Numerical Method for Solving Second-Order. Fuzzy Boundary Value Problems. by Using the RPSM

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1 International Mathematical Forum, Vol., 26, no. 4, HIKARI Ltd, Numerical Method for Solving Second-Order Fuzzy Boundary Value Problems by Using the RPSM Mazen Al Jazazi Department of Mathematics, Al-Albayt University, Mafraq 253, Jordan Copyright 26 Mazen Al Jazazi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to present analytic-numeric solutions of two-point, second-order fuzzy boundary value problems under strongly generalized differentiability based on the residual power series (RPS) method. The new approach provides the solution in the form of a rapidly convergent series with easily computable components, using symbolic computation software. The proposed method obtains the epansion of the solutions of the parameterized systems under appropriate guesses approimations. The proposed technique will be applied to a few test eamples in order to illustrate the accuracy, the efficiency, and the applicability of the method. Keywords: Residual power series method, Fuzzy differential equations Introduction The study of FDEs subjects to given fuzzy boundary conditions forms a suitable setting for the mathematical modeling of real-world problems in which uncertainty or vagueness pervades. In this paper, we introduce an iterative technique for numerically approimating solutions of FBVPs under the assumption of strongly generalized differentiability, which is y () = F (, y(), y ()), a b, (.)

2 644 Mazen Al Jazazi subject to the fuzzy boundary conditions y(a) = α, y(b) = β, (.2) where F [a, b] R F R F R F are continuous fuzzy-valued functions. The study of FDEs has gained importance in recent times; here, we are focusing our attention on the second-order, two-point fuzzy boundary value problems (FBVPs). First of all, approaches to FBVPs and other fuzzy equations can be of three types. The first approach assumes that even if only the boundary values are fuzzy, the solution is a fuzzy function and consequently the derivatives in the differential equation must be considered as fuzzy derivatives [, 2]. These can be either the use of the Hukuhara or the Seikkala derivatives for fuzzy-valued functions. In the second approach, the FBVP is transformed to a crisp one by interpreted it as a family of differential inclusions [3, 4]. The third approach based on the Zadeh s etension principle, where the associated crisp problem is solved and in the solution the boundary fuzzy values are substituted instead of the real constants, and in the final solution, arithmetic operations are considered to be operations on fuzzy numbers [5]. In various subjects of science and engineering, nonlinear evaluation fuzzy equations as well as their analytic and numerical solutions, are essentially important; therefore, FDEs are commonly solved approimately using numerical methods. On the other hand, many applications for different problems by using other numerical algorithms can be found in [6-4] This work is organized in four sections including the introduction. In the net chapter, we present some necessary definitions and preliminary results from the fuzzy calculus theory. The overview to the RPS method is utilized in Section 3. In Section 4, numerical eperiments and simulation results are presented and discussed. This work ends in Section 5 with some concluding remarks and future recommendations. 2 Preliminary notes Definition 2.[5] A fuzzy number u is a fuzzy subset of R with normal, conve, and upper semicontinuous membership function of bounded support. For each r (, ], set [u] r = {s R: u(s) r} and [u] = {s, R u(s) > } where { } denote the closure of { }. If uis a fuzzy number, then [u] r = [u (r), u 2 (r)], where u₁(r) = min{s s [u] r } and u₂(r) = ma{s s [u] r } for each r [,]. Theorem 2. [5] Suppose that u, u 2 : [,] R satisfy the following conditions; first, u₁ is a bounded increasing function and u₂ is a bounded decreasing function with u₁() u₂(); second, for each k (, ], u₁ and u₂ are left-hand continuous functions at r = k; third, u₁ and u₂ are right-hand continuous functions at r =. Then u: R [,] defined by u(s) = sup{r u₁ (r) s u₂(r)} is a fuzzy number with parameterizetion [u₁(r), u₂(r)]. Furthermore, ifu:

3 Numerical method for solving second-order fuzzy BVP 645 R [,] is a fuzzy number with parameterization[u₁(r), u₂(r)], then the functions u₁ and u₂ satisfy the aforementioned conditions. Definition 2.2 [7] The complete metric structure on R F is given by the Hausdorff distance mapping D: R F R F R + {} such that D(u, v) = sup r ma{ u r v r, u 2r v 2r }, (2.) for arbitrary fuzzy numbers u and v. Theorem 2.2. [8-2] If u and v are two fuzzy numbers, then for each r [, ], we have. [u + v] r = [u] r + [v] r = [u r + v r, u 2r + v 2r ]; 2. [λu] r = λ[u] r = [min{λu r, λu 2r }, ma{λu r, λu 2r }]; 3. [uv] r = [u] r [v] r = [min{u r v r, u r v 2r, u 2r v r, u 2r v 2r }, ma{u r v r, u r v 2r, u 2r v r, u 2r v 2r }]. Definition 2.3 Let u, v R F. If there eists an element w R F such that u = v + w, then wis called the Hukuhara difference of u and v, denoted byu v. Here, the sign stands always for Hukuhara difference and let us mention thatu v u + ( )v. Usually, we denote u + ( )vby u v, while u vstands for the Hukuhara difference. Definition 2.4 [2] Let y: [a, b] R F and fied [a, b]. We say that y is strongly generalized differentiable at, if there eists an element y ( ) R F such that either:. h > sufficiently close to, the H-differences y( + h) y( ), y( ) y( h) eist and y ( ) = lim h + lim, y( ) y( h) h + h y( + h) y( ) 2. h > sufficiently close to, the H-differences y( ) y( + h), y( h) y( ) eist andy ( ) = lim h + y( ) y( + h) h = lim h + h y( h) y( ) Definition 2.5 [5] Let y [a, b] R F. We say thatyis ()-differentiable on [a, b] if yis differentiable in the sense () of Definition 2.4 and its derivative is denoted D y. Similarly, we say that y is (2)-differentiable on [a, b] if y is differentiable in the sense (2) of Definition 2.4 and its derivative is denoted D 2 y. Theorem 2.3 [5] Let y: [a, b] R F, where [y()] r = [y r (), y 2r ()] for each r [,],. if y is ()-differentiable, then y r and y 2r are differentiable functions and [D y()] r = [y r (), y 2r ()], h. =

4 646 Mazen Al Jazazi 2. if y is (2)-differentiable, then y r and y 2r are differentiable functions and [D 2 y()] r = [y 2r (), y r ()]. For a given fuzzy-valued function y, we have two possibilities according to Definition 2.5in order to obtain the derivative of y as follows: D y() and D 2 y(). Anyhow, for each of these two derivative, we have again two possibilities of derivatives: D (D y()), D 2 (D y()), and D (D 2 y()), D 2 (D 2 y()), respectively. Definition 2.6 [2] Let y: [a, b] R F and n, m {,2}. We say that y is (n, m)- differentiable on [a, b] if D n y eist and its (m)-differentiable. The second 2 derivatives of y are denoted by D n,m y. Theorem2.4 Let D y [a, b] R F or D 2 y [a, b] R F, where [y()] r = [y r (), y 2r ()]for each r [,]:. D y is ()-differentiable, then y r and y 2r are differentiable functions and [D 2, y()] r = [y,r (), y 2,r ()], 2. D y is (2)-differentiable, then y r and y 2r are differentiable functions and [D 2,2 y()] r = [y 2,r (), y,r ()], 3. D 2 y is ()-differentiable, then y r and y 2r are differentiable functions and [D 2 2, y()] r = [y 2,r (), y,r ()], 4. D 2 y is (2)-differentiable, then y r and y 2r are differentiable functions and [D 2 2,2 y()] r = [y,r (), y 2,r ()]. For more details, we refer to [2-28] and references therein. 3 The Idea of the RPS Method Consider the following system of differential equations: y () = F (, y (), y 2 (), y (), y 2 ()), y 2 () = F 2 (, y (), y 2 (), y (), y 2 ()), (3.) with the boundary conditions y (a) = α, y 2 (a) = α 2 &y (b) = β, y 2 (b) = β 2. (3.2) First of all, we assume that the nonlinear system of Eq. (3.) satisfies the initial conditions y (a) = α, y 2 (a) = α 2 and y (a) = c, y 2 (a) = c 2, where the unknown constants c i can be determined later by substituting the boundary conditions y (b) = β, y 2 (b) = β 2 of Eq. (3.2) into the obtained series solutions. Suppose that these solutions take the form

