Fifth order Runge-Kutta method for solving first-order

Journal Nonlinear Analysis and Application 2016 No.1 (2016) 1-22 Available online at www.ispacs.com/jnaa Volume 2016, Issue 1, Year 2016 Article ID jnaa-00309, 22 Pages doi:10.5899/2016/jnaa-00309 Research Article

Fifth order Runge-Kutta method for solving first-order fully fuzzy differential equations under strongly generalized H-differentiability D. Vivek1∗ , K. Kanagarajan1 (1) Department of Mathematics, SRMV College of Arts and Science, Coimbatore - 641 020, Tamilnadu, India. c D. Vivek and K. Kanagarajan. This is an open access article distributed under the Creative Commons Attribution License, which Copyright 2016 ⃝ permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper we use fifth order Runge-Kutta method for solving fully fuzzy differential equations of the form ′ y (t) = a ⊗ y(t), y(0) = y0 , t ∈ [0, T ] under strongly generalized H-differentiability. The algorithm used here are based on cross product of two fuzzy numbers. Using cross product we can divide fully fuzzy differential equation (FFDE) into four different cases. We apply the results to a particular case of FFDE. The Convergence of this method is discussed and numerical examples are given to verify the reliability of this method. Keywords: Cross product, First-order fully fuzzy differential equation, Lipschitz condition, Strongly generalized H-differentiability,

Fifth order Runge-Kutta method.

1 Introduction The research of fuzzy differential equations (FDE) form an appropriate setting for mathematical modelling of real world problems in which uncertainties or imprecision pervades. The solutions of a FDE with fuzzy initial conditions are used in science and engineering fields, thus fuzzy initial value problem (FIVP) should be solved [4]. The term “fuzzy differential equation” was first coined in 1978 [24]. The idea of a fuzzy derivative was defined by Chang and Zadeh [2]. It was followed by Dubois and Prade [14], who used the extension principle. The brief sketch of FIVP was proposed by Seikkala and Kalava[22] and other researchers began to improve the fuzzy theory. There have been many ideas for the definition of fuzzy derivative to study FDE. The first and the most popular approach are using the Hukuhara differentiability for fuzzy-value functions. The strongly generalized differentiability was introduced Bede et.al (for e.g.[8],[7]). The strongly generalized derivative is defined for a longer class of fuzzy number valued functions than the Hukuhara derivative. The numerical ′ methods for solving FDEs y (x) = f (x, y) where x0 is real number and y(x0 ) = y0 are introduced in Allahviranloo [2], Abbasbandy et al. (see e.g.[3],[4],[7]) applied the concept of strongly generalized H-differentiability to solve linear first-order FDEs. It should be noted that in all mentioned numerical methods, finding a numerical solution for FDE is only possible with real coefficients. ′ In this paper, we find the numerical solution for FFDE in the form y (t) = a ⊗ y(t), y(0) = y0 , t ∈ [0, T ] where a is a fuzzy number. First, by choosing different types of derivatives and sign of a and y(t), FFDE is divided into four ∗ Corresponding

author. Email address: [email protected], Tel:+919787557676

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Journal of Nonlinear Analysis and Application 2016 No.1 (2016) 1-22 http://www.ispacs.com/journals/jnaa/2016/jnaa-00309/

