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Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind
SpringerPlus volumeÂ 5, ArticleÂ number:Â 387 (2016)
Abstract
In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.
Background
Integral equations find special applicability within many scientific and mathematical disciplines. Indeed modeling physical problems using integral equations with the exact parameters is not always easy but also impossible in the real problems. For this purpose, one way is using uncertainty measures for handling such lack of information. One of the most recent approaches is using fuzzy concept (Zadeh 1965). So instead of using deterministic models of integral equations, we can go in for fuzzy integral equations. Hence there occurs a need to develop mathematical models and numerical procedures that would appropriately treat general fuzzy integral equations and solve them. In this paper, we apply homotopy perturbation method (HPM) to solve both linear and nonlinear fuzzy Volterra integral equations of the second kind (FVIE2). Many research papers dealing with fuzzy integral equations exists in open literatures and some of them are reviewed and cited here for better understanding of the present analysis. We know that solving fuzzy integral equations requires appropriate definitions of fuzzy function and fuzzy integral of a fuzzy function. We refer to the reader (Dubois and Prade 1978) for basic arithmetic operations on fuzzy numbers and he also presented the elementary fuzzy calculus (Dubois and Prade 1982) based on extension principle. Alternative approaches were later suggested by others (Goetschel and Voxman 1986; Kaleva 1987). Various methods and applications of linear and nonlinear integral equations were also reported (Wazwaz 2011). The fuzzy integral equations (Friedman et al. 1999; Subrahmanyam and Sudarsanam 1996) is one of the most important fields of fuzzy set theory. The concepts of fuzzy integral equations and fuzzy integrodifferential equations have motivated a large amount of research works (Mirzaee et al. 2012;Â Hussain and Ali 2013a, b; Mosleh and Otadi 2013; Allahviranloo et al. 2014; Otadi and Mosleh 2014) in last decades because of its applications in scientific phenomena. Therefore investigating fuzzy integral equations by finding accurate and efficient methods for solving these equations has become an active research undertaking.
Homotopy perturbation method (He 1999) is a perturbation technique coupled with the homotopy technique was developed by He JH and was further improved by him (He 2000, 2003, 2004). In homotopy perturbation method, a complicated problem under study is continuously deformed into a simple problem which is easy to solve to obtain an approximate solution. HPM yields a very rapid convergence of the solution series in most cases, usually only few iterations lead to very accurate solutions. This method has been applied to many problems (Chun 2007; FilobelloNino et al. 2014b) and also in particular to integral equations (Abbasbandy 2007) and fuzzy integral equations (Matinfar and Saeidy 2010). In recent past, various analytical and numerical methods were used such as Adomian decomposition method, direct computation method, series solution method, successive approximation method and conversion to integral equations. However, these methods are not easy to use and require tedious calculations. But HPM was proved (SaberiNadjafi and Ghorbani 2009) to be an effective and reliable tool for handling most of the linear and nonlinear differential, ordinary and partial, as well as linear and nonlinear integral equations. The main reason consists in the fact that this gives flexibility in the choice of basis functions for the solution and does not involve linear inversion operators (as compared to the Adomian decomposition method), while still retaining a simplicity that makes the method easily understandable from the standpoint of generalÂ perturbation methods.
Fuzzy integral equations are important in studying and solving a large proportion of problems in various topics in applied mathematics and it also arise in many industrial fields, such as electromagnetic fields and thermal problems. It is wellknown that a large class of initial and boundary value problems can be converted to Volterra integral equations and also the problem of heat conduction with a variable heat transfer coefficient is reduced to the solution of Volterra integral equations of the second kind. Several authors applied HPM to solve linear and nonlinear partial differential equations of fractional order (Momani and Odibat 2007a, b), Volterra integral equations (Grover and Tomer 2011), singularly perturbed Volterra integral equations (Alnasr and Momani 2008) and nth order fuzzy linear differential equations (Tapaswini and Chakraverty 2013). The reader is also referred to two recent papers where the authors (FilobelloNino et al. 2014a; VazquezLeal and SarmientoReyes 2015) make use of HPM in their application oriented works. In the very recent studies, it has been found that homotopy perturbation method was also used to solve fractional fisherâ€™s equation (Hamdi Cherif et al. 2016) and a system of nonlinear chemistry problems (Ramesh Rao 2016). Taking into account all these specifications, the proposed work is mainly dealt with fuzzy Volterra integral equations. Up to authorsâ€™ knowledge no research has been carried out so far by constructing a numerical algorithm using HPM for solving fuzzy Volterra integral equations of the second kind and our work is aimed at these.
