Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Journal of Applied Mathematics Volume 212, Article ID 325473, 17 pages doi:1.1155/212/325473 Research Article Formulation and Solution of nth-order Derivative Fuzzy Integrodifferential Equation Using New Iterative Method with a Reliable Algorithm A. A. Hemeda Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt Correspondence should be addressed to A. A. Hemeda, aahemeda@yahoo.com Received 11 July 212; Revised 9 September 212; Accepted 1 September 212 Academic Editor: Fazlollah Soleymani Copyright q 212 A. A. Hemeda. 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. The nth-order derivative fuzzy integro-differential equation in parametric form is converted to its crisp form, and then the new iterative method with a reliable algorithm is used to obtain an approximate solution for this crisp form. The analysis is accompanied by numerical examples which confirm efficiency and power of this method in solving fuzzy integro-differential equations. 1. Introduction In 1975 Zadeh and then Dubois and Prade 1 introduced fuzzy numbers and fuzzy arithmetic. This concept propagated widely to various problems, for example, fuzzy linear systems 2, 3, fuzzy differential equations 4, 5, fuzzy integral equations 6 8, and fuzzy integro-differential equations 9 12. Additional topics can be found in 13, 14. Recently, several numerical methods were suggested to solve integro-differential equations, for example, Sine-Cosine wavelet used by Tavassoli Kajani et al. to obtain a solution of linear integro-differential equations 15 and variational iteration method used by Abbasbandy and Hashemi to formulate and solve fuzzy integro-differential equation 12, by Saberi-Nadjafi and Tamamgar for solving system of integro-differential equations 16, and by Shang and Han for solving nth-order integro-differential equations 17. Some other worthwhile works can be found in 18, 19. Recently, Daftardar-Gejji and Jafari 2, 21 proposed the new iterative method. This method has proven useful for solving a variety of linear and nonlinear equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer and fractional order, and system of equations as well. The new iterative method is simple to understand and easy to implement using computer packages and

2 Journal of Applied Mathematics yields better results 21 than the existing Adomian decomposition method 22, homotopy perturbation method 23, or variational iteration method 24. For more details, see 25 4. In the present work we apply the new iterative method with a reliable algorithm to solve the nth-order derivative fuzzy integro-differential equation in its crisp form. 2. Preliminaries In this section we set up the basic definitions of fuzzy numbers and fuzzy functions. Definition 2.1 see 2, 3, 12. A fuzzy number in parametric form is an ordered pair of functions ur, ur, r 1 which satisfy the following requirements: 1 ur is a bounded left continuous nondecreasing function over, 1, 2 ur is a bounded left continuous nonincreasing function over, 1, 3 ur ur, r 1. Remark 2.2 see 6, 12. Letur ur, ur, r 1 be a fuzzy number, and we can take u c r 1 ( ) ur ur, u d r 1 ( ) ur ur. 2.1 2 2 It is clear that u d r, ur u c r u d r and ur u c ru d r, and also a fuzzy number u E is said symmetric if u c r is independent of r for all r 1. Let f : a, b E for each partition P {t,...,t n } of a, b and for arbitrary ξ i t i 1 t i,1 i n suppose that R P n i1 fξ it i t i 1 and Δmax{ t i t i 1, i 1,...,n}. The definite integral of ft over a, b is b a ftdt lim Δ R P provided that this limit exists in the metric D. If the fuzzy function ft is continuous in the metric D, its definite integral exists. Also we have b a ft, rdt b a ft, rdt and b a ft, rdt b ft, rdt. a Definition 2.3 see 33. Letf : a, b R F and t a, b. We say that f is Hukuhara differentiable at t if there exists an element f t R F, such that for all h>sufficiently small, ft h H ft,ft H ft h and ft h H ft ft H ft h lim lim f t, h h h h 2.2 where H is Hukuhara difference, R F is the set of fuzzy numbers, and the limit is in the metric D. In parametric form, if f f, f, then f f, f, where f, f are Hukuhara differentiable of f, f, respectively. From 2.2, thenth-order Hukuhara differentiable, f n,off at t can be defined as in the following definition. Definition 2.4. Let f n 1 : a, b R F and t a, b where f n 1 is n 1th-orderHukuhara differentiable of f at t for all n>1. We say that f n is nth-order Hukuhara differentiable at