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differential equations. Since each of the divided differential equations satisfies the Lipschitz condition, they should have a unique solution and fifth order Runge-Kutta method is used to find the numerical solutions. The paper is organised as follows: In section 2, we recall some basic definitions. In section 3, cross product is defined and first-order fully fuzzy differential equation is introduced in section, 4. In section 5, fifth order Runge-Kutta method is presented in detail and its convergence is discussed. In section 6, the theory is illustrated by examples, in section 7, conclusion is drawn. 2 Basic concepts In this section, we recall the basic notation of fuzzy numbers, strongly generalized H-differentiability and the cross product. A non-empty subset A of R is called convex if and only if (1 − k)x + ky ∈ A for every x, y ∈ A and k ∈ [0, 1]. By Pk (R), we denote the family of all non-empty compact convex subsets of R. There are various definitions for the concepts of fuzzy numbers[14]. Definition 2.1. A fuzzy number is a function such as u : R → [0, 1] satisfying the following properties: (i) u is normal, i.e., there exist, x0 ∈ R with u(x0 ) = 1, (ii) u is a convex fuzzy set i.e. u(λ x + (1 − λ )y) ≥ min {u(x), u(y)} ∀x, y ∈ R, λ ∈ [0, 1], (iii) u is upper semi-continuous on R, (iv) {x ∈ R : u(x) > 0}is compact, where A denotes the closure of A. The set of all fuzzy real numbers is denoted by RF . Obviously,R ⊂ RF . For 0 < r ≤ 1, the r-level set is denoted by [u]r = {x ∈ R; u(x) ≥ r} and [u]0 = {x ∈ R; u(x) ≥ 0}. Then, it is well-known that for any r ∈ [0, 1], [u]r is a bounded closed interval. For u, v ∈ RF and λ ∈ R, the sum u + v and the product λ .u are defined by [u + v]r = [u]r + [v]r , [λ .u]r = λ [u]r , ∀r ∈ [0, 1], where [u]r + [v]r = {x + y : x ∈ [u]r , y ∈ [v]r } means the conventional addition of two intervals(subsets) of R, and λ [u]r = {λ x : x ∈ [u]r } means the conventional product between a scaler and a subset of R in[14]. Definition 2.2. The parametric form of a fuzzy number u(r) is a pair [u(r), u(r)] of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy the following conditions: (a) u(r) is a monotonically increasing left continuous function. (b) u(r) is a monotonically decreasing left continuous function. (c) u(r) ≤ u(r), 0 ≤ r ≤ 1. It should be noted that for a < b < c, a, b, c ∈ R, a triangular fuzzy number u = (a, b, c) is given such that u = b − (1 − r)(b − a) and u = b + (1 − r)(c − b) are the end point of the r-cut set for all 0 ≤ r ≤ 1. In this paper we use triangular fuzzy numbers. Here, u(r) = u(r) = b is denoted by [u]1 . For arbitrary fuzzy number [u]r = [u(r), u(r)] and [v]r = [v(r), v(r)] and k ∈ R, we define addition and multiplication as [u + v]r = [u]r + [v]r = [u(r) + v(r), u(r) + v(r)] and

{ r

[ku] =

[ku(r), ku(r)] if k ≥ 0, [ku(r), ku(r)] if k < 0,

(2.1)

(2.2)

respectively, for every r ∈ [0, 1]. We denote by −u = (−1)u ∈ RF the symmetricc of u ∈ RF . The producct u.v of fuzzy numbers u and v, based on Zadeh’s extension principle, is defined by [u.v]r = [min kr , max kr ]

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where kr = {u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)} . Definition 2.3. A fuzzy number u ∈ RF is said to be positive if u(1) ≥ 0, strict positive if u(1) > 0, negative if u(1) ≤ 0 and strict negative if u(1) < 0. We say that u and v have the same sign if they are both strict positive or both strict negative. If u is positive(negative) then −u is negative(positive). In this paper, we denote an arbitrary fuzzy number with compact support by a pair of functions (u(r), u(r)) , 0 ≤ r ≤ 1. Also, we use the Hausdorff distance between fuzzy numbers. This fuzzy number space as has beeen shown in Bede and Gal[8] can be embedded into the Banach space B = c[0, 1] × c[0, 1] with the usual metric defined as follows: Let RF be the set of all upper semi-continuous normal convex fuzzy numbers with bounded r-level sets. Since the r-cuts of fuzzy numbers are always closed and bounded, the intervals are written as [u]r = [u(r), u(r)], for all r. We denote by ω all of non empty convex compact sets. Recall that

