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Computation of semi-analytical solutions of fuzzy nonlinear integral equations

Abstract

In this article, we use a fuzzy number in its parametric form to solve a fuzzy nonlinear integral equation of the second kind in the crisp case. The main theme of this article is to find a semi-analytical solution of fuzzy nonlinear integral equations. A hybrid method of Laplace transform coupled with Adomian decomposition method is used to find the solution of the fuzzy nonlinear integral equations including fuzzy nonlinear Fredholm integral equation, fuzzy nonlinear Volterra integral equation, and fuzzy nonlinear singular integral equation of Abel type kernel. We also provide some suitable examples to better understand the proposed method.

Introduction

Recently, the notions of fuzzy differential equations (FDEs) and fuzzy integral equations (FIEs) with fuzzy control have attracted the researchers. Different definitions of the fuzzy derivative and fuzzy integral have been established and extended to fuzzy calculus. For the existence and uniqueness of the solution of these equations, sufficient conditions have been provided, and for the approximate solution different numerical algorithms have been designed. Before going into detail, some of the basic topics need to be presented here. Zadeh introduced the concept of fuzzy set [1], followed by that of fuzzy number, and implemented it in fuzzy control [2] and approximate reasoning [3, 4]. The researchers provided that a fuzzy number is a collection of α-cut. They also provided various arithmetic operations on fuzzy numbers [512] which are a base for fuzzy calculus. Various researchers implanted a fuzzy number in Banach and topological spaces [1320] and it has many applications in fuzzy linear systems [21, 22], fuzzy linear least squares problems [2327], fuzzy eigenvalues and eigenvectors [21, 28], and to characterize a compact set [17, 29]. Chang and Zadeh [30] introduced fuzzy mapping, and based on Zadeh’s extension principle [1] Dubois and Prade [31] developed elementary fuzzy calculus. Different definitions for fuzzy derivative and fuzzy integration were provided in [1, 12, 13, 15, 3133].

It is mentioned that integral equations have many applications in various scientific and mathematical disciplines. To model any physical problem by using integral equations with exact parameters is not an easy task, in the real world problems it is almost impossible. Therefore, various techniques are used to measure uncertainty in the data. To handle such problems, Zadeh introduced fuzzy set in 1965 [1]. Over time, many researchers, scientists, and mathematicians extended fuzzy set in various other structures and solved problems having unprecise and uncertain data. Recently, fuzzy set and system and fuzzy logic are used in the fields of fuzzy topological spaces [34], fuzzy metric spaces [35], fuzzy differential equations [36, 37], to control chaotic systems [38, 39], and in applied physics [40, 41]. The interest in fuzzy integral equations is growing very fast in many problems. So, deterministic models of integral equations are used instead of applying the fuzzy integral equation. Hence, a dire need is to develop a numerical procedure and mathematical models that would deal with the general fuzzy integral equations and solve them appropriately. Various applications and methods for solving linear and nonlinear integral equations are given in [42]. The fuzzy integral equation is one of the most dominant fields of fuzzy set theory [43, 44]. In the last few decades, the concepts of fuzzy integral equation and fuzzy integro-differential equation have stimulated researchers. Wu and Ma [19] provided the first application of fuzzy integral and investigated fuzzy Fredholm integral equation of the second kind (FF-2), while Park et al. [45] showed the existence of a solution of FIEs. A large amount of research work has been done to investigate FIEs [4657]. This is due to various applications of FIEs in the scientific field. Therefore, finding an efficient and accurate algorithm for investigating FIE has been one the hot areas of research in recent time. To achieve these goals, various methods and procedures were used to handle integral equations, for details, see [5860].

Motivated by the aforesaid work, in this article, we use a hybrid method of Laplace transform coupled with Adomian decomposition method to solve the fuzzy nonlinear integral equation of the type

$$ \omega (h,\gamma )=f(h,\gamma )+\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u) \omega (u,\gamma )\,du, $$
(1)

where \(f(h,\gamma )\) and \(G(h,u)\) are the fuzzy functions, and \(\omega (h,\gamma )\) is the unrevealed function, a nonlinear term that appeared under the integral, i.e., \(\ln (\omega (u,\gamma )),\exp (\omega (u,\gamma ))\), and \(\omega ^{2}(u,\gamma )\) etc., \(\gamma \in (0,1)\) is a fuzzy parameter, and μ is a constant parameter. The two-variable function \(G(h,u)\) is called kernel of fuzzy integral equations, \(\alpha _{1}(h)\) and \(\beta _{1}(h)\) are limits of integration. If both the limits of integration are real numbers, then Eq. (1) is called fuzzy nonlinear Fredholm integral equation. If one of its limits, say \(\alpha _{1}(h)\), is constant and the other limit, say \(\beta _{1}(h)\), is variable, then Eq. (1) is called fuzzy nonlinear Volterra integral equation; and if the kernel has a singularity in the domain of its integration, then Eq. (1) is called fuzzy nonlinear singular integral equation of Abel type kernel.