Journal of Applied Mathematics 3 t if there exists an element f n t R F such that for all h>sufficiently small, f n 1 t h H f n 1 t,f n 1 t H f n 1 t h and f n 1 t h H f n 1 t f n 1 t H f n 1 t h lim lim f n t, h h h h 2.3a n 1, 2,... In parametric form, as above, if f, f, then f n f n, f n, where f n, f n are nth-order Hukuhara differentiable of f, f, respectively. From 2.3a, in case n 1, we obtain 2.2. 3. Fuzzy Integro-Differential Equation Consider the nth-order derivative integro-differential equation 12 as follows: u n t ut λ ks, tu m sds gt, 3.1a where m, n N, m<n, t,t, λ R, gt is a known function and the kernel ks, t with the initial conditions as follows: u k h k, k, 1,...,n 1, h k R, 3.1b in the fuzzy case; that is, u and g be fuzzy functions. Let u n t, r ut, r ( ut, r, ut, r ), gt, r ( ) gt, r, gt, r, ( ) ( ) u n t, r, u n t, r, g n t, r g n t, r, g n t, r, 3.2 where all derivatives are, with respect to t, fuzzy functions. Therefore, related fuzzy integrodifferential equation of 3.1a can be written as follows: u n t, r ut, r λ ks, tu m s, rds gt, r, u n t, r ut, r λ ks, tu m s, rds gt, r, 3.3a 3.3b and its two crisp equations can be written as follows: u cn t, r u c t, r λ ks, tu cm s, rds g c t, r, u dn t, r u d t, r λ ks, tu dm s, rds g d t, r. 3.4a 3.4b

4 Journal of Applied Mathematics 4. New Iterative Method For simplicity, we present a review of the new iterative method 2, 21, 34 39, 41, and then we introduce a suitable algorithm of this method for solving the nth-order derivative fuzzy integro-differential equations. Consider the following general functional equation: u f Nu, 4.1 where N is a nonlinear operator from a Banach space B B, andf is a known function. We are looking for a solution u of 4.1 having the series form: u u i. i 4.2 The nonlinear operator N can be decomposed as follows: ( ) N u i Nu N i u j N i 1 u j. i i j j 4.3 From 4.2 and 4.3, 4.1 is equivalent to u i f Nu N i u j N i 1 u j. i i1 j j 4.4 We define the recurrence relation: u f, u 1 Nu, 4.5 u n1 Nu u 1 u n Nu u 1 u n 1, n 1, 2,... Then u 1 u n1 Nu u n, n 1, 2,..., u i f u i. i i1 4.6 The n-term approximate solution of 4.1 is given by u u u 1 u n 1.

Journal of Applied Mathematics 5 4.1. Reliable Algorithm After the above presentation of the new iterative method, we introduce a reliable algorithm of this method for solving any nth-order derivative fuzzy integro-differential equation. Consider the nth-order derivative integro-differential equation 3.1a and 3.1b defined as u n t ut λ ks, tu m sds gt, 4.7a with the following initial conditions: u k h k, k, 1,...,n 1, h k R. 4.7b The initial value problem 4.7a and 4.7b is equivalent to the following integral equation: ut u t It n ut λ ks, tu m sds u t Nu, 4.8 where u t is the solution of the nth-order differential equation: d n u gt, uk dtn h k, k, 1, 2,...,n 1, h k R, 4.9a Nu It n ut λ ks, tu m sds, 4.9b and It n is an integral operator of order n. Also, the two crisp equations 3.4a and 3.4b are equivalent to the two integral equations: u c t, r u c t, r In t u c t, r λ u d t, r u d t, r In t u d t, r λ ks, tu cm s, rds u c t, r Nuc, 4.1a ks, tu dm s, rds u d (u t, r N d), 4.1b where u c t, r and ud t, r are the solutions of the nth-order differential equations: d n u c dt n gc t, u ck h c k, k, 1, 2,...,n 1, hc k R, 4.1c d n u d dt n gd t, u dk h d k, k, 1, 2,...,n 1, hd k R. 4.1d

