Hindawi Journal of Applied Mathematics Volume 2017, Article ID 2417195, 12 pages https://doi.org/10.1155/2017/2417195
Research Article Computational Methods for Solving Linear Fuzzy Volterra Integral Equation Jihan Hamaydi and Naji Qatanani Department of Mathematics, An-Najah National University, Nablus, State of Palestine Correspondence should be addressed to Naji Qatanani;
[email protected] Received 21 February 2017; Accepted 18 April 2017; Published 28 May 2017 Academic Editor: Mehmet Sezer Copyright ยฉ 2017 Jihan Hamaydi and Naji Qatanani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.
1. Introduction Fuzzy integral equations of the second kind have attracted the attention of many scientists and researchers in recent years. These equations appear frequently in fuzzy control, fuzzy finance, approximate reasoning, and economic systems [1]. The concept of fuzzy sets was originally introduced by Zadeh [2] and led to the definition of fuzzy numbers and its implementation in fuzzy control [3] and approximate reasoning problems [4]. Dubois and Prade [5] were the first to introduce the concept of fuzzy functions. Alternative approaches were later suggested by Goetschel and Voxman [6], Kaleva [7], Nanda [8], and others. In recent years, numerous methods have been proposed for solving Volterra integral equations [9]. Tricomi [10] was the first to introduce the successive approximations method for nonlinear integral equations. Liao [11] employed the homotopy analysis method to solve nonlinear problems and then it has been applied by Abbasbandy [12] to solve fuzzy Volterra integral equations of the second kind. Babolian et al. [13] have used the orthogonal triangular functions as a direct method for numerically solving integral equations system. Jafarian et al. [14] have solved systems of linear integral equations by using Legendre wavelets. Kanwal and Liu [15] implemented the Taylor expansion approach for
solving integral equations while Xu [16] solved integral equations by the variational iteration method. In addition, Amawi and Qatanani [17, 18] have investigated the analytical and numerical treatment of fuzzy linear Fredholm integral equation. Hamaydi [19] has used various analytical and numerical methods to solve fuzzy Volterra integral equations. The paper is organized as follows: in Section 2, fuzzy Volterra integral equation of the second kind is introduced. The Taylor expansion method used to approximate solution of fuzzy Volterra integral equation is addressed in Section 3. In Section 4, we present the variational iteration method (VIM) which provides a sequence of functions that converges to the exact solution to the problem. The proposed methods are implemented using a numerical example with known exact solution by applying the MAPLE software in Section 5. Conclusions are given in Section 6.
2. Fuzzy Volterra Integral Equation A standard form of the Volterra integral equation of the second kind is given by [14] ๐ฅ
๐ข (๐ฅ) = ๐ (๐ฅ) + ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก) ๐๐ก, ๐
(1)
2
Journal of Applied Mathematics
where ๐ is a positive parameter, ๐(๐ฅ, ๐ก) is an arbitrary function called the kernel of the integral equation defined over square ๐บ : [๐, ๐] โ [๐, ๐], ๐(๐ฅ, ๐ก) = 0, ๐ โค ๐ก, ๐ฅ โค ๐, ๐ก > ๐ฅ, and ๐(๐ฅ) is a given function of ๐ฅ โ [๐, ๐]. If๐(๐ฅ) is a fuzzy function, then (1) is called a fuzzy Volterra integral equation of the second kind. Definition 1 (see [20]). The second kind fuzzy Volterra integral equations system is of the following form:
From (2), (3), and (4) we have the following system: ๐
๐๐,๐
๐ข๐ (๐ฅ, ๐ผ) = ๐๐ (๐ฅ, ๐ผ) + โ ๐ ๐,๐ (โซ ๐๐,๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก ๐=1
๐
๐ฅ
+ โซ ๐๐,๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก) , ๐๐,๐
๐
๐
๐ข๐ (๐ฅ, ๐ผ) = ๐๐ (๐ฅ, ๐ผ) + โ๐ ๐,๐ (โซ ๐๐,๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก
๐ฅ
๐ข๐ (๐ฅ) = ๐๐ (๐ฅ) + โ ๐ ๐๐ โซ ๐๐๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก) ๐๐ก, ๐=1
(5)
๐๐,๐
๐
(2)
๐
๐=1
๐ฅ
+ โซ ๐๐,๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก) . ๐๐,๐
where ๐ โค ๐ก โค ๐ฅ โค ๐ and ๐ ๐๐ =ฬธ 0 (for ๐, ๐ = 1, 2, 3, . . . , ๐) are real constants. Moreover, in system (2), the fuzzy function ๐๐ (๐ฅ) and kernel ๐๐๐ (๐ฅ, ๐ก) are given and assumed to be sufficiently differentiable with respect to all their arguments on the interval ๐ โค ๐ก, ๐ฅ โค ๐, and we assume that the kernel function ๐๐๐ (๐ฅ, ๐ก) โ ๐บ : [๐, ๐] โ [๐, ๐], and ๐ข๐ (๐ฅ) = [๐ข1 , ๐ข2 , . . . , ๐ข๐ ]๐ก is the solution to be determined. Now let (๐๐ (๐ฅ, ๐ผ), ๐๐ (๐ฅ, ๐ผ)) and (๐ข๐ (๐ฅ, ๐ผ), ๐ข๐ (๐ฅ, ๐ผ)), 0 โค ๐ผ โค 1, ๐ฅ โ [๐, ๐], be the parametric forms of ๐๐ (๐ฅ) and ๐ข๐ (๐ฅ), respectively; then the parametric forms of the fuzzy Volterra integral equations system are as follows: ๐
๐ฅ
๐ข๐ (๐ฅ, ๐ผ) = ๐๐ (๐ฅ, ๐ผ) + โ (๐ ๐๐ โซ ๐ (๐ก, ๐ผ) ๐๐ก) , ๐=1 ๐
๐
๐๐
๐ฅ
๐ข๐ (๐ฅ, ๐ผ) = ๐๐ (๐ฅ, ๐ผ) + โ (๐ ๐๐ โซ ๐๐๐ (๐ก, ๐ผ) ๐๐ก) , ๐=1
(3)
๐
๐ = 1, 2, . . . , ๐,
We seek the solution of system (5) in the form of ๓ตจ๓ตจ (๐) 1 ๐ ๐ข๐๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ (๐ก โ ๐ง)๐ ) , ๐ข๐,๐ (๐ก, ๐ผ) = โ ( ๓ตจ๓ตจ ๐! ๐ ๓ตจ๓ตจ๐ฅ=๐ง ๐=0 ๓ตจ (๐) ๐ 1 ๐ ๐ข๐๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ๓ตจ ๐ ๐ข๐,๐ (๐ก, ๐ผ) = โ ( ๓ตจ๓ตจ (๐ก โ ๐ง) ) ๓ตจ๓ตจ ๐! ๐๐ฅ๐ ๐=0 ๓ตจ๐ฅ=๐ง ๐
๐ โค ๐ฅ, ๐ง โค ๐, for ๐ = 1, . . . , ๐, which are the Taylor expansion of degree ๐ at ๐ฅ = ๐ง for the unknown functions ๐ข๐๐ (๐ฅ, ๐ผ) and ๐ข๐๐ (๐ฅ, ๐ผ), respectively. To obtain the solution in the form of expression (6) we find the ๐th derivative of each equation in system (5) with respect to ๐ฅ by using the Leibniz rule, (๐ = 0, 1, . . . , ๐ ) and obtain [21] (๐) ๐(๐) ๐ข๐๐ (๐ฅ, ๐ผ) ๐ ๐๐ (๐ฅ, ๐ผ) = ๐๐ฅ๐ ๐๐ฅ๐
where {๐๐๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) , ๐๐๐ (๐ฅ, ๐ก) โฅ 0 ๐๐๐ (๐ก, ๐ผ) = { ๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) , ๐๐๐ (๐ฅ, ๐ก) < 0, { ๐๐
๐ { ๐๐,๐ ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) + โ ๐ ๐,๐ {โซ ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก ๐๐ฅ๐ ๐ ๐=1 {
(4)
{๐๐๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) , ๐๐๐ (๐ฅ, ๐ก) โฅ 0 ๐ (๐ก, ๐ผ) = { ๐๐ ๐ ๐ก) ๐ข๐ (๐ก, ๐ผ) , ๐๐๐ (๐ฅ, ๐ก) < 0. { ๐๐ (๐ฅ,
3. Taylor Expansion Method This method depends on differentiating the fuzzy integral equation of the second kind ๐ โ times; then substitute the Taylor series expansion for the unknown function into the integral equation. As a result, we get a linear system for which the solution of this system yields the unknown Taylor coefficients of the solution functions.
