EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 290-313 ISSN 1307-5543 – ejpam.com Published by New York Business Global Solving fuzzy system of Volterra integro-differential equations by using Adomian decomposition method Mahasin Thabet Younis1, Waleed Al-Hayani1,∗ 1 Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Iraq Abstract. In this paper, the Adomian decomposition method and Modified Technique are success- fully applied to find the approximate solutions of the fuzzy system of Volterra integro-differential equations. The approximate solutions obtained have been improved by using the iteration of the integral equation and the numerical solution with the Simpson rule and Trapezoidal rule. These proposed methods gave excellent results close to the exact solution. The results show that the present method is very straightforward and effective. 2020 Mathematics Subject Classifications: 03E72, 26A42, 45J05 Key Words and Phrases: Fyzzy numbers, Fuzzy Systems of Volterra integro-differential equa- tions, Adomian decomposition method, Modified technique, Adomian polynomials 1. Introduction Integral equations are used in variety scientific and technical fields, especially in engineering fields, where they were considered from a period not briefly one of the most important tools in applied mathematics. Mathematical modeling problems common in the real world and in our lives are the result of analysis of differential equations, integral equations, integro-differential equations and statistical equations, and others. Many mathematical formulations of some physical phenomena contain, integro-differential equations, these equations appear in fluid mechanics, biological models, kinetic physics and chemical reactions, as well as, integro-differential equations appear in many physical processes. Like glass formation and nano-dynamics [1]. There are many effective ways to find approximate solutions and numerical and analytical solutions to linear and non- linear problems of integral equations and integro-differential equations used over decades Volterra integro-differential equations which has continuous kernel [1]. Approximation so- lution based on basis functions has been utilized to estimate solutions of integral equation in recent years. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1. Email addresses: mahasin thabet@uomosul.edu.iq (M. T. Younis), waleedalhayani@uomosul.edu.iq, waleedalhayani@yahoo.es (W. Al-Hayani) http://www.ejpam.com 290 © 2022 EJPAM All rights reserved. M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 291 Basic concept of fuzzy was first introduced by Professor Zadeh in 1965 after his publi- cations on fuzzy set theory [2, 3]. In the beginning of the 1980s, Adomian [4–6] proposed a new and fruitful method (so-called decomposition method) for solving linear and nonlinear (algebraic, differential, partial differential, integral, etc.) equations. It has been shown that this method yields a rapid convergence of the solutions series to linear and nonlinear deterministic and stochastic equations. Many recent years solving fuzzy integro-differential equations, fuzzy Volterra-Fredholm integral equation and fuzzy Volterra-Fredholm integro-differential equations require ap- propriate and applicable definitions of fuzzy function [7–10], the fuzzy systems of integral equations have attracted increasing attention, with regard to fuzzy control, have been developed mathematical models used in many problems of physics, biology, chemistry, engineering, and in other fields depend on an integral equation [11]. ADM is an analytical technique that uses Adomian polynomials to evaluate the ap- proximate solutions. This method does not simplify or discretize the problem, and it can be