The parametric case of Eq. (1) is

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+\mu \int _{ \alpha _{1}(h)}^{\beta _{1}(h)}\underline{G(h,u)\omega (u,\gamma )}\,du, \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+\mu \int _{ \alpha _{1}(h)}^{\beta _{1}(h)}\overline{G(h,u)\omega (u,\gamma )}\,du, \end{cases} $$
(2)

where \(\omega (h,\gamma )=(\underline{\omega }(h,\gamma ),\overline{\omega }(h, \gamma )), f(h,\gamma )=(\underline{f}(h,\gamma ),\overline{f}(h, \gamma ))\)

and

$$\begin{aligned} &\underline{G(h,u)\omega (u,\gamma )}= \textstyle\begin{cases} G(h,u)\underline{\omega }(h,\gamma ) & G(h,u)\geq 0, \\ G(h,u)\overline{\omega }(u,\gamma ) & G(h,u)< 0, \end{cases}\displaystyle \\ &\overline{G(h,u)\omega (u,\gamma )}= \textstyle\begin{cases} G(h,u)\overline{\omega }(h,\gamma ) & G(h,u)\geq 0, \\ G(h,u)\underline{\omega }(u,\gamma ) & G(h,u)< 0. \end{cases}\displaystyle \end{aligned}$$

By this method, we compute fuzzy solutions of the above integral equation (2) and provide various examples to demonstrate the procedure.

We organize the article as follows:

Section 2 presents the general procedure for nonlinear FIE. In Sect. 3, some test problems [61] are solved by the proposed method. In Sect. 4, the comparison and discussion of the proposed method with the existing techniques [6264] are given, while Sect. 5 is devoted to the conclusion of the article.

General procedure to handle fuzzy nonlinear integral equations

To solve the fuzzy nonlinear integral Eq. (1) in a fuzzy sense, the parametric form of Eq. (1) can be written into two integral equations as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+\mu \int _{ \alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\underline{\omega }^{n}(u,\gamma )\,du, \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+\mu \int _{ \alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\overline{\omega }^{n}(u,\gamma )\,du, \end{cases} $$
(3)

applying Laplace transform on Eq. (3)

$$ \textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\underline{f}(h,\gamma )]+L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\underline{\omega }^{n}(u, \gamma )\,du], \\ L[\overline{\omega }(h,\gamma )]=L[\overline{f}(h,\gamma )]+L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\overline{\omega }^{n}(u, \gamma )\,du], \end{cases} $$
(4)

operating inverse Laplace transform on Eq. (4), we obtain

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\underline{\omega }^{n}(u, \gamma )\,du]], \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)\overline{\omega }^{n}(u, \gamma )\,du]]. \end{cases} $$
(5)

Consider that the lower and upper fuzzy limit solutions of Eq. (5) can be extended by the Laplace decomposition algorithm into an infinite series as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\underline{\omega }_{n}(h, \gamma ), \\ \overline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\overline{\omega }_{n}(h, \gamma ), \end{cases} $$
(6)

and nonlinear lower and upper limit terms \((\underline{\omega }^{n}(h,\gamma ),\overline{\omega }^{n}(h,\gamma ))\) can be written as

$$ \textstyle\begin{cases} \underline{\omega }^{n}(u,\gamma )=\sum_{n=0}^{\infty }B_{n}(u,\gamma ), \\ \overline{\omega }^{n}(u,\gamma )=\sum_{n=0}^{\infty }\acute{B}_{n}(u, \gamma ), \end{cases} $$
(7)

where \((B_{n}(u,\gamma ),\acute{B_{n}}(u,\gamma ))\) are the Adomian polynomials [65]. By putting Eq. (6) and Eq. (7) in Eq. (5), we get

$$ \textstyle\begin{cases} \sum_{n=0}^{\infty }\underline{\omega }_{n}(h,\gamma )=\underline{f}(h, \gamma )+L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)[ \sum_{n=0}^{\infty }{B}_{n}(u,\gamma )]\,du]], \\ \sum_{n=0}^{\infty }\overline{\omega }_{n}(h,\gamma )=\overline{f}(h, \gamma )+L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{\beta _{1}(h)}G(h,u)[ \sum_{n=0}^{\infty }\acute{B}_{n}(u,\gamma )]\,du]]. \end{cases} $$
(8)

The Adomian polynomials in Eq. (7) can be generated by several means, and we use the following recursive formulation:

$$ \textstyle\begin{cases} B_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[N(\sum_{k=0}^{ \infty }\lambda ^{k}\underline{\omega }_{k})]_{\lambda =0}, \\ \acute{B}_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[N( \sum_{k=0}^{\infty }\lambda ^{k}\overline{\omega }_{k})]_{\lambda =0}, \end{cases} $$
(9)

for \(n=0,1,2,3,\ldots \) . Thus, in general, comparing the recursive relation of Eq. (8) termwise, we get for \(n=0\)

$$ \textstyle\begin{cases} \underline{\omega }_{0}(h,\gamma )=\underline{f}(h,\gamma ), \\ \overline{\omega }_{0}(h,\gamma )=\overline{f}(h,\gamma ), \end{cases} $$

at \(n=1\)

$$ \textstyle\begin{cases} \underline{\omega }_{1}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)B_{0}(u,\gamma )\,du]], \\ \overline{\omega }_{1}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)\acute{B}_{0}(u,\gamma )\,du]], \end{cases} $$

at \(n=2\)

$$ \textstyle\begin{cases} \underline{\omega }_{2}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)B_{1}(u,\gamma )\,du]], \\ \overline{\omega }_{2}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)\acute{B}_{1}(u,\gamma )\,du]], \end{cases} $$

and so on up to \(n+1\), we get

$$ \textstyle\begin{cases} \underline{\omega }_{n+1}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)B_{n}(u,\gamma )\,du]], \\ \overline{\omega }_{n+1}(h,\gamma )=L^{-1}[L[\mu \int _{\alpha _{1}(h)}^{ \beta _{1}(h)}G(h,u)\acute{B}_{n}(u,\gamma )\,du]], & n\geq 0, \end{cases} $$
(10)

where \((\underline{f}(h,\gamma ),\overline{f}(h,\gamma ))\) represents the source term.