(6)
(๐โ๐โ1) ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๐ข๐๐ (๐ฅ, ๐ผ)) +โ( ๓ตจ๓ตจ ๓ตจ๓ตจ๐ก=๐ฅ ๐๐ฅ๐ ๐=0 ๐โ1
+โซ
๐ฅ
๐(๐) ๐๐,๐ (๐ฅ, ๐ก)
๐๐,๐
๐(๐) ๐ข๐๐ (๐ฅ, ๐ผ) ๐๐ฅ๐
๐๐ฅ๐ =
} ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก} , }
๐(๐) ๐๐ (๐ฅ, ๐ผ) ๐๐ฅ๐
๐ { ๐๐,๐ ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) + โ ๐ ๐,๐ {โซ ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก ๐๐ฅ๐ ๐ ๐=1 {
Journal of Applied Mathematics
3
(๐โ๐โ1) ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๐ข๐๐ (๐ฅ, ๐ผ)) +โ( ๓ตจ๓ตจ ๓ตจ๓ตจ๐ก=๐ฅ ๐๐ฅ๐ ๐=0 ๐โ1
๐(๐) ๐๐,๐ (๐ฅ, ๐ก)
๐ฅ
+โซ
๐๐ฅ๐
๐๐,๐
Substituting (9) into (8) gives
} ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก} , } (7)
for ๐ = 0, 1, . . . , ๐ and ๐ = 1, 2, . . . , ๐. Using the Leibniz rule which is dealing with differentiation of product of functions, system (7) becomes ๐(๐) ๐๐ (๐ฅ, ๐ผ)
(๐)
๐ ๐ข๐๐ (๐ฅ, ๐ผ) = ๐๐ฅ๐
๐๐ฅ๐
๐
๐โ1
๐=1
๐=0
(๐) (๐,๐) (๐) ๐ข๐๐ (๐ง, ๐ผ) ๐ข(๐) (๐ง, ๐ผ) + โ { โ ๐ท๐,๐ ๐,๐ (๐ง, ๐ผ) = ๐๐ ๐
๐
๐=0
๐=0
(๐,๐)
(๐,๐) (๐) + โ๐ธ๐,๐ ๐ข๐๐ (๐ง, ๐ผ) + โ๐ธ๓ธ ๐,๐ ๐ข(๐) ๐๐ (๐ง, ๐ผ)} ,
๐ข(๐) ๐,๐ (๐ง, ๐ผ) = ๐๐
๐
+ โ ๐ ๐,๐
(๐)
๐
๐โ1
๐=1
๐=0
(10)
(๐,๐) (๐) ๐ข๐๐ (๐ง, ๐ผ) (๐ง, ๐ผ) + โ { โ ๐ท๐,๐
๐
๐
๐=0
๐=0
(๐,๐)
(๐,๐) (๐) + โ๐ธ๐,๐ ๐ข๐๐ (๐ง, ๐ผ) + โ๐ธ๓ธ ๐,๐ ๐ข(๐) ๐๐ (๐ง, ๐ผ)} ,
๐=1
(๐)
{ ๐๐,๐ ๐ ๐๐,๐ (๐ฅ, ๐ก) โ
{โซ ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก ๐๐ฅ๐ ๐ {
where
(๐โ๐โ๐โ1) ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) ๓ตจ๓ตจ๓ตจ๓ตจ ๐โ๐โ1 ๓ตจ๓ตจ ) +โ โ ( )( ๓ตจ๓ตจ ๐ ๓ตจ๓ตจ๐ก=๐ฅ ๐๐ฅ๐ ๐=0 ๐=0 ๐โ1 ๐โ๐โ1
(๐)
โ
(๐ข๐๐ (๐ฅ, ๐ผ))
+โซ
๐ฅ
(๐)
๐ ๐๐,๐ (๐ฅ, ๐ก) ๐๐ฅ๐
๐๐,๐
(๐)
(๐)
} ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก} , }
๐
(๐,๐) ๐ธ๐,๐
(8)
๐ ๐ข๐๐ (๐ฅ, ๐ผ) ๐ ๐๐ (๐ฅ, ๐ผ) = + โ๐ ๐,๐ ๐๐ฅ๐ ๐๐ฅ๐ ๐=1
(๐,๐) ๐ธ๓ธ ๐,๐
=
=
๐ ๐,๐ ๐! ๐ ๐,๐ ๐!
โซ
๐๐,๐
๐(๐) ๐๐,๐ ๐๐ฅ๐
๐ ๐
๐(๐) ๐๐,๐
๐๐,๐
๐๐ฅ๐
โซ
๓ตจ๓ตจ ๓ตจ๓ตจ (๐ฅ, ๐ก)๓ตจ๓ตจ๓ตจ (๐ก โ ๐ง)๐ ๐๐ก, ๓ตจ๓ตจ ๓ตจ๐ฅ=๐ง ๓ตจ๓ตจ ๓ตจ๓ตจ (๐ฅ, ๐ก)๓ตจ๓ตจ๓ตจ (๐ก โ ๐ง)๐ ๐๐ก, ๓ตจ๓ตจ ๓ตจ๐ฅ=๐ง (11)
(๐,๐) ๐ท๐,๐
{ ๐๐,๐ ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) โ
{โซ ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก ๐๐ฅ๐ ๐ {
๐โ๐
๐โ๐โ1
๐=0
๐
= ๐ ๐,๐ โ (
(๐โ๐โ๐โ1) ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) ๓ตจ๓ตจ๓ตจ๓ตจ ๐โ๐โ1 ๓ตจ๓ตจ ) +โ โ ( )( ๓ตจ๓ตจ ๐ ๓ตจ๓ตจ๐ก=๐ฅ ๐๐ฅ๐ ๐=0 ๐=0
)(
๐(๐) ๐๐,๐ ๐๐ฅ๐
(๐โ๐โ๐โ1) ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ , (๐ฅ, ๐ก)๓ตจ๓ตจ ) ๓ตจ๓ตจ ๓ตจ๐ฅ=๐ง
๐โ1 ๐โ๐โ1
(๐)
โ
(๐ข๐๐ (๐ฅ, ๐ผ))
+โซ
๐ฅ
๐(๐) ๐๐,๐ (๐ฅ, ๐ก)
๐๐,๐
๐๐ฅ
๐
๐, ๐ = 1, 2, . . . , ๐,
} ๐ข๐๐ (๐ก, ๐ผ) ๐๐ก} . }
๐โ1 (๐,๐) (๐) for ๐ = 0, โ๐ ๐=1 โ๐=0 ๐ท๐,๐ ๐ข๐๐ (๐ง, ๐ผ) = 0, for ๐ โค ๐, we have
Our objective is to determine the coefficients ๐ข๐๐ (๐) (๐ง, ๐ผ) and
(๐,๐) = 0, and ๐, ๐ = 0, 1, . . . , ๐ . ๐ท๐,๐ Consequently, (10) can be written in the matrix form:
๐ข๐๐ (๐) (๐ง, ๐ผ), for ๐ = 0, . . . , ๐ and ๐ = 1, . . . , ๐ in system (7); thus we expand ๐ข๐๐ (๐ก, ๐ผ) and ๐ข๐๐ (๐ก, ๐ผ) in Taylor's series at arbitrary point ๐ง: ๐ โค ๐ง โค ๐ ๓ตจ๓ตจ (๐) 1 ๐ ๐ข๐,๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ โ
(๐ก โ ๐ง)๐ ) , ๐ ๓ตจ๓ตจ ๐! ๐๐ฅ ๓ตจ๓ตจ๐ฅ=๐ง ๐=0 ๓ตจ (๐) ๐ 1 ๐ ๐ข๐,๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ๓ตจ ๐ ๐ข๐,๐ (๐ก, ๐ผ) = โ ( ๐ ๓ตจ๓ตจ๓ตจ โ
(๐ก โ ๐ง) ) , ๐! ๐๐ฅ ๓ตจ๓ตจ๐ฅ=๐ง ๐=0 ๐
where
๐ข๐,๐ (๐ก, ๐ผ) = โ (
๐ โค ๐ฅ, ๐ง โค ๐, 0 โค ๐ผ โค 1, for ๐ = 1, . . . , ๐.