ρ (x, A) = min ∥x − a∥ , a∈A

is the distance from a point x ∈ R to A ∈ ω and the Hausdorff separation

ρ (A, B) = max ρ (a, B). a∈A

Note that the notation is consistent, since ρ (a, B) = ρ ({a} , B). Now ρ is not metric. In fact, ρ (A, B) = 0 if and only if A ⊆ B. The Hausdorff metric dH on ω is defined by dH (A, B) = max {ρ (A, B), ρ (B, A)} . The metric d∞ is defined on RF as d∞ (u, v) = sup {dH ([u]r , [v]r ) : 0 < r ≤ 1} , u, v ∈ RF . For arbitrary (u, v) ∈ c[0, 1] × c[0, 1]. The following properties are well known:[16, 32] (i) d∞ (u + w, v + w) = d∞ (u, v), ∀u, v, w ∈ RF , (ii) d∞ (k.u, k.v) = |k| d∞ (u, v), ∀k ∈ R, u, v ∈ RF , (iii) d∞ (u + v, w + e) ≤ d∞ (u, w) + d∞ (v, e). Definition 2.4. [15] Let f : R → RF be a fuzzy-valued function. If for arbitrary fixed t0 ∈ R and ε > 0, δ > 0 such that |t − t0 | < δ ⇒ d∞ ( f (t), f (t0 )) < ε , f is said to be continuous. Definition 2.5. [7] Let x, y ∈ RF . If there exists z ∈ RF such that x = y + z, then z is called the H-difference of x and y, and it is denoted by x ⊖ y, t0 ∈ (a, b). We say that f is strongly generalized H-diffeentiability at t0 , if there exists an ′ element f (t0 ) ∈ RF . Definition 2.6. [7] Let f : (a, b) → RF and t0 ∈ (a, b). We say that f is strongly generalized H-differentiable at t0 , if ′ there exists an element f (t0 ) ∈ RF , such that (i) for all h > 0 sufficiently near to 0, there exist the H-difference f (t0 + h) ⊖ f (t0 ), f (t0 ) ⊖ f (t0 − h) and the limits hold. ′

f (t0 ) = lim

h→0+

f (t0 ) ⊖ f (t0 − h) f (t0 + h) ⊖ f (t0 ) = lim + h h h→0



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(ii) for all h < 0 sufficiently near to 0, there exists the H-difference f (t0 ) ⊖ f (t0 + h), f (t0 − h) ⊖ f (t0 ) and the limits ′

f (t0 ) = lim

h→0+

f (t0 ) ⊖ f (t0 + h) f (t0 − h) ⊖ f (t0 ) = lim h h h→0+

Notice that we say fuzzy-valued function f is (i)-differentiable if satisfy in the first form (i) in Definition(2.6), and we say f is (ii)-differentiable if satisfies in the second form (ii) in definition 2.6. As a special case when f is a fuzzy-valued function, we have the following results. Theorem 2.1. [25] Let f (t) be fuzzy-valued functions and denote f (t) = ( f (t; r), f (t; r)), for each r ∈ [0, 1]. Then ′

′

′

(a) if f (t) is (i)differentiable, then f (t; r) and f (t; r) have first-order derivative and f (t) = ( f (t; r), f (t; r)). ′

′

′

(b) if f (t) is (ii)differentiable, then f (t; r) and f (t; r) have first-order derivative and f (t) = ( f (t; r), f (t; r)). ′

Theorem 2.2. [25] Let f (t) and f (t) are two differentiable fuzzy-vlued functions and denote f (t) = ( f (t; r), f (t; r)), for each r ∈ [0, 1]. Then ′

′

(a) if f (t) and f (t) are (i)-differentiable, or f (t) and f (t) are (ii)-differentiable, then f (t; r) and f (t; r) have first′′