The paper is organized as follows: the fuzzy Volterra integral equations are recalled with the required theorem and the basic concept of HPM is provided. Our main results are stated in the numerical technique, which presents the detailed description of the proposed method. Besides we introduce the algorithm for solving FVIE2. Additionally test problems are included in the section â€˜Illustrative examplesâ€™ and concluding results are made over them. Finally, a brief conclusion is provided.
Remark
Some of the basic notations and definitions that are not described in this paper are standard and usual. One can refer to the authors (Dubois and Prade 1978; Goetschel and Voxman 1983, 1986; Puri and Ralescu 1983, 1986; Kaleva 1987) for the notion of fuzzy numbers and its arithmetic operations, Hausdorff distance between fuzzy numbers, continuity and existence of definite integral of fuzzy functions.
Fuzzy Volterra integral equations
The integral equations which are discussed in this section are the Volterra integral equations of the second kind. The Fredholm integral equation of the second kind (Hochstadt 1973) is given by
where Î»Â >Â 0, a and b are constants, k(x,Â t) is an arbitrary kernel function over the square aÂ â‰¤Â x, tÂ â‰¤Â b and f(x) is a function of aÂ â‰¤Â xÂ â‰¤Â b.
If the kernel satisfies k(x,Â t)Â =Â 0, xÂ >Â t,
we obtain the Volterra integral equation of the second kind and is given by
In the above equation a refers to constant and x is a variable. If f(x) is a crisp function, then the solutions of the following equations
are crisp as well, where \(\left( {\underline{f} \left( {x;\alpha } \right),\overline{f} \left( {x;\alpha } \right)} \right)\) is the parametric form of f(x). However, if f(x) is a fuzzy function these equations possess fuzzy solutions.
Let F: IÂ â†’Â E ^{1} be a fuzzy function. The integral of F over I, denoted by \(\int_{I} {F\left( x \right)dx} {\text{ or }}\int_{a}^{b} {F\left( x \right)dx}\), is defined levelwise by
for each 0Â â‰¤Â Î±Â â‰¤Â 1.
Now the parametric form of fuzzy Volterra integral equation of the second kind (FVIE2) are as follows (Ghanbari 2010)
Let (\(\underline{f} \left( {x,\alpha } \right)\), \(\overline{f} \left( {x;\alpha } \right)\)) and (\(\underline{F} \left( {x;\alpha } \right)\), \(\overline{F} \left( {x;\alpha } \right)\)), be the parametric forms of f(x) and F(x) respectively, 0Â â‰¤Â Î±Â â‰¤Â 1, aÂ â‰¤Â xÂ â‰¤Â b. Then the parametric forms of FVIE2 are given by
for 0Â â‰¤Â Î±Â â‰¤Â 1, where
and
The following theorem provides sufficient conditions for the existence of a unique solution to Eq.Â (1), where f(x) is a fuzzy function.
Theorem
(Wu and Ma 1990) Let \(k\left( {x, t} \right)\) be continuous for aÂ â‰¤Â x, tÂ â‰¤Â b, Î»Â >Â 0, and f(x) is a fuzzy continuous function of x, aÂ â‰¤Â xÂ â‰¤Â b. If \(\lambda < \frac{1}{{M\left( {b  a} \right)}}\) , where \(M = \begin{array}{*{20}c} {\hbox{max} } \\ {\left( {a \le x, t \le b} \right)} \\ \end{array} \left {k\left( {x,t} \right)} \right\), then the iterative procedure
converges to the unique solution of Eq.Â (1). Specifically,
where LÂ =Â Î»M(bÂ âˆ’Â a). This refers that F _{ k }(x) converges uniformly in x to F(x), i.e., given arbitrary É›Â >Â 0 we can find N such that
The proof this theorem can be easily extended for fuzzy Volterra integral equations of the second kind, i.e., for Eq.Â (2) where f(x) is a fuzzy function as well.