(๐ท + ๐ธ) ๐ = F,
(9)
๓ตจ๓ตจ ๐น = [โ๐ (๐ง, ๐ผ) , . . . , โ๐(๐) (๐ฅ, ๐ผ)๓ตจ๓ตจ๓ตจ , โ๐1 (๐ง, ๐ผ) , . . . , 1 1 ๓ตจ๐ฅ=๐ง ๓ตจ ๓ตจ๓ตจ (๐) ๓ตจ โ๐1 (๐ฅ, ๐ผ)๓ตจ๓ตจ๓ตจ๓ตจ , . . . , โ๐ (๐ง, ๐ผ) , . . . , โ๐(๐) (๐ฅ, ๐ผ)๓ตจ๓ตจ๓ตจ , ๐ ๐ ๓ตจ๐ฅ=๐ง ๓ตจ๐ฅ=๐ง ๐ก ๓ตจ๓ตจ (๐) โ๐๐ (๐ง, ๐ผ) , . . . , โ๐๐ (๐ฅ, ๐ผ)๓ตจ๓ตจ๓ตจ๓ตจ ] , ๓ตจ๐ฅ=๐ง
(12)
4
Journal of Applied Mathematics ๓ตจ๓ตจ ๓ตจ ๐ = [๐ข1๐ (๐ง, ๐ผ) , . . . , ๐ข(๐) 1๐ (๐ฅ, ๐ผ)๓ตจ๓ตจ๐ฅ=๐ง , ๐ข1๐ (๐ง, ๐ผ) , . . . , ๐ข(๐) 1๐
๓ตจ ๓ตจ๓ตจ ๓ตจ (๐ฅ, ๐ผ)๓ตจ๓ตจ๓ตจ๓ตจ๐ฅ=๐ง , . . . , ๐ข๐๐ (๐ง, ๐ผ) , . . . , ๐ข(๐) ๐๐ (๐ฅ, ๐ผ)๓ตจ๓ตจ๐ฅ=๐ง ,
(๐,๐)
๐ก ๓ตจ๓ตจ ๓ตจ ๐ข๐๐ (๐ง, ๐ผ) , . . . , ๐ข(๐) ๐๐ (๐ฅ, ๐ผ)๓ตจ๓ตจ๐ฅ=๐ง ] ,
๐ท(1,1) โ
โ
โ
๐ท(1,๐) [ ] [ ... ] ๐ท = [ ... ], d [ ] (๐,1) (๐,๐) โ
โ
โ
๐ท [๐ท ] ๐ธ
(1,1)
(๐,๐)
๐ท1,2 = ๐ท2,1
(๐,๐)
(๐,๐)
๐ท1,1 = ๐ท2,2
(1,๐)
โ
โ
โ
๐ธ
[ ] [ .. ] . ๐ธ = [ ... ] d . [ ] (๐,1) (๐,๐) โ
โ
โ
๐ธ [๐ธ ]
0 0 [ ๓ธ (๐,๐) [๐ 0 [ 1,0 [ [ . .. . =[ . [ . [ (๐,๐) [ ๓ธ (๐,๐) [๐ ๐ โ1,0 ๐๓ธ ๐ โ1,1 [ ๓ธ (๐,๐) ๓ธ (๐,๐) [ ๐ ๐ ,0 ๐ ๐ ,1 0 [ [0 [ [. =[ [ .. [ [0 [
โ
โ
โ
0
โ
โ
โ
0
d
.. .
0
โ
โ
โ
0
โ
โ
โ
(๐,๐) ๐๓ธ ๐ ,๐ โ1
0 โ
โ
โ
0 0 ] 0 โ
โ
โ
0 0] ] .. .. .. ] ] . d . .] ] 0 โ
โ
โ
0 0] ]
] 0] ] ] .. ] .] ], ] ] 0] ] 0]
.
[0 0 โ
โ
โ
0 0](๐ +1)ร(๐ +1)
(14) (13)
The Parochial matrices ๐ท(๐,๐) are defined by the following elements (see [14]):
(๐,๐)
(๐,๐)
๐ท1,1 ๐ท1,2 ๐ท(๐,๐) = [ (๐,๐) (๐,๐) ] , [๐ท2,1 ๐ท2,2 ] ๐ธ
(๐,๐)
(๐,๐)
(๐,๐)
(๐,๐)
(๐,๐)
๐ธ1,1 ๐ธ1,2
=[
๐, ๐ = 1, . . . , ๐, (๐,๐)
๐ธ1,1 = ๐ธ2,2 (๐,๐) ๐ โ [ 0,0 [ (๐,๐) [ ๐1,0
This method provides a sequence of functions, which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. Consider the following general nonlinear system [22]: ๐ฟ [๐ข (๐ฅ)] + ๐ [๐ข (๐ก)] = โ (๐ฅ) ,
1
[ [ . =[ [ .. [ [ (๐,๐) [ ๐๐ โ1,1 [ (๐,๐) [ ๐๐ ,0 (๐,๐)
Theorem 2 (see [14]). Let the kernel be bounded and belong to ๐บ:[๐, ๐] โ [๐, ๐] and ๐ข๐,๐ (๐ฅ, ๐ผ), ๐ข๐,๐ (๐ฅ, ๐ผ) (for ๐ = 1, . . . , ๐) are Taylor polynomials of degree ๐ , and their coefficients are computed by solving linear system (12); then these polynomials converge to the exact solution of fuzzy system (2), when ๐ โ +โ.
4. Variational Iteration Method
],
[๐ธ2,1 ๐ธ2,2 ]
(๐,๐)
3.1. Convergence Analysis. One can show that the above numerical method converges to the exact solution of the fuzzy system (2) (see [14] for more details).
(๐,๐) ๐0,1 (๐,๐) ๐1,1 โ
โ
โ
โ
1 โ
โ
โ
.. .
(๐,๐) ๐0,๐ โ1 (๐,๐) ๐1,๐ โ1
(๐,๐) ๐0,๐ (๐,๐) ๐1,๐
.. .