5 Numerical method for solving second-order fuzzy BVP 647 y () = α + c + e,m ( ) m, m=2 y 2 () = α 2 + c 2 + e 2,m ( ) m. m=2 (3.3) Obviously, when =, sincey i () satisfy the initial conditions of Eq. (3.2), we have y (a) = α and y 2 (a) = α 2. Anyhow, depending on the initial guesses approimations y (a) = α, y 2 (a) = α 2 and y (a) = c, y 2 (a) = c 2, we can calculate y i () for m = 2,3, and approimate the solutions y () and y 2 ()by the kth-truncated series y k () = α + c + e,m ( a) m, k m=2 k y k 2 () = α 2 + c 2 + e 2,m ( a) m. m=2 (3.4) Prior to applying the RPS technique, we rewrite system of BVP (3.) and (3.2) in the form of the following: y () F (, y (), y 2 (), y (), y 2 ()) =, y 2 () F 2 (, y (), y 2 (), y (), y 2 ()) =. (3.5) The subsisting of kth-truncated series y i k () of Eq. (3.4) into Eq. (3.5) leads to the following definition for the kth residual functions: Res k () = (y k ()) F (, y k (), y 2 k (), (y k ()), (y 2 k ()) ), Res 2 k () = (y 2 k ()) F 2 (, y k (), y 2 k (), (y k ()), (y 2 k ()) ). (3.6) Now, in order to obtain the 2nd-approimate solutions, we put k = 2 and substitute = a into Eq. (3.6) and using the fact that Res 2 (a) = Res 2 2 (a) =, to conclude the following values e,2 = 2 F (a, α + c a, α 2 + c 2 a, c, c 2 ) and e 2,2 = 2 F 2(a, α + c a, α 2 + c 2 a, c, c 2 ). Thus, using 2nd-truncated series the second approimation for system of BVP (3.) and (3.2) can be written as follows: y 2 () = α + c + 2 F (a, α + c a, α 2 + c 2 a, c, c 2 )( a) 2, y 2 2 () = α 2 + c F 2(a, α + c a, α 2 + c 2 a, c, c 2 )( a) 2. (3.7)

