari, R. Ezzati, S. Abbasbandy, Fuzzy block-pulse functions and its application to solve linear fuzzy Fredholm integral equations of the second kind, In: Advances in Computational Intelligence, Communications in Computer and Information Science, (2016) 821–832. |
[51] | S. Ziari, Iterative method for solving two-dimensional nonlinear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimation, Iran J. Fuzzy Syst., 15 (2018), 55–76. |
[52] |
S. Ziari, Towards the accuracy of iterative numerical methods for fuzzy Hammerstein-Fredholm integral equations, Fuzzy Set. Syst., 375 (2019), 161–178. https://doi.org/10.1016/j.fss.2018.09.006 doi: 10.1016/j.fss.2018.09.006
![]() |
[53] | S. Ziari, I. Perfilieva, S. Abbasbandy, Block-pulse functions in the method of successive approximations for nonlinear fuzzy Fredholm integral equations, Differ. Equ. Dyn. Syst., (2019), https://doi.org/10.1007/s12591-019-00482-y. |
[54] |
S. Ziari, A. M. Bica, An iterative numerical method to solve nonlinear fuzzy Volterra-Hammerstein integral equations, J. Intel. Fuzzy Syst., 37 (2019), 6717–6729. https://doi.org/10.3233/JIFS-190149 doi: 10.3233/JIFS-190149
![]() |
[55] | S. Ziari, A. M. Bica, R. Ezzati, Iterative fuzzy Bernstein polynomials method for nonlinear fuzzy Volterra integral equations, Comp. Appl. Math., 39 (2020). https://doi.org/10.1007/s40314-020-01361-x. |
[56] | S. Ziari, T. Allahviranloo, W. Pedrycz, An improved numerical iterative method for solving nonlinear fuzzy Fredholm integral equations via Picards method and generalized quadrature rule, Comp. Appl. Math., 40 (2021). https://doi.org/10.1007/s40314-021-01616-1. |
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