Test problems of nonlinear fuzzy integral equations

We solve a few examples by considering the proposed method to obtain an analytical approximate solution of fuzzy nonlinear integral equations of different types. Applications of the proposed method are more simple and upgrade its order.

Example 1

Consider the second kind fuzzy nonlinear Volterra integral equation in parametric form:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+\int _{0}^{h} \omega ^{2}(u,\gamma )\,du, \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+\int _{0}^{h} \omega ^{2}(u,\gamma )\,du, \end{cases} $$
(11)

where \(\mu =1,0\leq h\leq 1,0\leq u \leq h,G(h,u)=1, 0\leq \gamma \leq 1\) and nonhomogenous term is \((\underline{f}(h,\gamma ),\overline{f}(h,\gamma )=((\gamma ^{2}+ \gamma )h,(7-\gamma )h)\).

To solve Eq. (11) using the proposed method, first taking Laplace transform on Eq. (11), we have

$$ \textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\underline{f}(h,\gamma )]+L[\int _{0}^{h} \omega ^{2}(u,\gamma )\,du], \\ L[\overline{\omega }(h,\gamma )]=L[\overline{f}(h,\gamma )]+L[\int _{0}^{u} \omega ^{2}(u,\gamma )\,du]. \end{cases} $$
(12)

Operating the inverse Laplace transform on Eq. (12), we get

$$\begin{aligned} &\textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+L^{-1}[L[\int _{0}^{h} \omega ^{2}(u,\gamma )\,du]], \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+L^{-1}[L[\int _{0}^{h} \omega ^{2}(u,\gamma )\,du]], \end{cases}\displaystyle \end{aligned}$$
(13)
$$\begin{aligned} &\textstyle\begin{cases} \underline{\omega }(h,\gamma )=(\gamma ^{2}+\gamma )h+L^{-1}[L[\int _{0}^{h} \omega ^{2}(u,\gamma )\,du]], \\ \overline{\omega }(h,\gamma )=(7-\gamma )h+L^{-1}[L[\int _{0}^{h} \omega ^{2}(u,\gamma )\,du]]. \end{cases}\displaystyle \end{aligned}$$
(14)

Assume that the lower and upper limit fuzzy solutions of Eq. (14) are an infinite series as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\underline{\omega }_{n}(h, \gamma ), \\ \overline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\overline{\omega }_{n}(h, \gamma ). \end{cases} $$
(15)

Putting Eq. (15) in Eq. (14), we get

$$ \textstyle\begin{cases} \sum_{n=0}^{\infty }\underline{\omega }_{n}(h,\gamma )=(\gamma ^{2}+ \gamma )h+L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{\infty }B_{n}(u,\gamma )]]], \\ \sum_{n=0}^{\infty }\overline{\omega }_{n}(h,\gamma )=(7-\gamma )h+L^{-1}[L[ \int _{0}^{h}[\sum_{n=0}^{\infty }{\acute{B}_{n}(u,\gamma )}]]], \end{cases} $$
(16)

where \((B_{n},\acute{B}_{n})\) are Adomian polynomials which represent the nonlinear term and can be decomposed as follows:

$$ \textstyle\begin{cases} B_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[(\sum_{k=0}^{ \infty }\lambda ^{k}\underline{\omega }_{k})]^{2}_{\lambda =0}, \\ \acute{B}_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[( \sum_{k=0}^{\infty }\lambda ^{k}\overline{\omega }_{k})]^{2}_{\lambda =0}. \end{cases} $$
(17)

Now, we first solve the lower limit fuzzy solution of Eq. (11). So comparing the lower limit terms of Eq. (16) and solving, we obtain

$$ \textstyle\begin{cases} \underline{\omega }_{0}(h,\gamma )=(\gamma ^{2}+\gamma )h, \\ \underline{\omega }_{1}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{0}(u,\gamma )]\,du]]=\frac{1}{3}(\gamma ^{2}+\gamma )^{2}h^{3}, \\ \underline{\omega }_{2}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{1}(u,\gamma )]\,du]]=\frac{2}{15}(\gamma ^{2}+\gamma )^{3}h^{5}, \\ \underline{\omega }_{3}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{2}(u,\gamma )]\,du]]=\frac{17}{315}(\gamma ^{2}+\gamma )^{4}h^{7}, \\ \vdots \end{cases} $$
(18)

and for the upper limit fuzzy solution of Eq. (11), comparing the upper limit terms of Eq. (16) and solving

$$ \textstyle\begin{cases} \overline{\omega }_{0}(h,\gamma )=(7-\gamma )h, \\ \overline{\omega }_{1}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{0}(u,\gamma )]\,du]]=\frac{1}{3}(7-\gamma )^{2}h^{3}, \\ \overline{\omega }_{2}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{1}(u,\gamma )]\,du]]=\frac{2}{15}(7-\gamma )^{3}h^{5}, \\ \overline{\omega }_{3}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{2}(u,\gamma )]\,du]]=\frac{17}{315}(7-\gamma )^{4}h^{7}, \\ \vdots \end{cases} $$
(19)