′′

′′

order and second-order derivatives and f (t) = ( f (t; r), f (t; r)). ′

′

(b) if f (t) is (i)-differentiable and f (t) (ii)-differentiable, or f (t) is (ii)-differentiable and f (t) is (i)-differentiable, ′′

′′

′′

then f (t; r) and f (t; r) have first-order and second-order derivatives and f (t) = ( f (t; r), f (t; r)). ′

Lemma 2.1. [25] For x0 ∈ R, the fuzzy differential equation y = f (x, y), y(x0 ) = y0 ∈ RF , where f : R × R → RF is supposed to be continuous, is equivalent to the one of the integral equations ∫ x

y(x) = y0 +

x0

f (t, y(t))dt, ∀x ∈ [x0 , x1 ]

or ∫ x

y(0) = y(x) + (−1). x0

f (t, y(t))dt, ∀x ∈ [x0 , x1 ]

on some interval (x0 , x1 ) ⊂ R, under the srtongly differentiability condition (i) or (ii), respectively. Here, the equivalence between two equations means that any solution of an equation is a solution for the other one. ′

Remark 2.1. [8] In the case of strongly generalized differentiability, to the fuzzy differential equation y = f (x, y) we may attach two different integral equations, while in the case of H-differentiability, ∫we may attach only one. The second integral equations in Lemma 2.1 can be written in the form y(0) = y(x) + (−1). xx0 f (t, y(t))dt. The following theorems concern the existence of solutions of a fuzzy initial-value problem under generalized differentiability in[8]. Theorem 2.3. Suppose that following conditions hold: (a) Let R0 = [x0 , x0 + p] × B(y0 , q), p, q > 0, y0 ∈ RF where B(y0 , q) = {y ∈ RF : D(y, y0 ) ≤ q} denote a closed ball in RF and let f : R0 → RF be a continuous function such that d∞ (0, f (x, y)) ≤ M for all (x, y) ∈ R0 . (b) Let g : [x0 , x0 + p] × [0, q] → R, such that g(x, 0) ≡ 0 and 0 ≤ g(x, u) ≤ M1 , ∀x ∈ [x0 , x0 + p], 0 ≤ u ≤ q such that ′ g(x, u) is non-decreasing in u and the initial value problem u (x) = g(x, u(x)), u(x0 ) = 0 has only the solution u(x) ≡ 0 on [x0 , x0 + p].



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(c) We have d∞ ( f (x, y), f (x, z) ≤ g(x, d∞ (y, z)), ∀(x, y), (x, z) ∈ R0 and d∞ (y, z) ≤ q. (d) There exist d > 0 such that for x ∈ [x0 , x0 + d] the sequence y1n : [x0 , x0 + d] → RF given by y10 (x) = y0 , y1n+1 (x) = ∫ y0 ⊖ (−1). xx0 f (t, y1n )dt is defined for any n ∈ N. Then the fuzzy initial value problem {

′

y = f (x, y), y(x0 ) = y0 ,

(2.3) ′

has { two solutions } (one (i)-differentiable and the other on (ii)-differentiable) y, y : [x0 , x0 + d] → B(y0 , q) where r = q q min s, m , m1 , d and the successive iteration ∫ x

y0 (x) = y0 , yn+1 (x) = y0 +

x0

f (t, yn )dt,

and y10 = y0 , y1n+1 (x) = y0 ⊖ (−1).