6 Journal of Applied Mathematics Nu c and Nu d are the following two integral equations: Nu c It n u c t, r λ Nu c It n u d t, r λ ks, tu cm s, rds, 4.1e ks, tu cm s, rds. 4.1f We get the solution of 4.8 or the two 4.1a, 4.1b by employing the recurrence relation 4.5. 4.2. Convergence of the New Iterative Method Now, we introduce the condition of convergence of the new iterative method, which is proposed by Daftardar-Gejji and Jafari in 26 2, also called DJM 39. From4.3, the nonlinear operator N is decomposed as follows 39: Nu Nu Nu u 1 Nu Nu u 1 u 2 Nu u 1. 4.11 Let G Nu and G n N ( ) ( n n 1 u i N u i ), n 1, 2,... 4.12 i i Then Nu i G i. Set u f, 4.13 u n G n 1, n 1, 2,... 4.14 Then u u i i 4.15 is a solution of the general functional equation 4.1. Also, the recurrence relation 4.5 becomes u f, u n G n 1, n 1, 2,... 4.16

Journal of Applied Mathematics 7 Using Taylor series expansion for G i, i 1, 2,...,n, we have G 1 Nu u 1 Nu Nu N u u 1 N u u2 1 2! Nu N k u uk 1 k!, k1 G 2 Nu u 1 u 2 Nu u 1 N u u 1 u 2 N u u 1 u2 2 2! N ij u ui j 1 u 2 i! j!, i j1 4.17 G 3 N i1i 2 i 3 u ui 3 3 i 3! i 3 1i 2 i 1 u i 2 2 u i 1 1 i 2! i 1!. In general: G n i n 1 i n 1 i 1 N n k1 ik u n u i j j. 4.18 i j! j1 In the following theorem we state and prove the condition of convergence of the method. Theorem 4.1. If N is C in a neighborhood of u and { } N n u sup N n u h 1,...,h n : h i 1, 1 i n L, 4.19a for any n and for some real L> and u i M<1/e, i 1, 2,..., then the series n G n is absolutely convergent, and, moreover, G n LM n e n 1 e 1, n 1, 2,... 4.19b Proof. In view of 4.18, G n LM n i n 1 i n 1 n i 1 j1 u i j j LM n e n 1 e 1. i j! 4.19c Thus, the series n1 G n is dominated by the convergent series LMe 1 n1 Men 1, where M<1/e. Hence, n G n is absolutely convergent, due to the comparison test. For more details, see 39.

8 Journal of Applied Mathematics 5. Numerical Examples Example 5.1. Consider the following fuzzy integro-differential equation: u t, r λ ks, tus, rds gt, r, t, r 1, 5.1 where ut, r ut, r, ut, r 2rt,4 rt is the exact solution for 5.1, gt, r gt, r, gt, r 2r 2rt/3, r 2t rt/3, ks, t st, andλ 1, with the initial conditions u,r, u,r4. Taking g c t, r 1/2gt, r gt, r r/2 t rt/6, g d t, r 1/2gt, r gt, r 3r/2trt/2. The exact solution for related crisp equations is u c t, r 2 rt/2, u d t, r 2 3rt/2. At first, we identify u c t, r and u d t, r. From3.4a and 3.4b, 5.1, where n 1, we have u c t, r u d t, r stu c s, rds g c t, r, u c,r 2, 5.2a stu d s, rds g d t, r, u d,r 2. 5.2b From 4.1c, 4.1d, weobtain u c rt t, r 2 2 t2 2 rt2 12, 3rt ud t, r 2 2 t2 2 rt2 4. 5.3 Therefore, from 4.1a, 4.1b the fuzzy integro-differential equations 5.2a and 5.2b are equivalent to the following integral equations: u c t, r 2 rt 2 t2 2 rt2 12 I t stu c s, rds, u d t, r 2 3rt 2 t2 2 rt2 4 I t stu d s, rds. 5.4