putting Eq. (18) and Eq. (19) in Eq. (15), we get the approximate lower and upper limit fuzzy solutions as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=(\gamma ^{2}+\gamma )h+\frac{1}{3}( \gamma ^{2}+\gamma )^{2}h^{3}+\frac{2}{15}(\gamma ^{2}+\gamma )^{3}h^{5}+ \frac{17}{315}(\gamma ^{2}+\gamma )^{4}h^{7}+\cdots, \\ \overline{\omega }(h,\gamma )=(7-\gamma )h+\frac{1}{3}(7-\gamma )^{2}h^{3}+ \frac{2}{15}(7-\gamma )^{3}h^{5}+\frac{17}{315}(7-\gamma )^{4}h^{7}+ \cdots. \end{cases} $$
(20)

Example 2

Consider the second kind fuzzy nonlinear Fredholm integral equation in a parametric form:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+\int _{0}^{1} \underline{\omega }^{2}(u,\gamma )\,du, \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+\int _{0}^{1} \overline{\omega }^{2}(u,\gamma )\,du, \end{cases} $$
(21)

where \(0\leq h,u\leq 1,0\leq \gamma \leq 1,G(h,u)=1\) and \((\underline{f}(h,\gamma ),\overline{f}(h,\gamma ))=(\gamma,2- \gamma )\).

To solve Eq. (21) by the proposed method, taking Laplace transform, we have

$$ \textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\underline{f}(h,\gamma )]+L[\int _{0}^{1} \underline{\omega }^{2}(u,\gamma )\,du], \\ L[\overline{\omega }(h,\gamma )]=L[\overline{f}(h,\gamma )]+L[\int _{0}^{1} \overline{\omega }^{2}(u,\gamma )\,du]. \end{cases} $$
(22)

Now, applying the inverse Laplace transform on Eq. (22), we get

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\gamma +L^{-1}[L[\int _{0}^{1} \underline{\omega }^{2}(u,\gamma )\,du]], \\ \overline{\omega }(h,\gamma )=(2-\gamma )+L^{-1}[L[\int _{0}^{1} \overline{\omega }^{2}(u,\gamma )\,du]]. \end{cases} $$
(23)

Assume that the lower and upper limit fuzzy solutions of Eq. (23) are an infinite series as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\underline{\omega }_{n}(h, \gamma ), \\ \overline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\overline{\omega }_{n}(h, \gamma ). \end{cases} $$
(24)

Putting Eq. (24) in Eq. (23), we get

$$ \textstyle\begin{cases} \sum_{n=0}^{\infty }\underline{\omega }_{n}(h,\gamma )=\gamma +L^{-1}[L[ \int _{0}^{h}[\sum_{n=0}^{\infty }B_{n}(u,\gamma )]\,du]], \\ \sum_{n=0}^{\infty }\overline{\omega }_{n}(h,\gamma )=(2-\gamma )+L^{-1}[L[ \int _{0}^{h}[\sum_{n=0}^{\infty }{\acute{B}_{n}(u,\gamma )}]\,du]], \end{cases} $$
(25)

where \((B_{n},\acute{B}_{n})\) are Adomian polynomials which represent nonlinear term and can be solved as follows:

$$ \textstyle\begin{cases} B_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[(\sum_{k=0}^{ \infty }\lambda ^{k}\underline{\omega }_{k})]^{2}_{\lambda =0}, \\ \acute{B}_{n}(u,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[( \sum_{k=0}^{\infty }\lambda ^{k}\overline{\omega }_{k})]^{2}_{\lambda =0}. \end{cases} $$
(26)

Now first we solve the lower limit fuzzy solution of Eq. (21). So, comparing the lower limit case of Eq. (25) termwise and solving, we obtain

$$ \textstyle\begin{cases} \underline{\omega }_{0}(h,\gamma )=\gamma, \\ \underline{\omega }_{1}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{0}(u,\gamma )]\,du]]=\gamma ^{2}, \\ \underline{\omega }_{2}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{1}(u,\gamma )]\,du]]=2\gamma ^{3}, \\ \underline{\omega }_{3}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }B_{2}(u,\gamma )]\,du]]=5\gamma ^{4}, \\ \vdots \end{cases} $$
(27)

and for the upper limit fuzzy solutions of Eq. (21), comparing the upper limit case of Eq. (25) termwise and solving, we obtain

$$ \textstyle\begin{cases} \overline{\omega }_{0}(h,\gamma )=2-\gamma, \\ \overline{\omega }_{1}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{0}(u,\gamma )]\,du]]=(2-\gamma )^{2}, \\ \overline{\omega }_{2}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{1}(u,\gamma )]\,du]]=2(2-\gamma )^{3}, \\ \overline{\omega }_{3}(h,\gamma )=L^{-1}[L[\int _{0}^{h}[\sum_{n=0}^{ \infty }\acute{B}_{2}(u,\gamma )]\,du]]=5(2-\gamma )^{4}, \\ \vdots \end{cases} $$
(28)