.. .
d
(๐,๐)
โ
โ
โ
๐๐ โ1,๐ โ1 โ 1
(๐,๐)
โ
โ
โ
๐๐ โ1,1 ๐๐ ,1
(๐,๐)
(๐,๐)
๐๐ ,๐ โ1
(๐,๐)
๐๐ โ1,๐
] ] ] ] ] ], ] ] ] ] ]
(๐,๐) ๐๐ ,๐ โ 1]
[ [ ๓ธ (๐,๐) [ ๐ [ 1,0 [ [ .. =[ . [ [ (๐,๐) [ ๐๓ธ [ ๐ โ1,0 [ ๓ธ (๐,๐) [ ๐ ๐ ,0
๐ข๐+1 (๐ฅ) = ๐ข๐ (๐ฅ)
(16)
๐ข๐ (๐ง)] โ โ (๐ง)] ๐๐ง, + โซ ๐ (๐ง) [๐ฟ [๐ข๐ (๐ง)] + ๐ [ฬ
(๐,๐)
(๐,๐)
where ๐ฟ is a linear operator, ๐ is a nonlinear operator, and โ(๐ฅ) is a given continuous function. According to the variational iteration method, we can construct the following correction functional:
๐ฅ
๐ธ1,2 = ๐ธ2,1
๐๓ธ 0,0 โ 1
(15)
0
(๐,๐)
๐๓ธ 0,1
(๐,๐)
(๐,๐)
โ
โ
โ
๐๓ธ 0,๐ โ1
๐๓ธ 0,๐
๐๓ธ 1,1 โ 1 โ
โ
โ
๐๓ธ 1,๐ โ1
(๐,๐)
๐๓ธ 1,๐
(๐,๐)
.. . (๐,๐)
๐๓ธ ๐ โ1,1 (๐,๐) ๐๓ธ ๐ ,1
] ] ] ] ] ] .. .. ], d . . ] ] ๓ธ (๐,๐) ๓ธ (๐,๐) ] โ
โ
โ
๐ ๐ โ1,๐ โ1 โ 1 ๐ ๐ โ1,๐ ] ] ๓ธ (๐,๐) ๓ธ (๐,๐) โ
โ
โ
๐ ๐ ,๐ โ1 ๐ ๐ ,๐ โ 1] (๐,๐)
where ๐ = 0, 1, 3, . . ., ๐ is Lagrange multiplier which can be identified optimally via variational theory, ๐ข๐ is the ๐th approximate solution, and ๐ขฬ๐ is a restricted variation (i.e., ๐ฟฬ ๐ข๐ = 0). Next, we take the partial derivative to both sides of the Volterra integral equation (1) with respect to ๐ฅ, and we get ๐ข๓ธ (๐ฅ) = ๐๓ธ (๐ฅ) +
๐ ๐ฅ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก) ๐๐ก. ๐๐ฅ ๐
(17)
Journal of Applied Mathematics
5
Take
Now, using the variational iteration method and (22), we get the following iteration formulas:
๐ฅ
๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก) ๐๐ก, ๐๐ฅ ๐
(18)
as a restricted variation and use the variational iteration method in direction ๐ฅ, then we obtain the following iteration sequence:
๐ฅ
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ข๐ (๐ฅ, ๐ผ) โซ [๐ข๓ธ ๐ (๐ง, ๐ผ) โ ๐๓ธ (๐ง, ๐ผ) 0
โโซ
๐
๐ฅ
๐ง
๐ข๐+1 (๐ฅ) = ๐ข๐ (๐ฅ) + โซ ๐ (๐ง) [๐ข๐๓ธ (๐ง) โ ๐๓ธ (๐) 0
โ
๐ง
๐ โซ ๐ (๐ง, ๐ก) ๐ข๐ (๐ก) ๐๐ก] ๐๐ง, ๐๐ง ๐
โโซ (19)
(20)
therefore, we get the following stationary conditions:
1 + ๐ (๐ง)|๐ง=๐ฅ = 0;
(21)
hence, the Lagrange multiplier can be identified: ๐(๐ง) = โ1. If we substitute the value of the Lagrange multiplier into (19), we obtain the following iteration formula: ๐ฅ
๐ข๐+1 (๐ฅ) = ๐ข๐ (๐ฅ) โ โซ [๐ข๐๓ธ (๐ง) โ ๐๓ธ (๐ง) 0
๐ง
๐ โ โซ ๐ (๐ง, ๐ก) ๐ข๐ (๐ก) ๐๐ก] ๐๐ง. ๐๐ง ๐
(22)
โโซ
๐ถ
๐
โโซ
๐ง
๐
๐ข (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ)๐๐ก,
๐๐ (๐ง, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก โ ๐ (๐ง, ๐ง) ๐ข (๐ง, ๐ผ) ๐๐ง ๐๐ (๐ง, ๐ก) (๐ก, ๐ผ) ๐ข๐ (๐ก, ๐ผ) ๐๐ก] ๐๐ง, ๐๐ง
5. Numerical Examples and Results In this section, in order to examine the accuracy of the proposed methods, we have chosen one example of linear fuzzy integral equation of the second kind. Moreover, the numerical results will be compared with the exact solution. Example 3 (Taylor expansion method). Consider the following fuzzy linear Volterra integral equations: ๐ข (๐ฅ, ๐ผ) = (๐ผ) cot (๐ฅ) + โซ ๐๐ฅโ๐ก ๐ข (๐ก, ๐ผ) ๐๐ก, 0
๐ฅ
(23)
(26)
0 โค ๐ผ โค 1, 0 โค ๐ฅ โค
๐ . 4
The analytical solution of the above equation is given as
Suppose that the kernel ๐(๐ฅ, ๐ก) > 0 for ๐ โค ๐ก โค ๐ and ๐(๐ฅ, ๐ก) < 0 for ๐ โค ๐ก โค ๐ฅ, and then (23) becomes [12] ๐
3 1 2 ๐ข (๐ฅ, ๐ผ) = (๐ผ) ( cos (๐ฅ) + sin (๐ฅ) + ๐2๐ฅ ) , 5 5 5 3 1 2 ๐ข (๐ฅ, ๐ผ) = (2 โ ๐ผ) ( cos (๐ฅ) + sin (๐ฅ) + ๐2๐ฅ ) , 5 5 5
๐ข (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ) ๐๐ก ๐
๐ฅ
(24)
Expand the unknown functions in Taylor series at ๐ง = ๐/12 and implement the following algorithm.
๐
๐ฅ
+ ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ) ๐๐ก, ๐
for each 0 โค ๐ผ โค 1 and ๐ โค ๐ฅ โค ๐.
(27)
0 โค ๐ผ โค 1.
+ ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ) ๐๐ก, ๐ข (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ) ๐๐ก
(๐ง, ๐ผ) โ ๐ (๐ง, ๐ผ)
0
๐
๐
(25) ๓ธ
where ๐ = 0, 1, 2, . . . In virtue of (25), we can find a solution of (24) and hence obtain a fuzzy solution of the linear fuzzy Volterra integral equation of the second kind.
๐ข (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + ๐ โซ ๐ (๐ฅ, ๐ก) ๐ข (๐ก, ๐ผ)๐๐ก.
๐
[๐ข๓ธ ๐
๐ข (๐ฅ, ๐ผ) = (2 โ ๐ผ) cot (๐ฅ) + โซ ๐๐ฅโ๐ก ๐ข (๐ก, ๐ผ) ๐๐ก,
๐ฅ
๐ฅ
๐ฅ
๐ฅ
We apply the variational iteration method on fuzzy Volterra integral equation of the second kind:
๐
๐๐ (๐ง, ๐ก) (๐ก, ๐ผ) ๐ข๐ (๐ก, ๐ผ) ๐๐ก] ๐๐ง, ๐๐ง 0
๐ฅ
๓ตจ ๐๓ธ (๐ง)๓ตจ๓ตจ๓ตจ๓ตจ๐ง=๐ฅ = 0,
๐
๐๐ (๐ง, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก โ ๐ (๐ง, ๐ง) ๐ข๐ (๐ง, ๐ผ) ๐๐ง
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ข๐ (๐ฅ, ๐ผ) โ โซ
and finally we calculate variation with respect to ๐ข๐ ; notice that ๐ฟ๐ข๐ (0) = 0 yields ๓ตจ ๐ฟ๐ข๐+1 = ๐ฟ๐ข๐ + ๐๐ฟ๐ข๐ ๓ตจ๓ตจ๓ตจ๓ตจ๐ง=๐ฅ โ โซ ๐๓ธ (๐ง) ๐ฟ๐ข๐ ๐๐ง = 0; 0
๐ถ
Algorithm 4. (1) Input ๐, ๐, ๐ง, ๐, ๐, ๐๐,๐ (๐ฅ, ๐ก), ๐๐ (๐ฅ, ๐ผ), ๐๐ (๐ฅ, ๐ผ). (2) Input the Taylor expansion degree (๐ ).