Basic concept of homotopy perturbation method and its application to Volterra integral equations
For convenience of the reader, we present the review of the HPM. The essential idea of this method is to introduce a homotopy parameter, say p, which takes the values from 0 to 1. When pÂ =Â 0, the system of equation usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p gradually increases to 1, the system goes through a sequence of deformation, the solution of each of which is close to that at the previous stage of deformation. Eventually at pÂ =Â 1, the system takes the original form of the equation and final stage of deformation gives the desired solution.
To figure out how HPM works, consider the nonlinear integral equation
with boundary conditions \(B\left( {u, \frac{\partial u}{\partial n}} \right) = 0\), \(r \epsilon \varGamma\),
where L is a linear operator, N is a nonlinear operator, B is a boundary operator, Î“ is the boundary of the domain \(\varvec{\varOmega}\) and f(r) is a known analytic function. In order to use the HPM, a suitable construction of homotopy is of vital importance. If L(u)Â =Â 0 with some possible unknown parameter can best describe the original nonlinear system. Generally, a homotopy can be constructed as (He 1999, 2000)
or
where \(p \epsilon [0, 1]\) and \(r \epsilon {\varvec\varOmega }\).
The nonlinear Volterra integral equation is given by
where Î³(x) is an unknown function that will be determined, k(x,Â t) is the kernel of the integral equation, f(x) is an analytic function, R(Î³) and N(Î³) are linear and nonlinear function of Î³ respectively.
To illustrate the HPM, we reconstitute Eq.Â (14) as
with the solution u(x)Â =Â Î³(x) and we define the homotopy H(U,Â p) by
where F(u) is a functional operator with solution, say u _{0}, which can be obtained easily. We can choose a homotopy
and continuously trace an implicitly defined curve from a starting point \(H\left( {u_{0} , 0} \right)\) to a solution function H(Î³,Â 1). The embedding parameter p monotonically increases from 0 to 1 as the trivial problem F(u)Â =Â 0 continuously deformed to the original problem L(u)Â =Â 0. The embedded parameter \(p \epsilon [0, 1]\) can be considered as an expanding parameter.
When pÂ â†’Â 1, Eq.Â (17) corresponds to Eq.Â (15) and Eq.Â (18) becomes the approximate solution of Eq.Â (14), i.e.,
The above series is convergent in most of the cases and also the rate of convergence depends on L(u) (He 2000).
In other words, it is very natural to assume that the solution of [Eqs.Â (16) and (17)] can be expressed as
Equating the terms with identical powers of p, we get
Therefore when pÂ =Â 1, the approximate solution of above equation can be readily obtained as follows
The above series is convergent for most cases.
Proposed numerical technique for solving FVIE2 using HPM
As HPM is considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to small parameters as occur with traditional perturbation methods, because HPM neither requires small parameter nor linearization, but only few iterations to obtain highly accurate solution. Hence we introduce the recursive scheme for solving FVIE2 as follows
In view of homotopy, we can define the following convex homotopy,
where Î± is the fuzzy parameter (0Â â‰¤Â Î±Â â‰¤Â 1).
The solution of Eq.Â (23) is assumed in the following form
where \(\left( {\underline{V}_{i} , \bar{V}_{i} } \right) \forall i\) are unknown functions to be determined.
The initial approximation is taken as
Substituting Eqs.Â (24) and (25) in Eq.Â (23) reduces to
Equating the coefficients with like powers of p, we yield the following iterations
Finally, the solution of FVIE2 is given by
In general, we obtain the following iteration formulae
These relations [Eq.Â (31)] will enable us to determine the components \(\underline{u} \left( {x, \alpha } \right)\) and \(\bar{u}\left( {x, \alpha } \right)\) recursively for iÂ â‰¥Â 0.