Put Eq. (27) and Eq. (28) in Eq. (25) to get the approximate lower and upper limit fuzzy solutions:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\gamma +\gamma ^{2}+2\gamma ^{3}+5 \gamma ^{4}+\cdots, \\ \overline{\omega }(h,\gamma )=(2-\gamma )+(2-\gamma )^{2}+2(2-\gamma )^{3}+5(2- \gamma )^{4}+\cdots. \end{cases} $$
(29)

Example 3

Consider the second kind fuzzy nonlinear singular integral equation of Abel type kernel of parametric form:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\underline{f}(h,\gamma )+\int _{0}^{h} \frac{\underline{\omega }^{2}(u,\gamma )}{\sqrt{h-u}}\,du, \\ \overline{\omega }(h,\gamma )=\overline{f}(h,\gamma )+\int _{0}^{h} \frac{\overline{\omega }^{2}(u,\gamma )}{\sqrt{h-u}}\,du, \end{cases} $$
(30)

where \((\underline{f}(h,\gamma ),\overline{f}(h,\gamma ))=(h\gamma - \frac{16}{15}\gamma ^{2}h^{5/2},h(3-\gamma )-\frac{16}{15}(3-\gamma )^{2}h^{5/2})\) and \(0\leq h\leq 1, 0\leq u\leq h,0\leq \gamma \leq 1\).

To solve Eq. (30) by the proposed method, taking the Laplace transform on Eq. (30), we have

$$ \textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\gamma h-\frac{16}{15}\gamma ^{2}h^{5/2}]+L[ \int _{0}^{h}\frac{\underline{\omega }^{2}(u,\gamma )}{\sqrt{h-u}}\,du], \\ L[\overline{\omega }(h,\gamma )]=L[(3-\gamma )h-\frac{16}{15}(3- \gamma )^{2}h^{5/2}]+L[\int _{0}^{h} \frac{\overline{\omega }^{2}(u,\gamma )}{\sqrt{h-u}}\,du]. \end{cases} $$
(31)

Applying the fuzzy convolution theorem on Eq. (31), we obtain

$$\begin{aligned} &\textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\gamma h-\frac{16}{15}\gamma ^{2}h^{5/2}]+L[(h)^{-1/2}] \cdot L[\underline{\omega }^{2}(u,\gamma )], \\ L[\overline{\omega }(h,\gamma )]=L[(3-\gamma )h-\frac{16}{15}(3- \gamma )^{2}h^{5/2}]+L[(h)^{-1/2}]\cdot L[\overline{\omega }^{2}(u, \gamma )], \end{cases}\displaystyle \\ &\textstyle\begin{cases} L[\underline{\omega }(h,\gamma )]=L[\gamma h-\frac{16}{15}\gamma ^{2}h^{5/2}]+ \sqrt{\frac{\pi }{u}}L[\underline{\omega }^{2}(u,\gamma )], \\ L[\overline{\omega }(h,\gamma )]=L[(3-\gamma )h-\frac{16}{15}(3- \gamma )^{2}h^{5/2}]+\sqrt{\frac{\pi }{u}}L[\overline{\omega }^{2}(u, \gamma )]. \end{cases}\displaystyle \end{aligned}$$
(32)

Applying the inverse Laplace transform on Eq. (32), we obtain

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\gamma h-\frac{16}{15}\gamma ^{2}h^{5/2}+L^{-1}[ \sqrt{\frac{\pi }{u}}L[\underline{\omega }^{2}(u,\gamma )]], \\ \overline{\omega }(h,\gamma )=(3-\gamma )h-\frac{16}{15}(3-\gamma )^{2}h^{5/2}+L^{-1}[ \sqrt{\frac{\pi }{u}}L[\overline{\omega }^{2}(u,\gamma )]]. \end{cases} $$
(33)

Assume that the lower and upper limit fuzzy solutions of Eq. (33) are an infinite series as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\underline{\omega }_{n}(h, \gamma ), \\ \overline{\omega }(h,\gamma )=\sum_{n=0}^{\infty }\overline{\omega }_{n}(h, \gamma ). \end{cases} $$
(34)

Putting Eq. (34) in Eq. (33), we get

$$\begin{aligned} &\textstyle\begin{cases} \sum_{n=0}^{\infty }\underline{\omega }_{n}(h,\gamma )=\gamma h- \frac{16}{15}\gamma ^{2}h^{5/2}+L^{-1}[\sqrt{\frac{\pi }{u}}L[ \underline{\omega }^{2}(h,\gamma )]], \\ \sum_{n=0}^{\infty }\overline{\omega }_{n}(h,\gamma )=(3-\gamma )h- \frac{16}{15}(3-\gamma )^{2}h^{5/2}+L^{-1}[\sqrt{\frac{\pi }{u}}L[ \overline{\omega }^{2}(h,\gamma )]], \end{cases}\displaystyle \end{aligned}$$
(35)
$$\begin{aligned} &\textstyle\begin{cases} \sum_{n=0}^{\infty }\underline{\omega }_{n}(h,\gamma )=h\gamma - \frac{16}{15}\gamma ^{2}h^{5/2}+L^{-1}[\sqrt{\frac{\pi }{u}}L[\sum_{n=0}^{ \infty }B_{n}(h,\gamma )]], \\ \sum_{n=0}^{\infty }\overline{\omega }_{n}(h,\gamma )=(3-\gamma )h- \frac{16}{15}(3-\gamma )^{2}h^{5/2}+L^{-1}[\sqrt{\frac{\pi }{u}}L[ \sum_{n=0}^{\infty }{\acute{B}_{n}}(h,\gamma )]], \end{cases}\displaystyle \end{aligned}$$
(36)