used to solve both linear and non-linear problems [12]. The ADM has been used to solve Volterra integral equations, integro-differential equa- tions and Fredholm integro-differential equations of various types in the literature [13, 14]. Dalal Adnan and Eman Ahmed [15] have been examined the topic of population expan- sion. Arikglu and Ozkol [12] have been applied Differential Transform Method (DTM) on both integral equation and integro-differential equation system. Hassan and Peter [16] solve new iterative method with a reliable algorithm and applied to the systems of Volterra integro-differential equations. Berenguer et al [17] have been solved systems of integro- differential equations using numerical of fixed Point. Biazar and Aminikhah [18] have been used Variational iteration method (VIM) for solving nonlinear integral-differential equations. Bani Issaa et al [19] applied numerical methods to slove the fuzzy integro-differential equations of the second kind. Shabestari et al [20] have been solved Fuzzy Volterra Integro- differential equations of fractional order by Bernoulli Wavelet method. Das and Taluk- dar [21] solved fuzzy integro-differential equation by using fuzzy Laplace Transformation. Mikaeilvand et al [22] applied the differential transform method (DTM) to solve fuzzy integro-differential equation. The main objective of this article is to use the Standard Adomian Decomposition Method (ADM) and Modified Technique (MT) to solve the fuzzy systems of linear and non-linear four Volterra integro-differential equations (VIDEs) with a comparison between the both techniques and improved the approximate solutions obtained for the system of non-linear four VIDEs by using the iteration of the integral equation and the numerical solution with the Simpson rule and Trapezoidal rule. 2. Basic Concepts Fuzzy numbers are generalized classical real numbers, and we can define them as a fuzzy subset of the real line with some additional features. The concept of a fuzzy number is essential for fuzzy analysis, fuzzy integral equations, as well as a useful tool in a variety M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 292 of fuzzy set applications. The basic definitions of fuzzy numbers are the following: Definition 1. (Fuzzy set) [23]: A set à = {( t,MÃ(t) ) , t ∈ X } is called a fuzzy set where MÃ(t) is the membership function of fuzzy set A is defined by MÃ(t) : X → [0, 1], and the value of MÃ(t) is called the membership degree X. Definition 2. (Fuzzy number) [24, 25]: A fuzzy number is a map ũ : R → [a, b], which satisfying (i) ũ is upper semi-continuous function. (ii) ũ (t) = 0 outside some interval [a, d] . (iii) There are real numbers b, c such a ≤ b ≤ c ≤ d i) ũ (t) is a monotonic increasing function on [a, b] . ii) ũ (t) is a monotonic decreasing function on [c, d] . iii) ũ (t) = 1 for all t ∈ [b, c] . Definition 3. [25–27]: A fuzzy number ũ in a parametric form is a pair (u, ū) of a function u (r) , ū (r) , 0 ≤ r ≤ 1; which satisfy the requirements (i) u (r) is a left continuous function and bounded monotonic increasing. (ii) ū (r) is a left continuous function and bounded monotonic decreasing. (iii) u (r) ≤ ū (r) , 0 ≤ r ≤ 1 The triangular fuzzy number [2, 20, 28], which is defined as a fuzzy set in R and characterized by an ordered triple u = (a, b, c) ∈ R3 with a ≤ b ≤ c such that u (r) = a+ (b− a) r and ū (r) = c− (c− b) r, are the endpoints of r−level sets for all r ∈ [0, 1] is a popular fuzzy number: We can represent a crisp number x by (u (r) , ū (r)) = (x, x) , 0 ≤ r ≤ 1. By appropriate definitions, the fuzzy number space {u (r) ≤ ū (r)} becomes a convex cone E1 which could be isometrically and isomorphically into a Banach space [29]. Let x̃ = (x (r) , x̄ (r)), ỹ = (y (r) , ȳ (r)), 0 ≤ r ≤ 1, and k ∈ R. Then (i) x̃ = ỹ iff x (r) = y (r) , x̄ (r) = ȳ (r) . (ii) x̃+ ỹ = ( x (r) + y (r) , x̄ (r) + ȳ (r) ) . (iii) x̃− ỹ = ( x (r)− ȳ (r) , x̄ (r)− y (r) ) . (iv) kx̃ = { (kx, kx̄) , k ≥ 0 (kx̄, kx) , k < 0 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 293 Definition 4. (r−level) [12, 25, 27, 30]: Let E be the set of all fuzzy number on R, we denoted [u]r r−level set of a fuzzy number u ∈ E, 0 ≤ r ≤ 1, it is a mapping between close interval [0, 1] to power set of R, where [u]r = { [a (r) , b (r)], r ∈ [0, 1] cl (supp (u)) , r = 0 [u]r is closed and bounded interval [u (r) , u (r)] where u (r) denotes the left–hand endpoint of [u]r and ū (r) denotes the right–hand endpoint of [u]r since each u ∈ R can be observed as defined by ũ = { 1 if t = u 0 if t ̸= u′ where cl (supp (u)) = closure of support u and sup p (u) = {t : u (t) > 0} . Definition 5. [23]: Let u = (u (r) , u (r)) , v = (v (r) , v (r)) , 0 ≤ r ≤ 1 be two any arbitrary fuzzy numbers and k is scalar, we define the operation of a fuzzy number (i) (u+ v) (r) = (u (r) + v (r)) , (u+ v) (r) = (u (r) + v (r)) . (ii) (u− v) (r) = (u (r)− v (r)) , (u− v) (r) = (u (r)− v (r)) . (iii) kŭ = { (ku (r) , ku (r)) , k ≥ 0 (ku (r) , ku (r)) , k < 0 (iv) ũ.ṽ = { uv (r) = max {u (r) v (r) , u (r) v (r) , u (r) v (r) , u (r) v (r)} uv (r) = min {u (r) v (r) , u (r) v (r) , u (r) v (r) , u (r) v (r)} Definition 6. (The fuzzy Riemann integral) [31]: We say that 1- f̃ (t) is a fuzzy valued function if f̃ : X → F. 2- f̃ (t) is a closed fuzzy valued function if f̃ : X → Fcl. 3- f̃ (t) is a bounded fuzzy valued function if f̃ : X → Fb. We denote Ar = [∫ b a f̃L r (s) ds, ∫ b a f̃U r (s) ds ] . If f̃ (t) is a fuzzy valued function on [a, b] that is closed and bounded, then the fuzzy Riemann integral ∫ b a f̃ (s) ds, is a closed fuzzy number. Furthermore, the [u]r set of ∫ b a f̃ (s) ds is(∫ b a f̃ (s) ds ) r = [∫ b a f̃L r (s) ds, ∫ b a f̃U r (s) ds ] . 3. Fuzzy System of Volterra Integro-Differential Equations When a physical system is modeled in a differential sense, the result is a fuzzy integral equation or a fuzzy integro-differential equation, and hence the solution of fuzzy integro-differential equations is important in science and engineering. Analytically, non- linear fuzzy integro-differential equations are difficult to solve, and accurate solutions are rare [10]. M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 294 We consider the following fuzzy system of Volterra integro-differential equations (FS- VIDEs) of the second kind [17, 18]: ũ′′i (x, α) = f̃i (x, α) + 2∑ j=1 λij ∫ x 0 Kij (x, t) F̃ij (ũ (t, α)) dt, i = 1, 2 (1) with the initial conditions ũi (0, α) = ( ai (α) , ai (α) ) , ũ′i (0, α) = ( bi (α) , bi (α) ) , i = 1, 2 (2) where λij ̸= 0, i, j = 1, 2 are real constants, ũ (t, α) = (ũ1 (t, α) , ũ2 (t, α)) T , 0 ≤ t ≤ x, a ≤ x ≤ b, ũi (x, α) are unknown functions, f̃i (x, α) and the kernels Kij (x, t) are analyt- ical functions, F̃ij (ũ (t, α)) are linear or non-linear functional of the unknown functions ũi (x, α) . Under the appropriate conditions f̃i (x, α) and Kij (x, t), the FSVIDEs (1) have a unique continuous solution ũi (x, α) on the given interval [a, b] . The parametric form of the given FSVIDEs (1) can be written as u′′i (x, α) = fi (x, α) + 2∑ j=1 λij ∫ x 0 Kij (x, t)Fij (u (t, α)) dt, u′′i (x, α) = fi (x, α) + 2∑ j=1 λij ∫ x 0 Kij (x, t)Fij (u (t, α)) dt, i = 1, 2 (3) 4. Adomian Decomposition Method The Adomian decomposition method gives the solution of the FSVIDEs (1) as an infinite series usually converging to the closed form solution. To solve the FSVIDEs (1) and (2) by the Adomian’s technique, assume an infinite series solution for the unknowns functions ũi (x, α) = [ ui (x, α) , ui (x, α) ] , i = 1, 2 given by [19, 21, 23] as follows ui (x, α) = ∞∑ n=0 ui,n (x, α) , ui (x, α) = ∞∑ n=0 ui,n (x, α) , i = 1, 2 (4) and decomposing non-linear functions F̃ij (ũ (t, α)) = [ Fij (u (t, α)) , Fij (u (t, α)) ] , i, j = 1, 2 as Fij (u (t, α)) = ∞∑ n=0 Aij,n, Fij (u (t, α)) = ∞∑ n=0 Aij,n, i, j = 1, 2 (5) M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 295 where Ãij,n = [ Aij,n, Aij,n ] , n ≥ 0, are polynomials (so called Adomian polynomials) [3–6] of ũi,0 (t, α) , ũi,1 (t, α) , . . . , ũi,n (t, α) , i = 1, 2 given by Aij,n = 1 n! ∂n ∂λn [ F ( ∞∑ k=0 λiui,k (t, α) )] λ=0 , Aij,n = 1 n! ∂n ∂λn [ F ( ∞∑ k=0 λiui,k (t, α) )] λ=0 , i, j = 1, 2 n = 0, 1, 2, . . . (6) Applying the Adomian’s technique as in [4–6], the FSVIDEs (3) can be written as Lui (x, α) = fi (x, α) + 2∑ j=1 λij ∫ x 0 Kij (x, t)Nijdt, Lui (x, α) = fi (x, α) + 2∑ j=1 λij ∫ x 0 Kij (x, t)Nijdt, i = 1, 2 (7) where Lui (x, α) = d2ui dx2 , Lui (x, α) = d2ui dx2 , i = 1, 2 are the linear operators and Nij = Fij (u (t, α)) , Nij = Fij (u (t, α)) , i, j = 1, 2 are the non-linear operators. Operating on both sides of the FSVIDEs (7) with the inverse operator of L ( namely L−1 = ∫ x 0 ∫ x 0 [·] dtdt ) yields ui (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Nijdt ] , ui (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Nijdt ] , i = 1, 2 (8) Using (4) and (5) into the FSVIDEs (8) it follows that ∞∑ n=0 ui,n (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t) ∞∑ n=0 Aij,ndt ] , ∞∑ n=0 ui,n (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t) ∞∑ n=0 Aij,ndt ] , i = 1, 2 (9) Using the fuzzy Standard ADM, according to (9), the lower iterations (L) are then deter- mined in the following recursive way: ui,0 (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) , ui,n+1 (x, α) = L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,ndt ] , i = 1, 2 n = 0, 1, 2, . . . (10) M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 296 and the upper iterations (U) are ui,0 (x, α) = ui (0, α) + u′i (0, α)x+ L−1fi (x, α) , ui,n+1 (x, α) = L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,ndt ] , i = 1, 2 n = 0, 1, 2, . . . (11) Using the fuzzy Modified Technique (MT) [11], according to (9), Lower iterations (L) are then calculated in a recursive manner as follows: ui,0 (x, α) = ui (0, α) + u′i (0, α)x, ui,1 (x, α) = L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,0dt ] ui,n+1 (x, α) = L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,ndt ] , i = 1, 2 n = 1, 2, . . . (12) and the upper iterations (U) are ui,0 (x, α) = ui (0, α) + u′i (0, α)x, ui,1 (x, α) = L−1fi (x, α) + L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,0dt ] ui,n+1 (x, α) = L−1 [ 2∑ j=1 λij ∫ x 0 Kij (x, t)Aij,ndt ] , i = 1, 2 n = 1, 2, . . . (13) Thus, all components ( ũi,n (x, α) = [ ui,n (x, α) , ui,n (x, α) ] , i = 1, 2 ) of ũi (x, α) =[ ui (x, α) , ui (x, α) ] , i = 1, 2 can be calculated once the Ãij,n are given for i, j = 1, 2, . . . ,m and n = 0, 1, 2, . . . . We then define the n–term approximants to the solutions ũi (x, α) =[ ui (x, α) , ui (x, α) ] , i = 1, 2 by ϕi,n [ ui (x, α) ] = n−1∑ k=0 ui,k (x, α) , i = 1, 2 with lim n→∞ ϕi,n [ ui (x, α) ] = ui (x, α) , i = 1, 2 and by ϕi,n [ui (x, α)] = n−1∑ k=0 ui,k (x, α) , i = 1, 2 with lim n→∞ ϕi,n [ui (x, α)] = ui (x, α) , i = 1, 2. 