6
Journal of Applied Mathematics (3) Calculate
(9) Put ๐(๐) ๐๐,๐ (๐ฅ, ๐ผ) ๐๐ฅ๐ ๐(๐) ๐ (๐ฅ, ๐ผ) ๐
๐๐ฅ๐
, (๐,๐)
,
(28)
๐(๐) ๐๐ (๐ฅ, ๐ผ) , ๐๐ฅ๐
=
๐๐ ๐๐,๐
๐๐,๐
โซ
๐!
๐๐ฅ๐
๐
(๐,๐)
๐
(๐ฅ, ๐ก) โ
(๐ก โ ๐ง) ๐๐ก
(29)
(5) Calculate ๐ ๐,๐
=
๐!
โซ
๐๐ ๐๐,๐
๐๐,๐
๐๐ฅ๐
๐
๐
(๐ฅ, ๐ก) โ
(๐ก โ ๐ง) ๐๐ก
(30)
๐, ๐ = 1, . . . , ๐.
(๐,๐)
0 0 [ ๓ธ (๐,๐) [๐ 0 [ 2,1 [ [ . .. . =[ . [ . [ (๐,๐) [ ๓ธ (๐,๐) [ ๐ ๐,1 ๐๓ธ ๐,2 [ ๓ธ (๐,๐) ๓ธ (๐,๐) [๐ ๐+1,1 ๐ ๐+1,2
(๐,๐)
๐ท
(๐,๐)
๐ธ1,1 = ๐ธ2,2 (๐,๐)
(๐,๐)
๐ โ 1 ๐1,2 [ 1,1 (๐,๐) [ (๐,๐) ๐2,2 โ 1 [ ๐2,1 [ [ . .. =[ . [ .. [ [ (๐,๐) (๐,๐) [ ๐๐ ,1 ๐๐ ,2 [ (๐,๐)
(๐,๐)
[ ๐๐ +1,1
๐๐ +1,2
(๐,๐)
๐1,๐ +1
๐2,๐
(๐,๐)
๐2,๐ +1
.. .
.. .
โ
โ
โ
๐1,๐
โ
โ
โ
d
(๐,๐)
(๐,๐) โ
โ
โ
๐๐ ,๐ โ1
โ
โ
โ
(๐,๐)
๐๐ +1,๐
๐๐ ,๐ +1 (๐,๐)
0
โ
โ
โ
0
d
.. .
โ
โ
โ
0
0
(๐,๐)
โ
โ
โ
๐๓ธ ๐+1,๐
] 0] ] ] .. ] . .] ] ] ] 0] ] 0](๐ +1)ร(๐ +1)
(35)
(๐,๐)
(๐,๐)
๐ท1,1 ๐ท1,2
=[
]. (๐,๐) (๐,๐) ๐ท ๐ท 2,2 ] [ 2,1
(36)
(12) Put
(๐,๐) (๐,๐)
โ
โ
โ
(11) Denote
(6) Put (๐,๐)
(34)
๐ท๓ธ 1,2 = ๐ท๓ธ 2,1
๐, ๐ = 1, . . . , ๐.
(๐,๐) ๐๓ธ ๐+1,๐+1
.
(10) Put
(4) Calculate ๐ ๐,๐
0 โ
โ
โ
0 0 ] 0 โ
โ
โ
0 0] ] .. .. .. ] ] . d . .] ] 0 โ
โ
โ
0 0] ]
[0 0 โ
โ
โ
0 0](๐ +1)ร(๐ +1) ๐ = 0, 1, . . . , ๐ .
(๐,๐) ๐๐+1,๐+1
(๐,๐)
๐ท1,1 = ๐ท2,2
0 [ [0 [ [. =[ [ .. [ [0 [
] ] ] ] ] ] ] ] ] ] ]
F = [โ๐ (๐ง, ๐ผ) , . . . , โ๐(๐) (๐ง, ๐ผ) , 1
1
.
(31)
(๐ )
โ๐1 (๐ง, ๐ผ) , . . . , โ๐1 (๐ง, ๐ผ) , . . . , โ๐ (๐ง, ๐ผ) , . . . , โ๐(๐ ) (๐ง, ๐ผ) , ๐
๐๐ +1,๐ +1 โ 1](๐ +1)ร(๐ +1)
(37)
๐
(๐ )
๐ก
โ๐๐ (๐ง, ๐ผ) , . . . , โ๐๐ (๐ง, ๐ผ)] .
(7) (๐,๐)
(๐,๐)
๐ธ๓ธ 1,2 = ๐ธ๓ธ 2,1
(13) Put
(๐,๐)
(๐,๐)
๐๓ธ 1,1 โ 1
[ [ ๓ธ (๐,๐) [ ๐ [ 2,1 [ [ .. =[ . [ [ (๐,๐) [ ๐๓ธ [ ๐ ,1 [ ๓ธ (๐,๐) [ ๐ ๐ +1,1
๐๓ธ 1,2
(๐,๐)
โ
โ
โ
๐๓ธ 1,๐
๐๓ธ 2,2 โ 1 โ
โ
โ
๐๓ธ 2,๐
(๐,๐)
.. . (๐,๐)
๐๓ธ ๐ ,2
(๐,๐)
๐๓ธ ๐ +1,2
(๐,๐)
๐๓ธ 1,๐ +1
] ] ] ] (32) ] ] .. .. ], d . . ] ] ๓ธ (๐,๐) ๓ธ (๐,๐) โ
โ
โ
๐ ๐ ,๐ โ 1 ๐ ๐ ,๐ +1 ] ] ] (๐,๐) (๐,๐) โ
โ
โ
๐๓ธ ๐ +1,๐ ๐๓ธ ๐ +1,๐ +1 โ 1] (๐,๐)
(๐,๐)
๐๓ธ 2,๐ +1
where ๐, ๐ = 1, 2, . . . , ๐. (8) Calculate ๓ธ ๐๐+1,๐+1 (๐โ๐โ๐โ1) ๓ตจ๓ตจ (33) ๓ตจ๓ตจ ๐(๐) ๐๐,๐ (๐ฅ, ๐ก) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ ) = โ ( )( ๓ตจ๓ตจ . ๓ตจ ๐ ๓ตจ๓ตจ ๐๐ฅ ๐โ๐โ๐โ1 ๓ตจ๓ตจ๓ตจ๐ก=๐ฅ ๐=0 ๓ตจ๓ตจ๐ฅ=๐ง ๐โ๐โ1
๐โ๐โ1
(๐ ) ๐ข = [๐ข1๐ (๐ง, ๐ผ) , . . . , ๐ข(๐ ) 1๐ (๐ง, ๐ผ) , ๐ข1๐ (๐ง, ๐ผ) , . . . , ๐ข1๐ (๐ง, ๐ผ) ,
. . . , ๐ข๐๐ (๐ง, ๐ผ) , . . . , ๐ข(๐ ) ๐๐ (๐ง, ๐ผ) , ๐ข๐๐ (๐ง, ๐ผ) , . . . ,
(38)
๐ก
๐ข(๐ ) ๐๐ (๐ง, ๐ผ)] .