Algorithm of the approach
On the basis of our proposed results using HPM, which are discussed in the above section, we convert to a numerical HPM algorithm that is one of the main results of this paper. Hence, in an algorithmic form, HPM can be expressed and implemented as follows.
The Eqs.Â (5) and (6) with the iteration index iÂ â‰¥Â 0, and a appropriate value for the tolerance (here \(Tol = 10^{  5}\)).
Main steps
 Step 1::

For reckoning the initial data, let \(\underline{V}_{0} = \underline{V}\) and \(\bar{V}_{0}\)Â =Â \(\bar{V}\) where xÂ =Â 0
 Step 2::

Use the recursive relations [Eq.Â (31)] to compute the values of \(\underline{V}_{i + 1} \left( x \right)\) by using \(\underline{V}_{i} \left( x \right)\) and \(\bar{V}_{i + 1} \left( x \right)\) by using \(\bar{V}_{i} \left( x \right)\)
 Step 3::

If \(\left\{ {\hbox{max} \left {\underline{V}_{i + 1} \left( x \right)  \underline{V}_{i} \left( x \right)} \right < Tol, max\left { \bar{V}_{i + 1} \left( x \right)  \bar{V}_{i} \left( x \right)} \right < Tol} \right\}\), then go to step 4, else set iÂ =Â iÂ +Â 1 and go to step 2
 Step 4::

Print \(\underline{u} \left( x \right) = \sum\nolimits_{i = 0}^{\infty } {\underline{V}_{i} }\) and \(\bar{u}\left( x \right) = \sum\nolimits_{i = 0}^{\infty } {\bar{V}_{i} }\) as the approximates of the exact solutions
Illustrative examples
To give a clear overview and to demonstrate the efficiency of the homotopy perturbation method, we implement the proposed numerical technique by solving a linear FVIE2 and a nonlinear FVIE2 with known exact solutions.
Linear fuzzy Volterra integral equations of the second kind
Example 1
We consider the linear FVIE2 given by
where Î»Â =Â 1, aÂ =Â 0, k(x,Â t)Â =Â sinhx and
The exact solution in this case is given by
Here we have
To solve the given equation by HPM we construct a convex homotopy as follows
Now we make use of the iteration formulae [Eq.Â (31)] and applying the numerical algorithm, the HPM solution series with few iterative terms are as follows
\(p^{0} :( \underline{V}_{0} \left( {x,\alpha } \right), \bar{V}_{0} \left( {x,\alpha } \right)\)) is the initial fuzzy approximations,where
where
In the same way, the iterations \(p^{2} :\left( { \underline{V}_{2} \left( {x,\alpha } \right), \bar{V}_{2} \left( {x,\alpha } \right)} \right)\) and \(p^{3} :\left( {\underline{V}_{3} \left( {x,\alpha } \right), \bar{V}_{3} \left( {x,\alpha } \right)} \right)\) were also computed for calculating fuzzy approximate solutions which could not be stated here as the iterations contain very long expressions.
We approximate \(\underline{F} \left( {x,\alpha } \right)\) and \(\bar{F}\left( {x,\alpha } \right)\) by setting pÂ =Â 1 and obtain the following fuzzy solutions, which are given by
and
Error analysis
Absolute errors are computed as
The components of iteration formulae [Eq.Â (31)] were obtained by the Mathematica program (Mathematica package version 5.2) according to the numerical algorithm where the tolerance is given a suitable positive value. The numerical results of the obtained approximate solutions are compared with the exact solutions for different Î± values and are presented in TableÂ 1. Moreover exact and approximate solutions are shown graphically in Fig.Â 1.