Journal of Applied Mathematics 9 Let Nu c I t stuc s, rds,nu d I t stud s, rds. Therefore, from 4.5, we can obtain easily the following first few components of the new iterative solution for 5.1: u c rt t, r 2 2 t2 2 rt2 12, 3rt ud t, r 2 2 t2 2 rt2 4, u c 1 t, r N( u c ) 9t 2 ) 9rt2 16 96, ud 1 (u t, r N d 9t2 16 9rt2 32, u c 9t2 2t, r 128 9rt2 768, ud 9t2 9rt2 2t, r 128 256,. 5.5 u c 5 t, r 9t2 65536 9rt2 393216, ud 9t2 5t, r 65536 9rt2 13172, and so on. The 6-term approximate solution is 5 u c t, r u c n 2 rt 2 t 2 65536 rt 2 393216, n 5 u d t, r u d n 2 3rt 2 t 2 65536 rt 2 13172. n 5.6 It is clear that the iterations converge to the exact solution of the two crisp equations u c,u d as the number of iteration converges to, that is, lim n u c nt, r 2 rt/2 u c Exact,and lim n u d nt, r 2 3rt/2 u d Exact. From the relations between u, u, uc,u d,andu in Section 2, it follows immediately that this solution is the solution of the fuzzy integro-differential equation 5.1. Example 5.2. Consider the following fuzzy integro-differential equation: u t, r λ ks, tus, rds gt, r, t, r 1. 5.7 In this example ut, r ut, r, ut, r2rt,4rt is the exact solution for 5.7, gt, r gt, r, gt, r 2rt/3, 4rt/3, ks, t st, λ 1, and the initial conditions are u,r, u,r2r, u,r, u,r4r. Also, the exact solution for related crisp equations is u c t, r 3rt, u d t, r rt with g c t, r rt, g d t, r rt/3. At first, we identify u c t, r, u d t, r. From3.4a and 3.4b, 5.7, where n 2, we have u c t, r u d t, r stu c s, rds g c t, r, u c,r, u c,r 3r, 5.8a stu d s, rds g d t, r, u d,r, u d,r r. 5.8b

1 Journal of Applied Mathematics From 4.1c, 4.1d,weobtain u c rt3 t, r 3rt 6, ud rt3 t, r rt 18. 5.9 Therefore, from 4.1a, 4.1b, the fuzzy integro-differential equations 5.8a and 5.8b are equivalent to the following integral equations: u c t, r 3rt rt3 6 I2 t stu c s, rds, u d t, r rt rt3 18 I2 t stu d s, rds. 5.1 Let Nu c It 2 stuc s, rds, Nu d It 2 stud s, rds. Therefore, from 4.5, we can obtain easily the following first few components of the new iterative solution for 5.7: u c rt3 t, r 3rt 6, ud rt3 t, r rt 18, u c 1 t, r N( u c ) 31rt 3 ) 18, ud 1 (u t, r N d 31rt3 54, u c 31rt3 2t, r 54, ud 31rt2 2t, r 162,. 5.11 u c 5 t, r 31rt 3 1458, ud 5 t, r 29rt 2 4374, and so on. The 6-term approximate solution is u c t, r 5 u c rt 3 n 3rt 1458, ud t, r n 5 u d n rt n rt 3 4374. 5.12 It is clear that the iterations converge to the exact solution of u c,u d as n,thatis, lim n u c nt, r 3rt u c Exact, lim n u d nt, r rt u d Exact. As the above example the solution of 5.7 follows immediately. Example 5.3. The third example is the fuzzy integro-differential equation: u t, r λ ks, tu s, rds gt, r, t, r 1, 5.13 where ut, r ut, r, ut, r rt,4 3rt is the exact solution for 5.13, gt, r gt, r, gt, r rt/2, 3rt/2, ks, t st, λ 1, and the initial conditions are u,r,