where \((B_{n},\acute{B}_{n})\) are Adomian polynomials which represent the nonlinear terms and can be found as follows:

$$ \textstyle\begin{cases} B_{n}(h,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[(\sum_{k=0}^{ \infty }\lambda ^{k}\underline{\omega }_{k})]^{2}_{\lambda =0}, \\ \acute{B}_{n}(h,\gamma )=\frac{1}{n!}\frac{d^{n}}{d\lambda ^{n}}[( \sum_{k=0}^{\infty }\lambda ^{k}\overline{\omega }_{k})]^{2}_{\lambda =0}. \end{cases} $$
(37)

Now, first we solve the lower limit fuzzy solution of Eq. (30). So, comparing the lower limit terms of Eq. (36) and solving, we obtain

$$ \textstyle\begin{cases} \underline{\omega }_{0}(h,\gamma )=\gamma h-\frac{16}{15}\gamma ^{2}h^{5/2}, \\ \underline{\omega }_{1}(h,\gamma )=L^{-1}[\sqrt{\frac{\pi }{u}}[L[B_{0}(u, \gamma )]]]=\frac{16}{15}\gamma ^{2}h^{5/2}+\frac{16}{45}\gamma ^{4}h^{3}- \frac{7}{12}\gamma ^{3}\pi h^{4}, \\ \vdots \end{cases} $$
(38)

and for the upper limit solution of Eq. (30), comparing the upper limit terms of Eq. (36) and solving, we obtain

$$ \textstyle\begin{cases} \overline{\omega }_{0}(h,\gamma )=(3-\gamma )h-\frac{16}{15}(3- \gamma )^{2}h^{5/2}, \\ \overline{\omega }_{1}(h,\gamma )=L^{-1}[\sqrt{\frac{\pi }{u}}[L[\sum_{n=0}^{ \infty }\acute{B}_{0}(u,\gamma )]], \\ \phantom{\overline{\omega }_{1}(h,\gamma )}=\frac{16}{15}(3-\gamma )^{2}h^{5/2}+\frac{112}{90}(3-\gamma )^{2} \pi h^{3}-\frac{7}{12}(3-\gamma )^{3}\pi h^{4}, \\ \vdots \end{cases} $$
(39)

and so on. Compute the various terms of the series fuzzy solutions like we computed in Eq. (39) and in Eq. (34). After computation of the terms and putting them in Eq. (38), we cancel the noise terms and get the lower and upper limit solution as follows:

$$ \textstyle\begin{cases} \underline{\omega }(h,\gamma )=\gamma h, \\ \overline{\omega }(h,\gamma )=(3-\gamma )h. \end{cases} $$
(40)

Comparison with HAM and discussion

In HAM the convergence of solution in the form of series depends upon four factors, i.e., the initial guess, the auxiliary linear operator, the auxiliary function which we define for homotopy, and the auxiliary parameter ħ. Further, if we select \(\hbar =-1\), and the auxiliary function also equals 1, HPM is obtained. Hence, HPM is a special case of HAM whose convergence only depends upon two factors: the auxiliary linear operator and the initial guess. So, given the initial guess and the auxiliary linear operator, the HPM approach cannot provide other ways to ensure that the solution is convergent. On the other hand, the proposed method solution for both linear and nonlinear problems is obtained in the form of a series showing higher convergence of the method. Among all other analytical methods, the proposed method is an efficient analytical method to solve nonlinear problems of differential or integral equations. Therefore, a hybrid method is the combination of two powerful methods: Laplace transform and Adomian decomposition method. The mentioned method does not need any kind of discretization or linearization. It also does not need a predefined parameter which controls these schemes as needed in HAM. Therefore, our proposed method is an efficient analytical technique for treating those equations that represent nonlinear models. If we consider our proposed model without initial condition, it will converge to a particular solution.

Conclusion

In this article, different types of fuzzy nonlinear integral equations of Abel type kernel are solved by using a hybrid method of integral transform and decomposition technique. Further, it is shown that the solution of fuzzy nonlinear integral equations by the newly proposed algorithm is easily, effectively and accurately convergent. To show the efficiency of the proposed algorithm, some numerical examples are given, which give the exactness of the method. Shortly, some modification is required to extend the method for the system of fuzzy nonlinear integral equations, in which the coefficients are mixed.

References

  1. 1.

    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    MATH  Article  Google Scholar 

  2. 2.

    Chang, S.S.L., Zadeh, L.: On fuzzy mapping and control. IEEE Trans. Syst. Man Cybern. 2, 30–34 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8(3), 199–249 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Zadeh, L.A.: Linguistic variables, approximate reasoning and disposition. Med. Inform. 8, 173–186 (1983)

    Article  Google Scholar 

  5. 5.

    Negoita, C.V., Ralescu, D.A.: Applications of Fuzzy Sets to Systems Analysis. Wiley, New York (1975)

    Google Scholar 

  6. 6.

    Mizumoto, M., Tanaka, K.: The four operations of arithmetic on fuzzy numbers. Syst. Comput. Controls 7(5), 73–81 (1976)

    MathSciNet  Google Scholar 

  7. 7.