(14) Solve the following linear system (๐ธ + ๐ท)๐ข = F. (15) Estimate ๐ข(๐ง, ๐ผ), ๐ข(๐ง, ๐ผ) by computing Taylor expansion for ๐ข ๓ตจ๓ตจ (๐) ๐ 1 ๐ ๐ข๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ (๐ฅ โ ๐ง)๐ ) ๐ข๐,๐ = โ ( ๐ ๐! ๐๐ฅ ๓ตจ๓ตจ๓ตจ ๐=0 ๓ตจ๐ฅ=๐ง ๓ตจ (๐) ๐ (39) 1 ๐ ๐ข๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ๓ตจ ๐ ๐ข๐,๐ = โ ( ๓ตจ (๐ฅ โ ๐ง) ) , ๐ ๓ตจ ๓ตจ๓ตจ ๐! ๐๐ฅ ๐=0 ๓ตจ๐ฅ=๐ง ๐ โค ๐ฅ, ๐ง โค ๐, 0 โค ๐ผ โค 1, ๐ = 1, 2, . . . , ๐.
Journal of Applied Mathematics
7
We get the following results:
(1,1) (1,1) ๐ธ1,1 = ๐ธ2,2
(1,1) (1,1) = ๐ธ2,1 ๐ธ1,2
โ.04088 .00364 โ.00024 1.275 ร 10โ5 โ5.598 ร 10โ7
โ.70073 [ [ .29927 [ [ [ .29927 [ =[ [ .29927 [ [ [ .29927 [ [ .29927
] โ1.0409 .00364 โ.00024 1.275 ร 10โ5 โ5.598 ร 10โ7 ] ] ] โ.04088 โ.99636 โ.00024 1.275 ร 10โ5 โ5.598 ร 10โ7 ] ] ], โ5 โ7 ] โ.04088 .00364 โ1.0002 1.275 ร 10 โ5.598 ร 10 ] ] โ.04088 .00364 โ.00024 โ0.99999 โ5.598 ร 10โ7 ] ] โ.04088 .00364 โ.00024 1.275 ร 10โ5
๐ท=[
0 [ [1 [ [ [1 =[ [1 [ [ [1 [
0 0 0 0 0
๐ โ๐ผ cos ( ) [ ] 12 [ ] [ ] ๐ [ ] ) ๐ผ sin ( [ ] 12 [ ] [ ] [ ๐ผ cos ( ๐ ) ] [ ] [ ] 12 [ ] [ ] ๐ [ โ๐ผ sin ( ) ] [ ] 12 [ ] [ ] [ โ๐ผ cos ( ๐ ) ] [ ] 12 [ ] [ ] [ ] ๐ [ ] ) ๐ผ sin ( [ ] 12 [ ]. F= [ ] [โ (2 โ ๐ผ) cos ( ๐ )] [ 12 ] [ ] [ ] ๐ ] [ [ (2 โ ๐ผ) sin ( ) ] [ 12 ] [ ] [ ] [ (2 โ ๐ผ) cos ( ๐ ) ] [ 12 ] [ ] [ ] ๐ ] [ [ โ (2 โ ๐ผ) sin ( ) ] [ 12 ] [ ] [ ] [โ (2 โ ๐ผ) cos ( ๐ )] [ 12 ] [ ] [ ๐ ] (2 โ ๐ผ) sin ( ) [ 12 ]
0 [ [0 [ [ [0 =[ [0 [ [ [0 [ [0
0 0 0 0 0
] 0 0 0 0 0] ] ] 1 0 0 0 0] ], 1 1 0 0 0] ] ] 1 1 1 0 0] ] [1 1 1 1 1 0 ]
(1,1) (1,1) ๐ท1,2 ๐ท1,1 (1,1) (1,1) ๐ท2,1 ๐ท2,2
] 0 0 0 0 0] ] ] 0 0 0 0 0] ], 0 0 0 0 0] ] ] 0 0 0 0 0] ] 0 0 0 0 0]
],
(41)
F (๐ง, ๐ผ) = [โ๐ (๐ง, ๐ผ) , โ๐๓ธ (๐ง, ๐ผ) , โ๐๓ธ ๓ธ (๐ง, ๐ผ) , โ๐๓ธ ๓ธ ๓ธ (๐ง, ๐ผ) , โ๐(4) (๐ง, ๐ผ) , โ๐(5) (๐ง, ๐ผ) โ ๐ (๐ง, ๐ผ) , ๓ธ
๓ธ ๓ธ
๓ธ ๓ธ ๓ธ
(4)
โ๐ (๐ง, ๐ผ) , โ๐ (๐ง, ๐ผ) , โ๐ (๐ง, ๐ผ) , โ๐ (5)
โ๐
๐ก
(๐ง, ๐ผ)] ,
(40)
0 0 0 0 0 ] 0 0 0 0 0] ] ] 0 0 0 0 0] ]; 0 0 0 0 0] ] ] 0 0 0 0 0] ] [0 0 0 0 0 0]
(1,1) (1,1) ๐ธ1,1 ๐ธ1,2 ๐ธ = [ (1,1) (1,1) ] , ๐ธ2,1 ๐ธ2,2
(1,1) (1,1) ๐ท1,2 = ๐ท2,1
]
0 [ [0 [ [ [0 =[ [0 [ [ [0 [
hence
(1,1) (1,1) = ๐ท2,2 ๐ท1,1
โ1.000
Solving the linear system
(๐ง, ๐ผ) , (๐ท + ๐ธ) ๐ข = F,
(42)
8
Journal of Applied Mathematics 1
we obtain 1.30655494406977๐ผ [ ] [ 1.38836501660070๐ผ ] [ ] [ ] [ 2.06962325214024๐ผ ] [ ] [ 5.36399137514409๐ผ ] [ ] [ ] [ 11.4350895320246๐ผ ] [ ] [ ] [ 21.4350895320246๐ผ ] [ ] ๐ข (๐ง, ๐ผ) = [ ], [1.30655494406977 (2 โ ๐ผ)] [ ] [ ] [1.38836501660070 (2 โ ๐ผ)] [ ] [ ] [2.06962325214024 (2 โ ๐ผ)] [ ] [5.36399137514409 (2 โ ๐ผ)] [ ] [ ] [11.4350895320246 (2 โ ๐ผ)] [ ] [21.4350895320246 (2 โ ๐ผ)]
0.8 0.6 ํผ 0.4 0.2 0
1
2
3
U(x, ํผ) Exact Approximate
(43)
Figure 1: Exact and numerical solutions at ๐ฅ = ๐/8. 3.4 ร 10โ6 3.3 ร 10โ6 3.2 ร 10โ6 3.1 ร 10โ6 3. ร 10โ6 2.9 ร 10โ6 2.8 ร 10โ6 2.7 ร 10โ6 2.6 ร 10โ6 2.5 ร 10โ6 2.4 ร 10โ6 2.3 ร 10โ6 2.2 ร 10โ6 2.1 ร 10โ6 2. ร 10โ6 1.9 ร 10โ6 1.8 ร 10โ6
๓ตจ๓ตจ (๐) ๐ 1 ๐ ๐ข๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ ๐ ๐ ๓ตจ (๐ฅ โ ) ), ๐ข (๐ฅ, ๐ผ) = โ ( ๓ตจ ๓ตจ๓ตจ ๐! ๐๐ฅ๐ 12 ๓ตจ๓ตจ๐ฅ=๐ง ๐=0 ๓ตจ (๐) 1 ๐ ๐ข๐ (๐ฅ, ๐ผ) ๓ตจ๓ตจ๓ตจ๓ตจ ๐ ๐ ๐ข (๐ฅ, ๐ผ) = โ ( ๓ตจ๓ตจ (๐ฅ โ ) ) , ๐ ๓ตจ๓ตจ ๐! ๐๐ฅ 12 ๐=0 ๓ตจ๐ฅ=๐ง ๐
0โค๐ฅโค
0
๐ , 0 โค ๐ผ โค 1. 4
The approximate solution is ๐ข (๐ฅ, ๐ผ) = 0.9999816737๐ผ + 1.000399233๐ผ๐ฅ
0
+ 0.4962378250 + 0.5186782119๐ผ๐ฅ3
0.1
0.2
0.3
0.4
0.5 ํผ
0.6
0.7
0.8
0.9
1
Figure 2: Absolute error between the exact and numerical solutions.