Journal of Applied Mathematics 11 u,r r, u,r 4, u,r 3r. The exact solution for related crisp equations is u c t, r 2 rt, u d t, r 2 2rt with g c t, r rt/2, g d t, r rt. At first, we identify u c t, r, u d t, r, andfrom3.4a and 3.4b, 5.13, where n 2, we have u c t, r u d t, r stu c s, rds g c t, r, u c,r 2, u c,r r, 5.14a stu d s, rds g d t, r, u d,r 2, u d,r 2r. 5.14b From 4.1c, 4.1d, weobtain u c rt3 t, r 2 rt 12, ud rt3 t, r 2 2rt 6. 5.15 Therefore, from 4.1a and 4.1b, the fuzzy integro-differential equations 5.14a and 5.14b are equivalent to the integral equations: u c t, r 2 rt rt3 12 I2 t stu c s, rds, u d t, r 2 2rt rt3 6 I2 t stu d s, rds. 5.16 Let Nu c It 2 s, rds, Nu d I 2 stuc t s, rds. Therefore, from 4.5, we can stud obtain easily the following first few components of the new iterative solution for 5.13: u c rt3 t, r 2 rt 12, ud rt3 t, r 2 2rt 6, u c 1 t, r N( u c ) 7rt 3 ) 96, ud 1 (u t, r N d 7rt3 48, u c 2t, r 7rt3 768, ud 2t, r 7rt3 384, 5.17 u c 5. t, r 7rt3 393216, ud 7rt3 5t, r 19668, and so on. The 6-term approximate solution is u c t, r 5 u c rt 3 n 2 rt 393216, ud t, r n 5 u d n 2 2rt n rt 3 19668. 5.18

12 Journal of Applied Mathematics It is clear that the iterations converge to the exact solution of u c,u d as n,thatis, lim n u c nt, r 2 rt u c Exact, lim n u d nt, r 2 2rt u d Exact. Therefore the solution of 5.13 follows immediately. Example 5.4. Let us consider the following fuzzy integro-differential equation: u t, r ut, r λ ks, tus, rds gt, r, t, r 1, 5.19 with the exact solution ut, r ut, r, ut, r 3rt,8 rt, gt, r g, g 2rt,8 4t 2rt/3, ks, t st, λ 1, and the initial conditions u,r, u,r3r, u,r8, u,r r. The exact solution for related crisp equations is u c t, r 4 rt, u d t, r 4 2rt with g c t, r 4 2t 2rt/3, g d t, r 4 2t 4rt/3. From 3.4a, 3.4b, and5.19, where n 2, the two crisp equations are u c t, r u c t, r u d t, r u d t, r stu c s, rds g c t, r, u c,r 4, u c,r r, 5.2a stu d s, rds g d t, r, u d,r 4, u d,r 2r. 5.2b From 4.1c, 4.1d we obtain u c t, r 4 rt 2t2 t3 3 rt3 9, ud t, r 4 2rt 2t2 t3 3 2rt3 9. 5.21 Therefore, from 4.1a, 4.1b, the fuzzy integro-differential equations 5.2a and 5.2b are equivalent to the following integral equations: u c t, r 4 rt 2t 2 t3 rt3 3 9 I2 t u c t, r u d t, r 4 2rt 2t 2 t3 3 2rt3 9 I2 t u d t, r stu c s, rds, stu d s, rds. 5.22 Let Nu c It 2 u c t, r ( N u d) It 2 u d t, r stu c s, rds, stu d s, rds. 5.23

Journal of Applied Mathematics 13 Therefore, from 4.5, we can obtain the following first approximate solutions for 5.19: u c t, r 4 rt 2t2 t3 3 rt3 9, ud t, r 4 2rt 2t2 t3 3 2rt3 9, u c 1 t, r 2t2 73t3 18 29rt3 27 t4 6 t5 6 rt5 18, u d 1 t, r 2t2 73t3 18 29rt3 135 t4 6 t5 rt5 6 9, u c 2t, r 311t3 42 421rt3 1134 t4 6 73t5 29rt5 t6 36 54 18 t7 rt7 252 756, 5.24 u d 421rt3 2t, r 311t3 42 567 t4 6 73t5 36 29rt5 t6 27 18 t7 252 rt7 378,. and so on. In the same manner the rest of components can be obtained. The 5-term approximate solution is u c t, r 4 u c n 4 rt 2.8739t3 3143448.119rt3 943344 n.523t5 27216.23rt5 2412 u d t, r.23t7 1584.1rt7 95256.13t9 18864.1rt9.1t1 163296 972.1t11.1rt11 199584 598752, 4 u d n 4 2rt 2.8739t3 3143448 n.119rt3 4715172.523t5 27216.23rt5 126 5.25.23t7 1584.1rt7 47628.13t9.1rt9.1t1 18864 81648 972.1t11 199584.1rt11 299376. Now, let us define the absolute value of the nth-term error by e n t, r u t, r u n t, r, where u t, r is the exact solution and u n t, r is the nth-term approximate solution respectively. It is clear from previous above solutions that the smallest value of the absolute nth-term error is at t r, that is, e n, en, c, en, d,, n, 1,...,4 and the largest value is at t r 1 as shown in Table 1. Therefore the iterations converge to the exact solution of u c,u d, that is, lim n u c nt, r 4 rt u c Exact, lim n u d nt, r 4 2rt u d Exact, and the solution of 5.19 follows immediately. Example 5.5. The final example is the following fuzzy integro-differential equation: u t, r ut, r λ ks, tu s, rds gt, r, t, r 1, 5.26