∫ x x0

f (t, y1n )dt,

converge to these two solutions, respectively. We denote, the space of continuous functions y : I = [a, b] → RF by C(I, RF ).C(I, RF ) is a complete metric space with the distant { } H(x, y) = sup d∞ (x(t), y(t))e−ρ t , t∈I

where ρ ∈ R is fixed[17]. 3 The cross product In this section, we recall summary from the theoretical properties of the cross product of two fuzzy numbers. Let R∗F = {u ∈ RF : u is positive or negative}. First, Ban and Bede begin with a theorem which has been obtained using the stacking theorem[29], for more details see[27]. Theorem 3.1. If u and v are positive fuzzy numbers then w = u ⊗ v defined by [w]r = [w(r), w(r)], where { w(r) = u(r)v(r) + u(1)v(r) − u(1)v(1) w(r) = u(r)v(r) + u(1)v(r) − u(1)v(1)

(3.4)

for every r ∈ [0, 1], is a positive fuzzy number. Remark 3.1. Let u and v be two fuzzy numbers. 1.If u is positive and v is negative then u ⊗ v = −(u ⊗ (−v)) is a negative fuzzy number. 2.If u is negative and v is positive then u ⊗ v = −((−u) ⊗ v) is a negative fuzzy number. 3.If u and v are negative then u ⊗ v = (−u) ⊗ (−v) is a positive fuzzy number. Now they (Ban and Bede) defined the cross product as follows: Definition 3.1. (cross product) The binary operation ⊗ on R∗F introduced by Theorem 3.1 and Remark 3.1 is called cross product of two fuzzy numbers. Remark 3.2. The cross product is difined for any fuzzy numbers in R∧F = {u ∈ R∗F ; there exists an unique x0 ∈ R such that u(x0 ) = 1}, so implicitily for any triangular fuzzy number.



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Remark 3.3. The below formulas of calculus can be easily proved (r ∈ [0, 1]):   u(r)v(r) + u(1)v(r) − u(1)v(1), if u is positive and v is negative, u(r)v(r) + u(1)v(r) − u(1)v(1), if u is negative and v is positive, w(r) =  u(r)v(r) + u(1)v(r) − u(1)v(1), if u and v are positive,

(3.5)

and   u(r)v(r) + u(1)v(r) − u(1)v(1), if u is positive and v is negative, u(r)v(r) + u(1)v(r) − u(1)v(1), if u is negative and v is positive, w(r) =  u(r)v(r) + u(1)v(r) − u(1)v(1), if u and v are positive,

(3.6)

Remark 3.4. The cross product extends the scalar multiplication of fuzzy numbers. Indeed, if one of operands is the r real number k identified with its characteristic function then for all r ∈ [0, 1], kr = k = k and following the above formulas of calculus we get the results. 4 First-order fully fuzzy differential equation In this section, we are going to show that, FFDE satisfies in the Lipschitz condition and, therefore, has unique solution. An FFDE has the following equation: { ′ y (t) = f (t, a, y(t)) = a ⊗ y(t), (4.7) y(t0 ) = y0 , t ∈ [0, T ], where a and y0 are triangular fuzzy numbers in this paper. The Lipschitz condition for problem (4.7) is introduced in following lemma. Lemma 4.1. Let a be an triangular fuzzy number, for each t ∈ [0, 1] there exist M > 0 such that d∞ (a ⊗ y1 (t), a ⊗ y2 (t)) ≤ Md∞ (y1 (t), y2 (t)). Proof. See in[13]. Theorem 4.1. Suppose that t0 ∈ [0, 1] and let f : (0, T ) × RF × RF → RF be continuous, a mapping y : [0, T ] → RF is a solution to the initial-valued problem(4.7) if and only if y is continuous and satisfies one of the following conditions: (a) ∫ t

y(t) =

f (s, a, y(s))ds + k, t0

where y is (i)-differential. or (b) y(t) = ⊖(−1)

∫ t

f (s, a, y(s))ds + k t0

where y is (ii)-differential. Proof. Since f is continuous, it must be integrable[8]. So Based on definition (2.6), in each case of (a) or (b) we have ′ y (t) = f (t, a, y(t)) can be written. The following lemma is needed to prove the initial-value problem(4.7) has a unique solution. Lemma 4.2 [5] For arbitrary (u, v) ∈ c[0, 1] × c[0, 1] we have d∞ (u ⊖ v, u ⊖ w) = d∞ (v, w).