    Mizumoto, M., Tanaka, K.: Some properties of fuzzy numbers. Computer Science (1979)

  8. 8.

    Yager, R.R., Gupta, M.M., Ragade, R.K.: Advances in Fuzzy Set Theory and Applications. North-Holland, Amsterdam (1979)

    Google Scholar 

  9. 9.

    Nahmias, S.: Fuzzy variables. Fuzzy Sets Syst.. 1, 97–111 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Dubois, D., Prade, H.: Operations on fuzzy numbers. J. Syst. Sci. 9, 613–626 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)

    Google Scholar 

  12. 12.

    Goetscbel, R., Voxman, W.: Elementary calculus. Fuzzy Sets Syst. 18, 31–43 (1986)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Purl, M.L., Ralescu, D.: Differential for fuzzy function. J. Math. Anal. Appl. 91, 552–558 (1983)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Purl, M.L., Ralescu, D.: The concept of normality for fuzzy random variables. Ann. Probab. 13, 1373–1379 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24, 301–317 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Palm, R., Driankov, D.: Fuzzy inputs. Fuzzy Sets Syst. 70, 315–335 (1995)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. Fuzzy Sets Syst. 35, 241–249 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Congxin, W., Ming, M.: An embedding operator of fuzzy numbers and its application for fuzzy integrals. Syst. Sci. Math. Sci. 8, 193–199 (1990)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Congxin, W., Ming, M.: On embedding problem of fuzzy number spaces (Part: I, II and III). Fuzzy Sets Syst. 44, 33–38 (1991), 45, 189–202 (1992) and 46, 281–286 (1992)

  20. 20.

    Ming, M.: On embedding problem of fuzzy number spaces (part: IV and V). Fuzzy Sets Syst. 55, 313–318 (1993) and 58, 185–193 (1993)

  21. 21.

    Buckley, J.J.: Fuzzy eigenvalues and input-output analysis. Fuzzy Sets Syst. 34, 187–195 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Buckley, J.J.: Solving fuzzy equations. Fuzzy Sets Syst. 50, 1–14 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Ding, Z., Kandel, A.: Existence and stability of fuzzy differential equations. J. Fuzzy Math. 5, 681–697 (1997)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Heshmaty, B., Kandel, A.: Fuzzy linear regression and its applications to forecasting in uncertain environments. Fuzzy Sets Syst. 15, 159–191 (1985)

    MATH  Article  Google Scholar 

  25. 25.

    Vrba, J.: A note on inverses in arithmetic with fuzzy numbers. Fuzzy Sets Syst. 50, 267–278 (1992)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Yager, R.R.: Fuzzy prediction based upon regression models. Inf. Sci. 26, 45–63 (1982)

    MATH  Article  Google Scholar 

  27. 27.

    Renhong, Z., Govind, R.: Solutions of algebraic equations involving generalized fuzzy number. Inf. Sci. 56, 199–243 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Goetschel, R., Voxman, W.: Eigen fuzzy number sets. Fuzzy Sets Syst. 16, 75–85 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Diamond, P., Kloeden, P.E.: Characterization of compact sets of fuzzy sets. Fuzzy Sets Syst. 29, 341–348 (1989)

    MATH  Article  Google Scholar 

  30. 30.

    Ming, M.: Some notes on the characterization of compact sets in (\(E^{n}\), \(d_{p}\)). Fuzzy Sets Syst. 56, 297–301 (1993)

    Article  MATH  Google Scholar 

  31. 31.

    Dubois, D., Prade, H.: Towards fuzzy differential calculus (part: 1,2 and 3). Fuzzy Sets Syst. 8, 1–7 (1982), 105–116 (1982) and 225–233 (1982)

  32. 32.

    Matloka, M.: On fuzzy integrals. In: Proc. 2nd Polish Symp. on Interval and Fuzzy Mathematics, Politechnika Poznansk, pp. 167–170 (1987)

    Google Scholar 

  33. 33.

    Nanda, S.: On integration of fuzzy mappings. Fuzzy Sets Syst. 32, 95–101 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Caldas, M., Jafari, S.: h-compact fuzzy topological spaces. Chaos Solitons Fractals 25, 229–232 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Park, J.H.: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 22, 1039–1046 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Abbasbandy, S., Allahviranloo, T.: Numerical solution of fuzzy differential equation by Runge–Kutta method. Nonlinear Stud. 11, 117–129 (2004)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Abbasbandy, S., Nieto, J.J., Alavi, M.: Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fractals 26, 1337–1341 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Feng, G., Chen, G.: Adaptive control of discrete-time chaotic systems: a fuzzy control approach. Chaos Solitons Fractals 23, 459–467 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Jiang, W., Guo-Dong, Q., Bin, D.: H1 variable universe adaptive fuzzy control for chaotic system. Chaos Solitons Fractals 24, 1075–1086 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    El-Naschie, M.S.: A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos Solitons Fractals 19, 209–236 (2004)

    MATH  Article  Google Scholar 

  41. 41.

    Allahviranloo, T., Khezerloo, M., Ghanbari, M., Khezerloo, S.: The homotopy perturbation method for fuzzy Volterra integral equations. Int. J. Cogn. Inform. Nat. Intell. 8, 31–37 (2010)

    Google Scholar 

  42. 42.