6 648 Mazen Al Jazazi Similarly, to find the 3rd approimation, we put k = 3 and differentiate both sides of Eq. (3.6) with respect to and finally substitute = a, to get (Here, we putting w () = α + c + e,2 ( a) 2 + e,3 ( a) 3, w 2 () = α 2 + c 2 + e 2,2 ( a) 2 + e 2,3 ( a) 3, w 3 () = c + 2e,2 ( a) + 3e,3 ( a) 2, w 4 () = c 2 + 2e 2,2 ( a) + 3e 2,3 ( a) 2 ). d d Res 3 (a) = 6e,3 [ F (a, α + c a, α 2 + c 2 a, c, c 2 ) + c w F (a, α + c a, α 2 + c 2 a, c, c 2 ) + c 2 w 2 F (a, α + c a, α 2 + c 2 a, c, c 2 ) + 2e,2 w 3 F (a, α + c a, α 2 + c 2 a, c, c 2 ) + 2e 2,2 w 4 F (a, α + c a, α 2 + c 2 a, c, c 2 )], d d Res 2 3 (a) = 6e 2,3 [ F 2(a, α + c a, α 2 + c 2 a, c, c 2 ) +c F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) + c 2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) 2 +2e,2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) + 2e 2,2 F 3 w 2 (a, α + c a, α 2 + c 2 a, c, c 2 )]. 4 (3.8) (3.9) In fact d d Res 3 (a) = d d Res 2 3 (a) =. Thus, one can write e,3 = 6 [ F (a, α + c a, α 2 + c 2 a, c, c 2 ) + c F w (a, α + c a, α 2 + c 2 a, c, c 2 ) + c 2 F w (a, α + c a, α 2 + c 2 a, c, c 2 ) 2 + 2e,2 F w (a, α + c a, α 2 + c 2 a, c, c 2 ) 3 + 2e 2,2 F w (a, α + c a, α 2 + c 2 a, c, c 2 )], 4 e 2,3 = 6 [ F 2(a, α + c a, α 2 + c 2 a, c, c 2 ) + c F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) + c 2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) 2 + 2e,2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) 3 + 2e 2,2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 )]. 4 (3.) (3.)

7 Numerical method for solving second-order fuzzy BVP 649 Hence, using 3rd-truncated series, the third approimation for system of BVP (3.) and (3.2) can be written as y 3 () = α + c + 2 F (a, α + c a, α 2 + c 2 a, c, c 2 )( a) [ F (a, α + c a, α 2 + c 2 a, c, c 2 ) + c w F (a, α + c a, α 2 + c 2 a, c, c 2 ) + c 2 w 2 F (a, α + c a, α 2 + c 2 a, c, c 2 ) + 2e,2 w 3 F (a, α + c a, α 2 + c 2 a, c, c 2 ) + 2e 2,2 w 4 F (a, α + c a, α 2 + c 2 a, c, c 2 )] ( a) 3, (3.2) y 2 3 () = α 2 + c F 2(a, α + c a, α 2 + c 2 a, c, c 2 )( a) [ F 2(a, α + c a, α 2 + c 2 a, c, c 2 ) + c F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) (3.3) + c 2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) 2 + 2e,2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 ) 3 + 2e 2,2 F w 2 (a, α + c a, α 2 + c 2 a, c, c 2 )] ( a) 3. 4 Finally, if we substitute the boundary conditions y (b) = β, y 2 (b) = β 2 of Eq. (3.) into Eq. (3.), then we obtain a system of nonlinear equations in the variables c, c 2, which can be easy solved using one of the symbolic computation software. This procedure can be repeated till the arbitrary order coefficients of RPS solutions for system of BVP (3.) and (3.2) are obtained. For more details about RPSM and other numerical schemes, we refer to [29-34]. 4 Numerical Results Eample Consider the following FDE: y () = y () + δ +,, (4.) subject to the boundary conditions y() =, y() = δ, (4.2) where δ is triangular fuzzy number having membership function δ(s) = ma(, s ), s R. (4.3) Case : The system of the ODEs corresponding to (,)-differentiability is y r () = y r () + r&y 2r () = y 2r () + 2 r, (4.4)