Nonlinear fuzzy Volterra integral equations of the second kind
Example 2
We consider the nonlinear FVIE2 given by
where Î»Â =Â 1, aÂ =Â 0, k(x,Â t)Â =Â x ^{2}(1Â âˆ’Â 2t) and
The exact solution in this case is given by
Here we have
To solve the given equation by HPM we construct a convex homotopy as follows
Now we make use of the iteration formulae [Eq.Â (31)] and applying the numerical algorithm, the HPM solution series with few iterative terms are as follows
\(p^{0} :\left( {\underline{V}_{0} \left( {x,\alpha } \right), \bar{V}_{0} \left( {x,\alpha } \right)} \right)\) is the initial fuzzy approximations,
where
where
In the same way, the iterations \(p^{2} :\left( {\underline{V}_{2} \left( {x,\alpha } \right), \bar{V}_{2} \left( {x,\alpha } \right)} \right)\) and \(p^{3} :\left( {\underline{V}_{3} \left( {x,\alpha } \right), \bar{V}_{3} \left( {x,\alpha } \right)} \right)\) were also computed for calculating fuzzy approximate solutions which could not be stated here as the iterations contain very long expressions.
We approximate \(\underline{F} \left( {x,\alpha } \right)\) and \(\bar{F}\left( {x,\alpha } \right)\) by setting pÂ =Â 1 and the obtained fuzzy solutions are given by
and
Error analysis
Absolute errors are computed as
By the same way of the previous example, we obtained the components of iteration formulae [Eq.Â (31)] by using the Mathematica program (Mathematica package version 5.2) according to the numerical algorithm where the tolerance is given a suitable positive value. The numerical results of the obtained approximate solutions are compared with the exact solutions for different Î± values and are presented in TableÂ 2. Moreover exact and approximate solutions are shown graphically in Fig.Â 2.
Concluding remarks on numerical results
By looking into the computed values presented above (TablesÂ 1, 2), one may conclude that the obtained approximate solutions by applying the proposed method (HPM) are in high agreement with the exact solutions. Furthermore, from the graphical representation (Figs.Â 1, 2), it is apparent that the plot of exact and approximate solutions merely coincide due to the occurrence of very minimum amount of errors (\(\underline{E} \left( {x,\alpha } \right)\) and \(\bar{E}\left( {x,\alpha } \right)\)). The fuzzy approximate solutions are calculated at four iterations for the numerical examples considered. The results presented clearly shows that the iterative steps number 3â€“4 in the numerical algorithm accompany for the convergence of the fuzzy solution so that the computations are minimized and the accuracy of the approximate values is also achieved with exceptionally few iteration and overall errors are reduced.
Conclusions
In this paper, we developed a simple yet versatile numerical technique using HPM and successfully applied it by means of the proposed algorithm for solving the linear and nonlinear fuzzy Volterra integral equations of the second kind. Mathematica has been used for computations in this paper. The examples analyzed illustrates that the answers are trusty and reveals the effectiveness of the proposed method. The reliability of HPM due to its precise results and the reduction in computation gives HPM a wider applicability. Moreover, it avoids the tedious work needed by the traditional numerical methods. Advantage of the proposed method lies in the free selection of initial approximation in a straightforward manner and also it overcomes the drawbacks of handling larger equations. This method does not involve any discretization of variables and hence it is free from rounding off errors. One can apply this method to higher order equations also.
Further research can be focused on studying the existence and convergence of HPM for linear and nonlinear FVIE2. Finally we conclude that this work provides the applicable computational technique will aid the practical applications already used in the design of various fuzzy dynamical systems and it is an issue of considerable importance at this current trend due to its versatile nature.