Lemma 4.2. FFDE (4.7) has a unique solution for each case. Proof. see [5].



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5 Fifth order Runge-Kutta method and its convergence In this section, we describe our purpose approach for solving FFDE(4.7), then we analyze the convergence of this method. In the beginning that the discrete equally spaced grid points {t0 .t1 , ........tN = T } , h = NT be a partition for interval [0, 1]. Then, Runge-kutta method of order five to approximate the solution of (7) is as follows: w(ti+1 ; w(ti+1 ; r)) = w(ti ; w(ti ; r)) + ∆w

(5.8)

where ∆w, ∆w =

1 (k1 + 4k4 + k5 ) . 6

 k1 = h.a ⊗ w(ti ; w(ti ; r))    k  h   k2 = h.a ⊗ w(ti + 3 ; w(ti ; r) + 31 ) h k3 = h.a ⊗ w(ti + 3 ; w(ti ; r) + k61 + k62 )    k4 = h.a ⊗ w(ti + h3 ; w(ti ; r) + k81 + 38 k3 )    k = h.a ⊗ w(t + h ; w(t ; r) + k1 − 3 k + 2k ) i i 4 5 3 2 2 3 Since in constituting the cross product a ⊗ w(ti ), the signs of a and w(ti ) are important, to determine w(ti+1 ) the sign of w(ti ) should be found in each step. Then, we consider four cases as follows: Case (1): In this case, we assume that y(t) is (i)-differentiable and a ≥ 0. Then, the Runge-kutta method to approximate w(ti+1 ) is as follows: If w(ti ) ≥ 0: { w(ti+1 ; r) = w(ti ; r) + ∆w. (5.9) w(ti+1 ; r) = w(ti ; r) + ∆w. where, {

( ) ∆w = 61 (k1 + 4k4 + k5 ) ∆w = 61 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )  4   ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )  5   ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )



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 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [    k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + k31 ) + a(1)w(ti + h3 ; w(ti ; r) + k31 )    ]   k1 h  −a(1)w(t + ; w(t ; 1) + )  i i  3 3  [   k1 h  k3 = h. a(r)w(ti + 3 ; w(ti ; 1) + 6 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )    ]   3k3 k1 h  + ; w(t ; 1) + + ) −a(1)w(t  i i  3 8 8  [   k1 h 3   k5 = h. a(r)w(ti + 3 ; w(ti ; 1) + 2 − 2 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) and if w(ti ) < 0: {

where,

{

w(ti+1 ; r) = w(ti ; r) + ∆w. w(ti+1 ; r) = w(ti ; r) + ∆w.

(5.10)

( ) ∆w = 16 (k1 + 4k4 + k5 ) ∆w = 16 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [   k1 k1 h h  k = h. a(r)w(t  i + 3 ; w(ti ; 1) + 3 ) + a(1)w(ti + 3 ; w(ti ; r) + 3 ) 2   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k31 )    [    k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h ; w(ti ; 1) + k1 + 3k3 ) + a(1)w(ti + h ; w(ti ; r) + k1 + 3k3 )   3 8 8 ] 3 8 8    3k3 k1 h  −a(1)w(ti + 3 ; w(ti ; 1) + 8 + 8 )    [    k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )    ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 )



9

Case (2): In this case, we assume that y(t) is (i)-differentiable and a < 0. Then, the Runge-kutta method to approximate w(ti+1 ) is as follows: If w(ti ) ≥ 0: { w(ti+1 ; r) = w(ti ; r) + ∆w. (5.11) w(ti+1 ; r) = w(ti ; r) + ∆w. where, {

) ( ∆w = 16 (k1 + 4k4 + k5 ) ∆w = 16 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k1 k1  h h  k = h. a(r)w(t  i + 3 ; w(ti ; 1) + 3 ) + a(1)w(ti + 3 ; w(ti ; r) + 3 ) 2   ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [    k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + k31 ) + a(1)w(ti + h3 ; w(ti ; r) + k31 )    ]    −a(1)w(ti + h3 ; w(ti ; 1) + k31 )    [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )  3   ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )  4   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 )    [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )  5   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) and if w(ti ) < 0 {