    Wazwaz, A.M.: Linear and Nonlinear Integral Equations: Methods and Application. Springer, Berlin (2011)

    Google Scholar 

  43. 43.

    Friedman, M., Ma, M., Kandel, A.: Numerical solution of fuzzy differential and integral equations. Fuzzy Sets Syst. 106, 35–48 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Subrahmanyam, P.V., Sudarsanam, S.K.: A note on fuzzy Volterra integral equations. Fuzzy Sets Syst. 81(2), 188–191 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Park, J.Y., Kwan, Y.C., Jeong, J.V.: Existence of solutions of fuzzy integral equations in Banach spaces. Fuzzy Sets Syst. 72, 373–378 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Mirzaee, F., Paripour, M., Yari, M.K.: Numerical solution of Fredholm fuzzy integral equations of the second kind via direct method using triangular functions. J. Hyperstruct. 1(2), 46–60 (2012)

    MATH  Google Scholar 

  47. 47.

    Allahviranloo, T., Ghanbari, B.: On the fuzzy fractional differential equation with interval Atangana–Baleanu fractional derivative approach. Chaos Solitons Fractals 130, 109397 (2020)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Song, S., Liu, Q., Xu, Q.: Existence and comparison theorems to Volterra fuzzy integral equation in \({(E^{n}, D)}^{1}\). Fuzzy Sets Syst. 104(2), 315–321 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  49. 49.

    Sahu, P.K., Ray, S.S.: A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein–Volterra delay integral equations. Fuzzy Sets Syst. 309, 131–144 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Park, J.Y., Jeong, J.U.: On the existence and uniqueness of solutions of fuzzy Volterra–Fredholm integral equations. Fuzzy Sets Syst. 115(3), 425–431 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Bica, A.M., Ziari, S.: Open fuzzy cubature rule with application to nonlinear fuzzy Volterra integral equations in two dimensions. Fuzzy Sets Syst. 358, 108–131 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Bica, A.M.: One-sided fuzzy numbers and applications to integral equations from epidemiology. Fuzzy Sets Syst. 219, 27–48 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Lakshmikantham, V., Mohapatra, R.N.: Theory of Fuzzy Differential Equations and Applications. Taylor & Francis, London (2003)

    Google Scholar 

  54. 54.

    Friedman, M., Ming, M., Kandel, A.: Numerical solution of fuzzy differential and integral equations. Fuzzy Sets Syst. 106, 35–48 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Wu, H.C.: The improper fuzzy Riemann integral and its numerical integration. Inf. Sci. 111, 109–137 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Eman, A., Hussain, W., Ali, A.: Homotopy analysis method for solving fuzzy integro-differential equations. Mod. Appl. Sci. 7(3), 15–25 (2013)

    MathSciNet  Google Scholar 

  57. 57.

    Eman, A., Hussain, W., Ali, A.: Homotopy analysis method for solving nonlinear fuzzy integral equations. Int. J. Appl. Math. 28(1), 1177–1189 (2013)

    Google Scholar 

  58. 58.

    Mosleh, M., Otadi, M.: Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. J. Adv. Inf. Technol. 4(3), 148–155 (2013)

    Google Scholar 

  59. 59.

    Allahviranloo, T., Ghanbari, M., Nuraei, R.: An application of a semi-analytical method on linear fuzzy Volterra integral equations. J. Fuzzy Set Valued Anal. 2014, 1–15 (2014)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Otadi, M., Mosleh, M.: Numerical solution of fuzzy nonlinear integral equations of the second kind. Iran. J. Fuzzy Syst. 1(1), 135–145 (2014)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Ahmad, J., Nosher, H.: Numerical treatment of nonlinear fuzzy integral equations by homotopy perturbation method. J. Inf. Eng. Appl. 6(10), 32–43 (2016)

    Google Scholar 

  62. 62.

    Liao, S.: Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169, 1186–1194 (2005)

    MathSciNet  MATH  Google Scholar 

  63. 63.

    Shah, R., Khan, H., Arif, M., Kumam, P.: Application of Laplace Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 21, 35 (2019)

    MathSciNet  Article  Google Scholar 

  64. 64.

    Manafianheris, J.: Solving the integro-differential equations using the modified Laplace Adomian decomposition method. J. Math. Ext. 6(1), 41–55 (2012)

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Adomian, G.: Frontier Problem of Physics: The Decomposition Method. Kluwer Academic, Boston (1994)

    Google Scholar 

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Acknowledgements

We are thankful to the reviewers for their careful reading and constructive suggestions which improved this paper very much.

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Abbreviations

Fuzzy differential equation (FDE); Fuzzy integral equation (FIE); Fuzzy Fredholm integral equation of the second kind (FF-2); Adomian decomposition method (ADM); Laplace Adomian decomposition method (LADM); Homotopy perturbation method (HPM).

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Ullah, Z., Ullah, A., Shah, K. et al. Computation of semi-analytical solutions of fuzzy nonlinear integral equations. Adv Differ Equ 2020, 542 (2020). https://doi.org/10.1186/s13662-020-02989-z

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Keywords

  • Semi-analytical solution
  • Fuzzy nonlinear Fredholm integral equation
  • Fuzzy nonlinear Volterra integral equation
  • Fuzzy singular integral equation
  • Fuzzy number