5. Applications and numerical results The fuzzy systems of linear and non-linear Volterra integro-differential equations are displayed in the following two problems, and we use ADM and MT to achieve ap- proximate exact solutions. In the problem 2. The maximum errors are defined as follows to demonstrate the great accuracy of the solution outcomes when compared to the exact solutions: L∞ [a, b] = ∥uExact (xi, α)− ϕi,n (xi, α)∥∞ , M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 297 L2,S [a, b] = √√√√ n∑ i=0 [uExact (xi, α)− ϕi,n (xi, α)] 2, L2,I [a, b] = √∫ b a [uExact (x, α)− ϕi,n (x, α)] 2 dx, where n = 1, 2, . . . represents the number of iterations. Moreover, we give the error residual in the non-linear problem. The computations for the problems were done with a precision of 40 digits using the Maple 18 package. Problem 1. Let us consider the following linear fuzzy system of Volterra integro-differential equations (LFSVIDEs) of the second kind ũ′′1 (x, α) = f̃1 (x, α) + ∫ x 0 [ũ1 (t, α) + (x− t) ũ2 (t, α)] dt, ũ′′2 (x, α) = f̃2 (x, α) + ∫ x 0 [(x− t) ũ1 (t, α)− ũ2 (t, α)] dt, (14) with the initial conditions ũ1 (0, α) = (3− α, 2α) , ũ′1 (0, α) = ( 2− α, α2 ) , ũ2 (0, α) = (3− 2α, 2α− 1) , ũ′2 (0, α) = ( 4− 2α, 3α2 − α ) , where f̃1 (x, α) = [ f1 (x, α) , f1 (x, α) ] and f̃2 (x, α) = [ f2 (x, α) , f2 (x, α) ] are given by f1 (x, α) = (3− α) + (1− 2α) ex + ( 1− 2α3 ) cosx, f1 (x, α) = ( 5α2 − 3 ) − αex − ( 3α2 − 2α ) cosx, f2 (x, α) = (3− 2α)x− (1− 2α) ex + ( 2− 3α5 ) sinx, f2 (x, α) = ( 3α5 − 2 ) x+ ( 2α2 − 1 ) ex − α sinx, The exact solutions of the LFSVIDEs (14) are ũ1 (x) = ex + cosx and ũ2 (x) = ex + sinx. Operating by the same way proceeding (3)-(9) as above, and applying the fuzzy Stan- dard ADM, the lower iterations (L) are then determined in the following recursive way:{ u1,0 (x, α) = (3− α) + (2− α)x+ L−1f1 (x, α) , u2,0 (x, α) = (3− 2α) + (4− 2α)x+ L−1f2 (x, α) , u1,n+1 (x, α) = L−1 [∫ x 0 [ u1,n (t, α) + (x− t)u2,n (t, α) ] dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 [ (x− t)u1,n (t, α)− u2,n (t, α) ] dt ] , n = 0, 1, 2, . . . (15) M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 298 and the upper iterations (U) are{ u1,0 (x, α) = 2α+ α2x+ L−1f1 (x, α) , u2,0 (x, α) = (2α− 1) + ( 3α2 − α ) x+ L−1f2 (x, α) ,{ u1,n+1 (x, α) = L−1 [∫ x 0 [u1,n (t, α) + (x− t)u2,n (t, α)] dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 [(x− t)u1,n (t, α)− u2,n (t, α)] dt ] , n = 0, 1, 2, . . . (16) Applying the fuzzy Modified Technique (MT) [11], the lower iterations (L) are then determined in the following recursive way:{ u1,0 (x, α) = (3− α) + (2− α)x, u2,0 (x, α) = (3− 2α) + (4− 2α)x, u1,1 (x, α) = L−1f1 (x, α) + L−1 [∫ x 0 [ u1,0 (t, α) + (x− t)u2,0 (t, α) ] dt ] , u2,1 (x, α) = L−1f2 (x, α) + L−1 [∫ x 0 [ (x− t)u1,0 (t, α)− u2,0 (t, α) ] dt ] , u1,n+1 (x, α) = L−1 [∫ x 0 [ u1,n (t, α) + (x− t)u2,n (t, α) ] dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 [ (x− t)u1,n (t, α)− u2,n (t, α) ] dt ] , n = 1, 2, 3, . . . (17) and the upper iterations (U) are{ u1,0 (x, α) = 2α+ α2x, u2,0 (x, α) = (2α− 1) + ( 3α2 − α ) x,{ u1,1 (x, α) = L−1f1 (x, α) + L−1 [∫ x 0 [u1,0 (t, α) + (x− t)u2,0 (t, α)] dt ] , u2,1 (x, α) = L−1f2 (x, α) + L−1 [∫ x 0 [(x− t)u1,0 (t, α)− u2,0 (t, α)] dt ] ,{ u1,n+1 (x, α) = L−1 [∫ x 0 [u1,n (t, α) + (x− t)u2,n (t, α)] dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 [(x− t)u1,n (t, α)− u2,n (t, α)] dt ] , n = 1, 2, 3, . . . (18) In Tables 1–4 we shown the maximum errors between the exact solutions and by fuzzy standard ADM and fuzzy MT for different norms on the interval [0, 1], where n represents the number of iterations. Table 1. Norm Error for ũ1 (x, α) (Problem 1) when α = 0.3 L∞ [0, 1] L2,S [0, 1] L2,I [0, 1] n i ADM MT ADM MT ADM MT 3 L 8.861E − 06 1.112E − 04 9.560E − 06 1.224E − 04 2.017E − 06 2.681E − 05 U 1.499E − 06 6.271E − 05 1.620E − 06 6.916E − 05 3.435E − 07 1.519E − 05 4 L 2.637E − 08 1.289E − 07 4.215E − 08 1.428E − 07 1.177E − 08 3.098E − 08 U 1.547E − 08 8.198E − 08 3.198E − 08 9.351E − 08 9.480E − 09 2.103E − 08 5 L 2.035E − 08 2.040E − 08 3.765E − 08 3.768E − 08 1.095E − 08 1.096E − 08 U 1.672E − 08 1.675E − 08 3.280E − 8 3.282E − 08 9.631E − 09 9.634E − 09 