14 Journal of Applied Mathematics Table 1 n 1 2 3 4 en c 1.777778 7.9629629 2 3.39565 4 4.588 5 5.67 7 en d 1.444444 7.47474 2 1.543214 4 4.361 5 3.96 7 with the exact solution ut, r ut, r, ut, r 3rt,4 rt, gt, r g, g 2rt,4 2rt/3, ks, t s 2 t, λ 1 and the initial values u,r, u,r3r, u,r4, u,rr. The exact solution for related crisp equations is u c t, r 22rt, u d t, r 2 rt with g c t, r 2 4rt/3, g d t, r 2 2rt/3. As above, where n 2, the two crisp equations are u c t, r u c t, r u d t, r u d t, r s 2 tu c s, rds g c t, r, u c,r 2, u c,r 2r, s 2 tu d s, rds g d t, r, u d,r 2, u d,r r. 5.27a 5.27b As the above examples, we obtain u c t, r 2 2rt t2 2rt3 9, ud t, r 2 rt t2 rt3 9. 5.28 Therefore, the fuzzy integro-differential equations 5.27a and 5.27b are equivalent to the following integral equations: u c t, r 2 2rt t 2 2rt3 9 I2 t u c t, r u d t, r 2 rt t 2 rt3 9 I2 t u d t, r s 2 tu c s, rds, s 2 tu d s, rds. 5.29 Let Nu c It 2 u c t, r ( N u d) It 2 u d t, r s 2 tu c s, rds, s 2 tu d s, rds. 5.3

Journal of Applied Mathematics 15 Table 2 n 1 2 3 4 en c 1.2222222 1.1111111 2 2.2222222 3 1.75733 4 4.3324 6 en d 8.888889 1 5.5555556 3 2.23412699 3 1.33892 4 1.19612 6 Therefore, we obtain u c t, r 2 2rt t2 2rt3 9, ud t, r 2 rt t2 rt3 9, u c 1 t, r t2 t3 12 rt3 5 t4 12 rt5 9, u d 1 t, r t2 t3 rt3 12 1 t4 rt5 12 18, u c 91t3 2t, r 18 43rt3 t4 189 12 t5 rt5 t6 rt7 24 1 36 378, 5.31 u d 91t3 43rt3 t4 2t, r 18 378 12 t5 24 rt5 t6 2 36 rt7 756,. and so on. The 5-term approximate solution is u c t, r 4 u c n 2 2rt.337t3 27216 5.753rt3 1571724 n.73t5 648.41rt5 126 u d t, r.1t7 972.17rt7 15876.1t9 72576.1rt9.1t1.1t11 27216 18144 299376, 4 u d n 2 rt.337t3 27216 n 5.753rt3 3143448.73t5 648.41rt5 2412 5.32.1t7 972.17rt7 31752.1t9.1rt9.1t1 72576 54432 18144.1t11 598752. As the previous example, it is clear from the obtained results that the smallest value of the absolute nth-term error is e n, e c n,, e d n,,, n, 1,...,4, and the largest value is at t r 1 as shown in Table 2. Therefore, the iterations converge to the exact solution of u c,u d, that is, lim n u c nt, r 2 2rt u c Exact, lim n u d nt, r 2 rt u d Exact. Therefore, the solution of 5.26 follows immediately.

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