(5.12)

where, {

( ) ∆w = 16 (k1 + 4k4 + k5 ) ∆w = 16 k1 + 4k4 + k5



10

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    5 ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [   k1 k1 h h  k = h. a(r)w(t  i + 3 ; w(ti ; 1) + 3 ) + a(1)w(ti + 3 ; w(ti ; r) + 3 ) 2   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k31 )    [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )  3   ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )  4   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 )    [    k = h. a(r)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )  5   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) Case (3): In this case, we assume that y(t) is (ii)-differentiable and a ≥ 0. Then, the Runge-kutta method to approximate w(ti+1 ) is as follows: If w(ti ) ≥ 0: { w(ti+1 ; r) = w(ti ; r) + ∆w. (5.13) w(ti+1 ; r) = w(ti ; r) + ∆w. where,

{

( ) ∆w = 16 (k1 + 4k4 + k5 ) ∆w = 16 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k1 k1  h h  k = h. a(r)w(t  i + 3 ; w(ti ; 1) + 3 ) + a(1)w(ti + 3 ; w(ti ; r) + 3 ) 2   ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )



11

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [    k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + k31 ) + a(1)w(ti + h3 ; w(ti ; r) + k31 )    ]   k1 h  −a(1)w(t + ; w(t ; 1) + )  i i  3 3  [   k1 h  k3 = h. a(r)w(ti + 3 ; w(ti ; 1) + 6 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )    ]   3k3 k1 h  + ; w(t ; 1) + + ) −a(1)w(t  i i  3 8 8  [   k1 h 3   k5 = h. a(r)w(ti + 3 ; w(ti ; 1) + 2 − 2 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) and if w(ti ) < 0 {

where,

{


(5.14)

( ) ∆w = 16 (k1 + 4k4 + k5 ) ∆w = 16 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [   k1 k1 h h  k = h. a(r)w(t  i + 3 ; w(ti ; 1) + 3 ) + a(1)w(ti + 3 ; w(ti ; r) + 3 ) 2   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k31 )    [    k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h ; w(ti ; 1) + k1 + 3k3 ) + a(1)w(ti + h ; w(ti ; r) + k1 + 3k3 )   3 8 8 ] 3 8 8    3k3 k1 h  −a(1)w(ti + 3 ; w(ti ; 1) + 8 + 8 )    [    k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )    ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 )



12

Case (4): In this case, we assume that y(t) is (ii)-differentiable and a < 0. Then, the Runge-kutta method to approximate w(ti+1 ) is as follows: If w(ti ) ≥ 0: { w(ti+1 ; r) = w(ti ; r) + ∆w. (5.15) w(ti+1 ; r) = w(ti ; r) + ∆w. where, {

) ( ∆w = 61 (k1 + 4k4 + k5 ) ∆w = 61 k1 + 4k4 + k5

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [    k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + k31 ) + a(1)w(ti + h3 ; w(ti ; r) + k31 )    ]   k1 h  + w(t ; 1) + −a(1)w(t ; )  i i  3 3  [   k1 h  k3 = h. a(r)w(ti + 3 ; w(ti ; 1) + 6 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )    ]   3k3 k1 h  −a(1)w(t + ; w(t ; 1) + + )  i i  3 8 8  [   k1 h 3   k5 = h. a(r)w(ti + 3 ; w(ti ; 1) + 2 − 2 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) and if w(ti ) < 0 {