8 65 Mazen Al Jazazi subject to the boundary conditions y r () =, y 2r () = &y r () = r, y 2r () = r. (4.5) The corresponding (,)-system has the analytic solutions y r () = e [ e + r( 2 + 2e + e )], (4.6) y 2r () = e [ 3 + 3e + 2 2e r( 2 + 2e + e )]. (4.7) Case 2: The system of the ODEs corresponding to(,2)-differentiability is y r () = y 2r () + 2 r&y 2r () = y r () + r, (4.8) subject to the boundary conditions y r () =, y 2r () = &y r () = r, y 2r () = r. (4.9) The corresponding (,2)- system has the analytic solutions y r () = e e [ 2(r )e + e2 e + (2 + r( 2 + )) + e ( + r)], (4.) y 2r () = e e [2(r )e + e2 + e + (2 + r( 2 + ) 2) + e ( (2 r))]. (4.) Case 3: The system of the ODEs corresponding to (2,)-differentiability is y r () = y r () + 2 r&y 2r () = y 2r () + r, (4.2) subject to the boundary conditions y r () =, y 2r () = &y r () = r, y 2r () = r. (4.3) The corresponding (2,)-system has the analytic solutions y r () = e [ + e + (2 r) (2 r)e]. (4.4) y 2r () = e [ + e + r re]. (4.5) Case 4: The system of the ODEs corresponding to (2,2)-differentiability is y r () = y 2r () + r&y 2r () = y r () + 2 r, (4.6) subject to the boundary conditions y r () =, y 2r () = &y r () = r, y 2r () = r, (4.7) The corresponding (2,2) - system has the analytic solutions y r () = e [ + e + (2 r) (2 r)e], (4.8)

9 Numerical method for solving second-order fuzzy BVP 65 y 2r () = e [ + e + r re]. (4.9) Hence, the following results will be obtained: The (,)-RPS solutions of (,)-system are given as y r () = ( r ) + ( r ) 2 + 4r + ( ) 9 2r + ( ), (4.2) y 2r () = ( r ) + ( r ) ( r ) ( r ). (4.2) The (,2)-RPS solutions of (,2)-system are given as y r () = ( r ) + ( r ) 2 + 4r + ( ) ( r ), (4.22)

10 652 Mazen Al Jazazi () = ( r ) + ( r ) ( r ) 9 2r + ( ). y 2r (4.23) y r The (2,)-RPS solutions of (2,)-system are given as () = (r ) (4.24) , y 2r () = ( r) (4.25) The (2,2)-RPS solutions of (2,2)-system are given as () = (r ) , (4.26) y r y 2r () = ( r) (4.27) The absolute errors of numerically approimating y r () by y r () for the (,)- system have been calculated for various and r as shown in Tables -5, while in Tables 6- the absolute errors have been tabulated for the (,2)-system. Table : Numerical results of (,)-system at r = for Eample 4. y y 2, () y, () y, () 2, ()

11 Numerical method for solving second-order fuzzy BVP 653 Table 2: Numerical results of (,)-system at r =.25 for Eample y,.25 () y,.25 () y 2,.25 () y 2,.25 () Table 3: Numerical results of (,)-system at r =.5 for Eample y,.5 () y,.5 () y 2,.5 () y 2,.5 () Table 4: Numerical results of (,)-system at r =.75 for Eample 4. y 2,.75 () y,.75 () y,.75 () y 2,.75 () Table 5: Numerical results of (,)-system at r = for Eample 4. y y 2, () y, () y, () 2, () Table 6: Numerical results of (,2)-system at r = for Eample 4. y, () y, () y 2, () y 2, ()

12 654 Mazen Al Jazazi Table 7: Numerical results of (,2)-system at r =.25 for Eample 4. y,.25 () y,.25 () y 2,.25 () y 2,.25 () Table 8: Numerical results of (,2)-system at r =.5 for Eample 4. y,.5 () y,.5 () y 2,.5 () y 2,.5 () Table 9: Numerical results of (,2)-system at r =.75 for Eample 4. y,.75 () y,.75 () y 2,.75 () y 2,.75 () Table : Numerical results of (,2)-system at r = for Eample 4. y, () y, () y 2, () y 2, () References [] V. Lakshmikantham, K. N. Murty and J. Turner, Two-point boundary value problems associated with non-linear fuzzy differential equations, Mathematical Inequalities & Applications, 4 (2),