+ 0.2403470047๐ผ๐ฅ4 + 0.1803786182๐ผ๐ฅ5 , ๐ข (๐ฅ, ๐ผ) = 2.000407015๐ฅ + 1.738708618๐ผ + 0.3524026447๐ฅ3 + 0.1975243281๐ฅ4 + 0.1905848255๐ฅ5 โ 0.7387269443
(44)
4
+ 0.04282267662๐ผ๐ฅ
5
โ 0.01020620726๐ผ๐ฅ
โ 1.000007782๐ผ๐ฅ + 0.9961625156๐ฅ2 โ 0.4999246903๐ผ๐ฅ2 + 0.1662755673๐ผ๐ฅ3 . Figure 1 compares both the exact and numerical solutions for the fuzzy integral equation (26) using the Taylor expansion method at ๐ฅ = ๐/8. Moreover, Figure 2 shows the absolute error between the exact and numerical solutions of this example.
Example 5 (variation alliteration method). Consider the linear fuzzy Volterra integral equation (26) with the exact solution (27). In the view of the variational iteration method, we construct a correction functional in the following form: ๐ฅ
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + โซ ๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก, 0
(45)
๐ฅ
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + โซ ๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก, 0
where ๐ = 0, 1, 2, . . . , ๐ . Start with the initial approximation in (45); that is, ๐ข0 (๐ฅ, ๐ผ) = (๐ผ) cos ๐ฅ, ๐ข0 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ,
(46) 0 โค ๐ผ โค 1.
Then successive approximations ๐ข๐ (๐ฅ, ๐ผ)๓ธ ๐ will be generated. In this example we calculate the 6th order of approximate
Journal of Applied Mathematics
9
solution using the variational iteration method by MAPLE software. We have carried out the following algorithm.
1 1 1 ๐ข3 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐๐ฅ โ ๐ผ๐๐ฅ โ cos ๐ฅ 2 4 2 +
Algorithm 6.
1 1 1 โ ๐ผ๐๐ฅ ๐ฅ + ๐๐ฅ ๐ฅ2 โ ๐ผ๐๐ฅ ๐ฅ2 , 2 2 4
(1) Input ๐, ๐, ๐, ๐(๐ฅ, ๐ก), ๐ , ๐(๐ฅ, ๐ผ), ๐(๐ฅ, ๐ผ). (2) Input ๐ข0 (๐ฅ, ๐ผ), ๐ข0 (๐ฅ, ๐ผ).
1 1 ๐ข4 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐๐ฅ โ ๐ผ๐๐ฅ โ ๐ผ๐๐ฅ ๐ฅ3 2 12
(3) For ๐ = 1 to ๐ + 1, compute
1 1 1 + ๐๐ฅ ๐ฅ + ๐๐ฅ ๐ฅ2 + ๐๐ฅ ๐ฅ3 โ cos ๐ฅ 2 2 6
๐ฅ
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + โซ ๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก, 0
๐ฅ
๐ข๐+1 (๐ฅ, ๐ผ) = ๐ (๐ฅ, ๐ผ) + โซ ๐ (๐ฅ, ๐ก) ๐ข๐ (๐ก, ๐ผ) ๐๐ก. 0
We obtain the following results: 1 1 1 ๐ข1 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ sin ๐ฅ, 2 2 2 1 1 1 ๐ข2 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ๐๐ฅ ๐ฅ, 2 2 2 3 1 1 1 ๐ข3 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ sin ๐ฅ + ๐ผ๐๐ฅ ๐ฅ 4 4 4 2 1 + ๐ผ๐๐ฅ ๐ฅ2 , 2 1 1 1 1 ๐ข4 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ sin ๐ฅ + ๐ผ๐๐ฅ ๐ฅ 2 2 4 4 1 1 + ๐ผ๐๐ฅ ๐ฅ2 + ๐ผ๐๐ฅ ๐ฅ3 , 4 12 5 1 3 1 ๐ข5 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ sin ๐ฅ + ๐ผ๐๐ฅ ๐ฅ 8 8 8 2
1 1 1 sin ๐ฅ + ๐ผ cos ๐ฅ๐ โ ๐ผ sin ๐ฅ + ๐๐ฅ ๐ฅ 2 4 4
(47)
+
1 1 1 sin ๐ฅ โ ๐ผ๐๐ฅ ๐ฅ โ ๐ผ๐๐ฅ ๐ฅ2 2 4 4
1 1 + ๐ผ cos ๐ฅ โ ๐ผ sin ๐ฅ, 2 4 3 3 1 ๐ข5 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐๐ฅ โ ๐ผ๐๐ฅ + ๐๐ฅ ๐ฅ2 4 8 4 3 1 + ๐ผ cos ๐ฅ + ๐๐ฅ ๐ฅ3 8 6 +
1 3 1 sin ๐ฅ + ๐๐ฅ ๐ฅ โ cos ๐ฅ โ ๐ผ๐๐ฅ ๐ฅ4 4 4 48
1 1 1 โ ๐ผ sin ๐ฅ โ ๐ผ๐๐ฅ ๐ฅ3 โ ๐ผ๐๐ฅ ๐ฅ2 8 12 8 1 1 โ ๐ผ๐๐ฅ ๐ฅ + ๐๐ฅ ๐ฅ4 , 2 24 3 3 1 ๐ข6 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐๐ฅ โ ๐ผ๐๐ฅ + ๐๐ฅ ๐ฅ2 4 8 2 3 1 + ๐ผ cos ๐ฅ + ๐๐ฅ ๐ฅ3 8 12 +
1 3 3 1 ๐ฅ 5 sin ๐ฅ + ๐๐ฅ ๐ฅ โ cos ๐ฅ + ๐ ๐ฅ 2 4 4 120
โ
1 1 1 ๐ผ๐๐ฅ ๐ฅ5 โ ๐ผ๐๐ฅ ๐ฅ4 โ ๐ผ sin ๐ฅ 240 48 4
1 1 1 + ๐ผ๐๐ฅ ๐ฅ2 + ๐ผ๐๐ฅ ๐ฅ3 + ๐ผ๐๐ฅ ๐ฅ4 4 24 48
โ
1 ๐ฅ 3 1 ๐ฅ 2 3 ๐ฅ ๐ผ๐ ๐ฅ โ ๐ผ๐ ๐ฅ โ ๐ผ๐ ๐ฅ 24 4 8
1 + ๐ผ๐๐ฅ ๐ฅ5 , 240
+
1 ๐ฅ 4 ๐ ๐ฅ. 24
1 1 1 + ๐ผ๐๐ฅ ๐ฅ2 + ๐ผ๐๐ฅ ๐ฅ3 + ๐ผ๐๐ฅ ๐ฅ4 , 8 12 48 5 1 3 3 ๐ข6 (๐ฅ, ๐ผ) = ๐ผ cos ๐ฅ + ๐ผ๐๐ฅ + ๐ผ sin ๐ฅ + ๐ผ๐๐ฅ ๐ฅ 8 8 4 8
1 ๐ข1 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐ โ ๐ผ๐๐ฅ โ cos ๐ฅ + sin ๐ฅ 2 ๐ฅ
1 1 + ๐ผ cos ๐ฅ โ ๐ผ sin ๐ฅ, 2 2 1 ๐ข2 (๐ฅ, ๐ผ) = (2 โ ๐ผ) cos ๐ฅ + ๐๐ฅ โ ๐ผ๐๐ฅ โ cos ๐ฅ 2 1 1 + ๐ผ cos ๐ฅ + ๐๐ฅ ๐ฅ โ ๐ผ๐๐ฅ ๐ฅ, 2 2
(48) Figure 3 shows a comparison between the exact and the numerical solutions for the fuzzy integral equation (26) by the variational iteration method (VIM) at ๐ฅ = ๐/8. Moreover, Figure 4 shows the absolute error between the exact and numerical solutions of this example. Table 1 shows a comparison between the exact and the numerical solutions for the fuzzy integral equation (26) using the Taylor expansion method and the variational iteration method (VIM) at ๐ฅ = ๐/8.