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 299 Table 2. Norm Error for ũ2 (x, α) (Problem 1) when α = 0.3 L∞ [0, 1] L2,S [0, 1] L2,I [0, 1] n i ADM MT ADM MT ADM MT 3 L 6.781E − 06 5.881E − 06 7.323E − 06 6.567E − 06 1.549E − 06 1.472E − 06 U 1.305E − 06 2.815E − 05 1.407E − 06 3.087E − 05 2.961E − 07 6.720E − 06 4 L 3.318E − 08 4.353E − 08 5.997E − 08 6.799E − 08 1.729E − 08 1.877E − 08 U 5.533E − 09 1.646E − 08 1.044E − 08 1.087E − 08 2.975E − 09 3.614E − 09 5 L 2.757E − 08 2.757E − 08 5.616E − 08 5.616E − 08 1.660E − 08 1.660E − 08 U 4.671E − 08 4.683E − 09 9.892E09 9.898E − 09 2.876E − 09 2.877E − 09 Table 3. Norm Error for ũ1 (x, α) (Problem 1) when α = 0.9 L∞ [0, 1] L2,S [0, 1] L2,I [0, 1] n i ADM MT ADM MT ADM MT 3 L 6.565E − 06 2.533E − 05 7.086E − 06 2.783E − 05 1.496E − 06 6.075E − 06 U 5.477E − 06 9.163E − 06 5.912E − 06 1.018E − 05 1.248E − 06 20264E − 06 4 L 1.009E − 08 3.163E − 08 1.427E − 08 3.544E − 08 3.781E − 09 7.792E − 09 U 3.448E − 09 1.362E − 08 7.534E − 09 1.711E − 08 2.384E − 09 4.220E − 09 5 L 5.478E − 09 5.488E − 09 1.068E − 08 1.068E − 08 3.139E − 09 3.140E − 09 U 4.725E − 09 4.731E − 09 9.424E − 09 9.428E − 09 2.774E − 09 7.792E − 09 Table 4. Norm Error for ũ2 (x, α) (Problem 1) when α = 0.9 L∞ [0, 1] L2,S [0, 1] L2,I [0, 1] n i ADM MT ADM MT ADM MT 3 L 3.348E − 06 2.314E − 05 3.617E − 06 2.545E − 05 7.657E − 07 5.565E − 06 U 2.144E − 06 1.220E − 05 2.320E − 06 1.352E − 05 4.492E − 07 2.996E − 06 4 L 1.216E − 07 1.118E − 8 2.087E − 08 1.733E − 8 5.934E − 09 4.481E − 09 U 8.131E − 10 1.502E − 8 1.572E − 09 1.609E − 08 4.182E − 10 3.314E − 09 5 L 9.265E − 09 9.276E − 09 1.887E − 08 1.887E − 08 5.575E − 09 5.576E − 09 U 1.130E − 09 1.123E − 09 1.982E − 09 1.978E − 09 5.553E − 10 5.545E − 10 In the following Figures (1–12) shows both the exact solutions of (ũi (x, α) , i = 1, 2) and the Standard ADM (ϕi,3 (x) , i = 1, 2) for the interval 0 ≤ t ≤ 1. We represent the exact solutions with a continuous line and use the symbol ◦ for the ADM ϕi,3 (x), i = 1, 2. M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 300 Figure 1: ϕ1,3 (x, α) , u1 (x., α) , α = 0.3 Figure 2: ϕ1,3 (x, α) , u1 (x, α) , α = 0.3 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 301 Figure 3: ϕ2,3 (x, α) , u2 (x, α) , α = 0.3 Figure 4: ϕ2,3 (x, α) , u2 (x, α) , α = 0.3 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 302 Figure 5: ϕ1,3 (x, α) , u1 (x, α) , α = 0.6 Figure 6: ϕ1,3 (x, α) , u1 (x, α) , α = 0.6 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 303 Figure 7: ϕ2,3 (x, α) , u2 (x, α) , α = 0.6 Figure 8: ϕ2,3 (x, α) , u2 (x, α) , α = 0.6 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 304 Figure 9: ϕ1,3 (x, α) , u1 (x, α) , α = 0.9 Figure 10: ϕ1,3 (x, α) , u1 (x, α) , α = 0.9 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 305 Figure 11: ϕ2,3 (x, α) , u2 (x, α) , α = 0.9 Figure 12: ϕ2,3 (x, α) , u2 (x, α) , α = 0.9 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 306 Problem 2. We consider the following non-linear fuzzy system of Volterra integro-differential equations (NFSVIDEs) of the second kind ũ′′1 (x, α) = f̃1 (x, α) + ∫ x 0 [ ũ21 (t, α) + ũ22 (t, α) ] dt, ũ′′2 (x, α) = f̃2 (x, α) + ∫ x 0 [ ũ21 (t, α)− ũ22 (t, α) ] dt, (19) with the initial conditions ũ1 (0, α) = (0, 0) , ũ′1 (0, α) = (2− α, α) , ũ2 (0, α) = (0, 0) , ũ′2 (0, α) = ( 2α− α2, 3− 2α ) , where f̃1 (x, α) = [ f1 (x, α) , f1 (x, α) ] and f̃2 (x, α) = [ f2 (x, α) , f2 (x, α) ] are given by f1 (x, α) = (9− 3α)x+ ( 1− 3α2 ) 3 x3 + (2− 4α) 7 x7, f1 (x, α) = ( 1 + 5α2 ) x− ( α3 + 3 ) 6 x3 − ( α2 + 1 ) 7 x7, f2 (x, α) = − ( 7− α2 ) x− ( 17α2 − α ) 20 x5, f2 (x, α) = − (2 + 4α)x− 4α 5 x5, The exact solutions of the NFSVIDEs (19) are ũ1 (x) = x+ x3 and ũ2 (x) = x− x3. Operating by the same way proceeding (3)-(9) as