Abbreviations
 HPM:

homotopy perturbation method
 FVIE2:

fuzzy Volterra integral equation of the second kind
References
Abbasbandy S (2007) Application of Heâ€™s homotopy perturbation method to functional integral equations. Chaos, Solitons Fractals 31(5):1243â€“1247
Allahviranloo T et al (2014) An application of a semianalytical method on linear fuzzy Volterra integral equations. J Fuzzy Set Valued Anal 2014:1â€“15
Alnasr MH, Momani Shaher (2008) Application of homotopy perturbation method to singularly perturbed Volterra integral equations. J Appl Sci 8(6):1073â€“1078
Chun C (2007) Integration using Heâ€™s homotopy perturbation method. Chaos, Solitons Fractals 34(4):1130â€“1134
Dubois D, Prade H (1978) Operations on fuzzy numbers. J Syst Sci 9:613â€“626
Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets Syst 8:1â€“7
FilobelloNino U et al (2014a) A handy approximate solution for a squeezing flow between two infinite plates by using of laplace transformhomotopy perturbation method. SpringerPlus 3:421
FilobelloNino U et al (2014b) Nonlinearities distribution laplace transform homotopy perturbation method. SpringerPlus 3:594
Friedman M, Ma M, Kandel A (1999) Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst 106:35â€“48
Ghanbari M (2010) Numerical solution of fuzzy linear Volterra integral equations of the second kind by homotopy analysis method. Int J Ind Math 2(2):73â€“87
Goetschel R, Voxman W (1983) Topological properties of fuzzy numbers. Fuzzy Sets Syst 10:87â€“99 Â
Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31â€“43
Grover M, Tomer A (2011) A new approach to evaluate second kind of Volterra integral equation with homotopy perturbation method and variational iteration method. IRACSTEng Sci Technol: Int J 1(1):24â€“29
Hamdi Cherif M et al (2016) Homotopy perturbation method for solving the fractional fisherâ€™s equation. Int J Anal Appl 10(1):9â€“16
He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178(3â€“4):257â€“262
He JH (2000) A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 35(1):37â€“43
He JH (2003) Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 135(1):73â€“79
He JH (2004) Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 156(2):527â€“539
Hochstadt H (1973) Integral equations. Wiley, Newyork
Hussain Eman A, Ali Ayad W (2013a) Homotopy analysis method for solving fuzzy integrodifferential equations. Mod Appl Sci 7(3):15â€“25
Hussain Eman A, Ali Ayad W (2013b) Homotopy analysis method for solving nonlinear fuzzy integral equations. Int J Appl Math 28(1):1177â€“1189
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24(2):301â€“317
Matinfar M, Saeidy M (2010) Application of homotopy perturbation method for fuzzy integral equations. J Math Comput Sci 1(4):377â€“385
Mirzaee F et al (2012) Numerical solution of Fredholm fuzzy integral equations of the second kind via direct method using triangular functions. J Hyperstruct 1(2):46â€“60
Momani S, Odibat Z (2007a) Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput Math Appl 54(7â€“8):910â€“919
Momani S, Odibat Z (2007b) Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys Lett A 365(5â€“6):345â€“350
Mosleh M, Otadi M (2013) Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. J Adv Inf Technol 4(3):148â€“155
Otadi M, Mosleh M (2014) Numerical solutions of fuzzy nonlinear integral equations of the second kind. Iran J Fuzzy Syst 1(1):135â€“145
Puri ML, Ralescu D (1983) Differential for fuzzy function. J Math Anal Appl 91:552â€“558
Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409â€“422
Ramesh Rao TR (2016) A comparison between the differential transform method and homotopy perturbation method for a system of non linear chemistry problems. Indian J Appl Res 6(2):171â€“180
SaberiNadjafi J, Ghorbani A (2009) Heâ€™s homotopy perturbation method: an effective tool for solving integral and integrodifferential equations. Comput Math Appl 58:2379â€“2390
Subrahmanyam PV, Sudarsanam SK (1996) A note on fuzzy Volterra intergral equations. Fuzzy Sets Syst 81(2):188â€“191
Tapaswini S, Chakraverty S (2013) Numerical solution of nth order fuzzy linear differential equations by homotopy perturbation method. Int J Comput Appl 64(6):5â€“10
VazquezLeal H, SarmientoReyes A (2015) GHM method for obtaining rational solutions of nonlinear differential equations. SpringerPlus 4:241. doi:10.1186/s400640151011x
Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Springer, Berlin, Heidelberg, Newyork
Wu C, Ma M (1990) On the integrals, series and integral equations of fuzzy set valued functions. J Harbin Inst Technol 21:11â€“19
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338â€“353
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Narayanamoorthy, S., Sathiyapriya, S.P. Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind. SpringerPlus 5, 387 (2016). https://doi.org/10.1186/s4006401620383
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DOI: https://doi.org/10.1186/s4006401620383