Taylor numerical solution ๐ข(๐ฅ, ๐ผ)
0 0.15081747 0.30163494 0.45245241 0.60326988 0.75408735 0.90490482 1.05572230 1.20653977 1.35735724 1.50817471
Exact solution ๐ข(๐ฅ, ๐ผ)
0 0.15081764 0.30163528 0.45245292 0.60327057 0.75408821 0.90490585 1.05572349 1.20654114 1.35735878 1.50817642
๐ผ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 0.15081760 0.30163520 0.45245280 0.60327040 0.75408800 0.90490560 1.05572320 1.20654080 1.35735840 1.50817600
Variation numerical solution ๐ข(๐ฅ, ๐ผ) 3.01635285 2.86553521 2.71471756 2.56389992 2.41308228 2.26226464 2.11144699 1.96062935 1.80981171 1.65899406 1.50817642
Exact solution ๐ข(๐ฅ, ๐ผ) 3.016349432 2.865531960 2.714714489 2.563897017 2.413079546 2.262262074 2.111444602 1.960627131 1.809809659 1.658992188 1.508174716
Taylor numerical solution ๐ข(๐ฅ, ๐ผ) 3.01635200 2.86553440 2.71471680 2.56389920 2.41308160 2.26226400 2.11144640 1.96062880 1.80981120 1.65899360 1.50817600
Variation numerical solution ๐ข(๐ฅ, ๐ผ)
Error = ๐ท(๐ขexact , ๐ขapprox ) (using Taylor expansion) 3.4210 ร 10โ6 3.2500 ร 10โ6 3.0790 ร 10โ6 2.9080 ร 10โ6 2.7360 ร 10โ6 2.5660 ร 10โ6 2.3950 ร 10โ6 2.2230 ร 10โ6 2.0520 ร 10โ6 1.8806 ร 10โ6 1.7104 ร 10โ6
Table 1: Absolute error between the exact and numerical solutions using the Taylor expansion method and the variational iteration method. Error = ๐ท(๐ขexact , ๐ขapprox ) (using variational iteration) 8.4600 ร 10 โ7 8.0400 ร 10โ7 7.6100 ร 10โ7 7.1900 ร 10โ7 6.7700 ร 10โ7 6.3600 ร 10โ7 5.9300 ร 10โ7 5.5000 ร 10โ7 5.0700 ร 10โ7 4.6500 ร 10โ7 4.2400 ร 10โ7
10 Journal of Applied Mathematics
Journal of Applied Mathematics
11
1
References
0.8 0.6 ํผ 0.4 0.2 0
0
1
2 u6 (x, ํผ)
3
4
Exact Approximate
Figure 3: Exact and numerical solutions at ๐ฅ = ๐/8.
ร โ 7
8. ร 10
7
.5 10 .5
10
7
.
10
7
.5
10
7
0.1
0.5
0.9
Figure 4: Absolute error between the exact and numerical solutions.
6. Conclusion In this article, Taylor expansion and variational iteration methods are proposed to solve a fuzzy linear Volterra integral equation of the second kind. The results of the example show that the convergence and accuracy of both methods were in a good agreement with the analytical solution. According to comparison of numerical results, mentioned in tables and figures, we conclude that the variational iteration method provides more accurate results and therefore is more advantageous.
Conflicts of Interest The authors declare that they have no conflicts of interest.
[1] T. Allahviranloo, S. Salahshour, M. Homayoun-Nejad, and D. Baleanu, โGeneral solutions of fully fuzzy linear systems,โ Abstract and Applied Analysis, vol. 2013, Article ID 593274, 2013. [2] L. A. Zadeh, โFuzzy sets,โ Information and Control, vol. 8, no. 3, pp. 338โ353, 1965. [3] S. S. L. Chang and L. A. Zadeh, โOn fuzzy mapping and control,โ IEEE Transactions on Systems, Man and Cybernetics, vol. 2, no. 1, pp. 30โ34, 1972. [4] L. A. Zadeh, โLinguistic variables, approximate reasoning and dispositions.,โ Medical Informatics, vol. 8, no. 3, pp. 173โ186, 1983. [5] D. Dubois and H. Prade, โOperations on fuzzy numbers,โ International Journal of Systems Science. Principles and Applications of Systems and Integration, vol. 9, no. 6, pp. 613โ626, 1978. [6] J. Goetschel and W. Voxman, โElementary fuzzy calculus,โ Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31โ43, 1986. [7] O. Kaleva, โFuzzy differential equations,โ Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301โ317, 1987. [8] S. Nanda, โOn integration of fuzzy mappings,โ Fuzzy Sets and Systems, vol. 32, no. 1, pp. 95โ101, 1989. [9] M. Friedman, M. Ma, and A. Kandel, โNumerical solutions of fuzzy differential and integral equations,โ Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35โ48, 1999. [10] F. G. Tricomi, Integral Equations, Dover Publications, New York, NY, USA, 1982. [11] S. Liao, โBeyond Perturbation: introduction to the homotopy analysis method,โ in Modern Mechanics and Mathematics, vol. 2, pp. 1โ336, Chapman and Hall/CRC Press, Boca Raton, Fla, USA, 2003. [12] S. Abbasbandy, โNumerical solutions of the integral equations: homotopy perturbation method and Adomianโs decomposition method,โ Applied Mathematics and Computation, vol. 173, no. 1, pp. 493โ500, 2006. [13] E. Babolian, H. Sadeghi Goghary, and S. Abbasbandy, โNumerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method,โ Applied Mathematics and Computation, vol. 161, no. 3, pp. 733โ744, 2005. [14] A. Jafarian, S. Measoomy Nia, and S. Tavan, โA numerical scheme to solve fuzzy linear volterra integral equations system,โ Journal of Applied Mathematics, vol. 2012, Article ID 216923, 17 pages, 2012. [15] R. P. Kanwal and K. C. Liu, โA Taylor expansion approach for solving integral equations,โ International Journal of Mathematical Education in Science and Technology, vol. 20, no. 3, pp. 411โ 414, 1989. [16] L. Xu, โVariational iteration method for solving integral equations,โ Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1071โ1078, 2007. [17] M. Amawi, Fuzzy Fredholm Integral Equation of the Second Kind (M. S. Thesis), An-Najah National University, 2014. [18] M. Amawi and N. Qatanani, โNumerical methods for solving fuzzy Fredholm integral equation of the second kind,โ International Journal of Applied Mathematics, vol. 28, no. 3, pp. 177โ195, 2015. [19] J. Hamaydi, Analytical and Numerical Methods for Solving Linear Fuzzy Volterra Integral Equation of the Second Kind (M. S. Thesis), Najah National University, 2016. [20] M. Ghanbari, โNumerical solution of fuzzy linear Volterra integral equations of the second kind by homotopy analysis
12 method,โ International Journal of Industrial Mathematics (IJIM), vol. 2, no. 2, pp. 73โ87, 2010. [21] H. H. Sorkun and S. YalcCinbac, โApproximate solutions of linear Volterra integral equation systems with variable coefficients,โ Applied Mathematical Modeling, vol. 34, no. 11, pp. 3451โ 3464, 2010. [22] M. Gholami and B. Ghanbari, โVariational iteration method for solving Voltarra and Fredholm integral equations of the second kind,โ General Mathematics Notes, vol. 2, no. 1, pp. 143โ148, 2011.
Journal of Applied Mathematics
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at https://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
#HRBQDSDฤฎ,@SGDL@SHBR
Journal of
Volume 201
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014