above, and applying the fuzzy Stan- dard ADM, the lower iterations (L) are then determined in the following recursive way:{ u1,0 (x, α) = (2− α)x+ L−1f1 (x, α) , u2,0 (x, α) = ( 2α− α2 ) x+ L−1f2 (x, α) , u1,n+1 (x, α) = L−1 [∫ x 0 ( A1,n +A2,n ) dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 ( A1,n −A2,n ) dt ] , n = 0, 1, 2, . . . (20) and the upper iterations (U) are{ u1,0 (x, α) = αx+ L−1f1 (x, α) , u2,0 (x, α) = (3− 2α)x+ L−1f2 (x, α) ,{ u1,n+1 (x, α) = L−1 [∫ x 0 ( A1,n +A2,n ) dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 ( A1,n −A2,n ) dt ] , n = 0, 1, 2, . . . (21) M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 307 where the non-linear terms defined by the series u21,n (t, α) = ∞∑ n=0 A1,n, u21,n (t, α) = ∞∑ n=0 A1,n, u22,n (t, α) = ∞∑ n=0 A2,n, u22,n (t, α) = ∞∑ n=0 A2,n, (22) the corresponding Adomian polynomials [ A1,n, A1,n ] and [ A2,n, A2,n ] [3, 4] are A1,n = ∞∑ n=0 u1,iu1,n−i, A1,n = ∞∑ n=0 u1,iu1,n−i, n ≥ i, n = 0, 1, 2, . . . A2,n = ∞∑ n=0 u2,iu2,n−i, A2,n = ∞∑ n=0 u2,iu2,n−i, n ≥ i, n = 0, 1, 2, . . . Applying the fuzzy Modified Technique (MT) [11], the lower iterations (L) are then determined in the following recursive way:{ u1,0 (x, α) = (2− α)x, u2,0 (x, α) = ( 2α− α2 ) x, u1,1 (x, α) = L−1f1 (x, α) + L−1 [∫ x 0 ( A1,0 +A2,0 ) dt ] , u2,1 (x, α) = L−1f2 (x, α) + L−1 [∫ x 0 ( A1,0 −A2,0 ) dt ] , u1,n+1 (x, α) = L−1 [∫ x 0 ( A1,n +A2,n ) dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 ( A1,n −A2,n ) dt ] , n = 1, 2, 3, . . . (23) and the upper iterations (U) are{ u1,0 (x, α) = αx, u2,0 (x, α) = (3− 2α)x,{ u1,1 (x, α) = L−1f1 (x, α) + L−1 [∫ x 0 ( A1,0 +A2,0 ) dt ] , u2,1 (x, α) = L−1f2 (x, α) + L−1 [∫ x 0 ( A1,0 −A2,0 ) dt ] ,{ u1,n+1 (x, α) = L−1 [∫ x 0 ( A1,n +A2,n ) dt ] , u2,n+1 (x, α) = L−1 [∫ x 0 ( A1,n −A2,n ) dt ] , n = 1, 2, 3, . . . (24) In Tables 5–8 display a comparison of the numerical results applying the ADM (n = 4), Iteration of the Integral Equation (IIE) (23), (24) and the numerical solution of (23), (24) with the Simpson rule (SIMP) and the trapezoidal rule (TRAP) on the interval [0, 1]. Twenty points have been used in the Simpson and trapezoidal methods. In Tables 9 and 10 we present the maximum error residual obtained by fuzzy standard ADM and fuzzy MT on the interval [0, 1], where n represents the number of iterations. M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 308 Table 5. Numerical results for ũ1 (x, α) (Problem 2) when α = 0.3 x i ADM IIE SIMP TRAP 0.0 L 0.0 0.0 0.0 0.0 U 0.0 0.0 0.0 0.0 0.2 L 0.350820906 0.350820906 0.350820906 0.350820978 U 0.061956312 0.061956312 0.061956312 0.061956441 0.4 L 0.767091047 0.767091047 0.767091047 0.767093546 U 0.136187326 0.136187326 0.136187326 0.136191319 0.6 L 1.317168717 1.317168717 1.317168718 1.317190739 U 0.237475634 0.237475634 0.237475634 0.237504395 0.8 L 2.076937697 2.076937698 2.076937706 2.077053136 U 0.384710630 0.384710628 0.384710630 0.384822684 1.0 L 3.139469824 3.139469864 3.139469904 3.139936553 U 0.600309591 0.600309524 0.600309527 0.600616856 Table 6. Numerical results for ũ2 (x, α) (Problem 2) when α = 0.3 x i ADM IIE SIMP TRAP 0.0 L 0.0 0.0 0.0 0.0 U 0.0 0.0 0.0 0.0 0.2 L 0.092801026 0.092801026 0.092801026 0.092801087 U 0.475703185 0.475703185 0.475703185 0.475703060 0.4 L 0.130785108 0.130785108 0.130785108 0.130787377 U 0.924910864 0.924910864 0.924910864 0.924907020 0.6 L 0.061390455 0.061390455 0.061390456 0.061411601 U 1.317657321 1.317657321 1.317657320 1.317629935 0.8 L −0.161630983 −0.161630982 −0.161630975 −0.161517388 U 1.617533199 1.617533199 1.617533198 1.617428686 1.0 L −0.570121458 −0.570121448 −0.570121413 −0.569668950 U 1.779759247 1.779759262 1.779759263 1.779483691 M. T. Younis, W. Al-Hayani / Eur. J. Pure Appl. Math, 15 (1) (2022), 290-313 309 Table 7. Numerical results for ũ1 (x, α) (Problem 2) when α = 0.9 x i ADM IIE SIMP TRAP 0.0 L 0.0 0.0 0.0 0.0 U 0.0 0.0 0.0 0.0 0.2 L 0.228404070 0.228404070 0.228404070 0.228404119 U 0.186735344 0.186735344 0.186735344 0.186735394 0.4 L 0.507332123 0.507332123 0.507332123 0.507333716 U 0.413926676 0.413926676 0.413926676 0.413928238 0.6 L 0.887832878 0.887832878 0.887832879 0.887845675 U 0.722198170 0.722198170 0.722198171 0.722209943 0.8 L 1.422181142 1.422181142 1.422181145 1.422242142 U 1.152251625 1.152251626 1.152251628 1.152302769 1.0 L 2.165166483 2.165166491 2.165166509 2.165395782 U 1.744176060 1.744176064 1.744176075 1.744349048 Table 8. Numerical results for ũ2 (x, α) (Problem 2) when α = 0.9 x i ADM IIE SIMP TRAP 0.0 L 0.0 0.0 0.0 0.0 U 0.0 0.0 0.0 0.0 0.2 L 0.189747962 0.189747962 0.189747962 0.189747969 U 0.232529982 0.232529982 0.232529982 0.232529970 0.4 L 0.330021457 0.330021457 0.330021457 0.330021918 U 0.420160289 0.420160289 0.420160289 0.420160096 0.6 L 0.371610554 0.371610554 0.371610555 0.371616888 U 0.517601598 0.517601598 0.517601598 0.517602534 0.8 L 0.266194321 0.266194321 0.266194325 0.266237859 U 0.478812319 0.478812319 0.478812322 0.478830230 1.0 L −0.032275868 −0.032275864 −0.032275846 −0.032075986 U 0.256634701 0.256634705 0.256634717 0.256743055 Table 9. Error Residual for ũ1 (x, α) (Problem 2) α = 0.3 α = 0.6 α = 0.9 n i ADM MT ADM MT ADM MT 3 L 1.128E − 03 4.673E − 01 6.123E − 04 3.920E − 01 2.906E − 04 3.152E − 01 U 1.600E − 03 7.019E − 02 4.431E − 04 1.227E − 01 1.895E − 04 2.319E − 01 4 L 1.285E − 05 1.989E − 03 6.062E − 06 8.645E − 04 2.495E − 06 2.060E − 03 U 1.698E − 05 2.298E − 03 1.955E − 06 2.962E − 03 1.285E − 06 2.724E − 03 5 L 1.050E − 07 1.255E − 03 4.649E − 08 9.859E − 04 1.720E − 08 7.284E − 04 U 1.847E − 07 1.159E − 04 2.712E − 08 2.320E − 04 9.465E − 08 4.912E − 04 6 L 7.390E − 10 1.156E − 05 3.058E − 10 9.433E − 07 9.920E − 11 3.428E − 06 U 1.350E − 09 3.118E − 06 7.110E − 11 5.122E − 06 5.014E − 11 5.347E − 06 REFERENCES 310 Table 10. Error Residual for ũ2 (x, α) (Problem 2) α = 0.3 α = 0.6 α = 0.9 n i ADM MT ADM MT ADM MT 3 L 3.751E − 04 1.147E − 01 2.955E − 04 5.029E − 02 1.979E − 04 5.800E − 03 U 3.081E − 04 4.793E − 02 3.550E − 05 5.832E − 02 1.570E − 04 3.058E − 02 4 L 2.525E − 06 1.333E − 02 2.198E − 06 9.011E − 03 1.211E − 06 5.735E − 03 U 4.310E − 06 1.150E − 03 1.956E − 06 1.755E − 03 9.418E − 06 3.528E − 03 5 L 4.271E − 08 1.581E − 03 1.928E − 08 1.080E − 03 7.010E − 09 6.880E − 04 U 5.304E − 08 2.046E − 05 7.163E − 09 2.097E − 03 4.217E − 09 3.332E − 04 6 L 4.399E − 09 2.319E − 05 1.501E − 10 1.066E − 05 4.090E − 11 3.979E − 06 U 8.205E − 10 2.289E − 07 3.357E − 11 5.404E − 07 1.672E − 11 6.306E − 07 6. Conclusions The results obtained from the two given problems, have been shown that the Ado- mian decomposition method and modified technique are powerful and efficient techniques to find the approximate solution for both linear and non-linear fuzzy system of Volterra integro-differential equations of the second kind according to the numerical results, men- tioned in the tables and graphs. Therefore, increasing the number of iterations in the Adomian decomposition method and modified technique, make the approximate solution tends to the exact solution. For the non-linear problem, we conclude that the numerical results by using the numerical solution with the Simpson rule close to the exact solution more than the numerical solution with the trapezoidal rule. Finally, we conclude that the methods are extremely quick and simple to converge for solving any equation or system for any value of α, and it provides more accurate results and therefore is more advantageous. 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