13 Numerical method for solving second-order fuzzy BVP 655 [2] D. O'Regana, V. Lakshmikantham, J.J. Nieto, Initial and boundary value problems for fuzzy differential equations, Nonlinear Analysis, 54 (23), [3] E. Hüllermeier, An approach to modelling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based System, 5 (997), [4] D. Li, M. Chen, X. Xue, Two-point boundary value problems of uncertain dynamical systems, Fuzzy Sets and Systems, 79 (2), [5] X. Guo, D. Shang, X. Lu, Fuzzy approimate solutions of second-order fuzzy linear boundary value problems, Boundary Value Problems, 23 (23), [6] A. El-Ajou, O. Abu Arqub and M. Al-Smadi, A general form of the generalized Taylor's formula with some applications, Applied Mathematical and Computation, 256 (25), [7] I. Komashynska and M. Al-Smadi, Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type, Journal of Applied Mathematics, 24 (24), Article ID 45959, -. [8] I. Komashynska, M. Al-Smadi, O. Abu Arqub, S. Momani, An Efficient Analytical Method for Solving Singular Initial Value Problems of Nonlinear Systems, Applied Mathematics & Information Sciences, (26), [9] M. Al-Smadi, A. Freihat, O. Abu Arqub and N. Shawagfeh, A novel multistep generalized differential transform method for solving fractional-order Lu chaotic and hyperchaotic systems, Journal of Computational Analysis and Applications, 9 (25), no. 4, [] M. Al-Smadi and Z. Altawallbeh, Solution of System of Fredholm Integro- Differential Equations by RKHS Method, International Journal of Contemporary Mathematical Sciences, 8 (23), no., [] M. Al-Smadi, O. Abu Arqub and A. El-Ajou, A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems,

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15 Numerical method for solving second-order fuzzy BVP 657 [22] M. Al-Smadi, O. Abu Arqub and S. Momani, A Computational Method for Two-Point Boundary Value Problems of Fourth-Order Mied Integrodifferential Equations, Mathematical Problems in Engineering, 23 (23), Article ID 83274, -. [23] M. Al-Smadi, A. Freihat, M. Abu Hammad, S. Momani and O. Abu Arqub, Analytical approimations of partial differential equations of fractional order with multistep approach, Journal of Computational and Theoretical Nanoscience, 26, in press. [24] O. Abu Arqub, M. Al-Smadi and N. Shawagfeh, Solving Fredholm integrodifferential equations using reproducing kernel Hilbert space method, Applied Mathematics and Computation, 29 (23), [25] S. Momani, A. Freihat and M. AL-Smadi, Analytical study of fractionalorder multiple chaotic FitzHugh-Nagumo neurons model using multi-step generalized differential transform method, Abstract and Applied Analysis, 24 (24), Article ID , -. [26] O. Abu Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Computing and Applications, [27] O. Abu Arqub, M. Al-Smadi, S. Momani, T. Hayat, Numerical Solutions of Fuzzy Differential Equations using Reproducing Kernel Hilbert Space Method, Soft Computing, (25), [28] G. Gumah, K. Moaddy, M. AL-Smadi and I. Hashim, Solutions of Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method, Journal of Function Spaces, 26 (26), Article ID , -. [29] O. Abu Arqub, Reproducing kernel algorithm for the analytical-numerical solutions of nonlinear systems of singular periodic boundary value problems, Mathematical Problems in Engineering, 25 (25), -3, Article ID [3] K. Moaddy, M. AL-Smadi and I. Hashim, A Novel Representation of the Eact Solution for Differential Algebraic Equations System Using Residual Power-Series Method, Discrete Dynamics in Nature and Society, 25 (25), -2, Article ID

16 658 Mazen Al Jazazi [3] R. Abu-Gdairi, M. Al-Smadi and G. Gumah, An Epansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme, Journal of Mathematics and Statistics, (25), no. 2, [32] M. Al-Smadi, Solving initial value problems by residual power series method, Theoretical Mathematics and Applications, 3 (23), no., [33] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh and S. Momani, Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method, Applied Mathematical and Computation, in press. [34] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems, Soft Computing, 26, in press. Received: April 4, 26; Published: July 5, 26