(5.16)

where, {

( ) ∆w = 61 (k1 + 4k4 + k5 ) ∆w = 61 k1 + 4k4 + k5



13

 k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [  k k   k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + 31 ) + a(1)w(ti + h3 ; w(ti ; r) + 31 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 31 )    [  k k k k   k3 = h. a(r)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) + a(1)w(ti + h3 ; w(ti ; r) + 61 + 62 )    ]  k k −a(1)w(ti + h3 ; w(ti ; 1) + 61 + 62 ) [  3k 3k k k   k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + 81 + 83 ) + a(1)w(ti + h3 ; w(ti ; r) + 81 + 83 )    ]  3k k   −a(1)w(ti + h3 ; w(ti ; 1) + 81 + 83 )    [  k k   k5 = h. a(r)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + 21 − 32 k3 + 2k4 )    ]  k   −a(1)w(ti + h3 ; w(ti ; 1) + 21 − 32 k3 + 2k4 )  k1 = h. [a(r)w(t  i ; w(ti ; 1)) + a(1)w(ti ; w(ti ; r)) − a(1)w(ti ; w(ti ; 1))]  [    k2 = h. a(r)w(ti + h3 ; w(ti ; 1) + k31 ) + a(1)w(ti + h3 ; w(ti ; r) + k31 )    ]   k1 h  −a(1)w(t + ; w(t ; 1) + )  i i  3 3  [   k1 h  k3 = h. a(r)w(ti + 3 ; w(ti ; 1) + 6 + k62 ) + a(1)w(ti + h3 ; w(ti ; r) + k61 + k62 )    ]  −a(1)w(ti + h3 ; w(ti ; 1) + k61 + k62 ) [    k4 = h. a(r)w(ti + h3 ; w(ti ; 1) + k81 + 3k83 ) + a(1)w(ti + h3 ; w(ti ; r) + k81 + 3k83 )    ]   3k3 k1 h  −a(1)w(t + ; w(t ; 1) + + )  i i  3 8 8  [   k1 3 h   k5 = h. a(r)w(ti + 3 ; w(ti ; 1) + 2 − 2 k3 + 2k4 ) + a(1)w(ti + h3 ; w(ti ; r) + k21 − 32 k3 + 2k4 )   ]    −a(1)w(ti + h3 ; w(ti ; 1) + k21 − 32 k3 + 2k4 ) Lemma 5.1. Let a sequence of numbers {W n }Nn=0 satisfy |W n+1 | ≤ A|W n | + B, for some give positive constants A and B. Then, |W n | ≤ An |W 0 | + B

An − 1 A−1

Proof. See detailed. in[1]. Theorem 5.1. [20] The numerical fifth order Runge-kutta method(4.8) is convergent to the solution of FFDE (4.7) in each case. T Proof. Suppose that the discrete equally spaced grid points {t0 ,t1 , .........tN } = T, h = [ N is a partition ] for interval r [0, 1]. If the exact and approximate solutions at ti , 0 ≤ i ≤ N are denoted by [y(ti )] = y(ti ; r), y(ti ; r) and [w(ti )]r = [w(ti ; r), w(ti ; r)], respectively. It is sufficient to show

lim w(tN ; r) = y(tN ; r),

h→0

lim w(tN ; r) = y(tN ; r).

h→0

We proof this method for y(t) is (i)-differentiable and proof of the other cases is similar. By using Taylor theorem we get h w(ti+1 ; r) = w(ti ; r) + [a(r)w(ti ; r) + a(1)w(ti ; r) − a(1)w(ti ; 1)] 6 ] 11987 6 h[ h MP5 + O(h7 ), y(ti+1 ; r) = y(ti ; r) + a(r)y(ti ; r) + a(1)y(ti ; r) − a(1)y(ti ; 1) + 6 12960



14

where ti ≤ ε i , ε ≤ ti+1 . With assumption a ≥ 0, w(ti ) ≥ 0 and y(ti ) < 0, the relations w(ti+1 ; r), w(ti+1 ; r) can be transformed as follows (proof of the other cases is similar): Therefore, ) h( a(r)w(ti ; 1) − a(r)y(ti ; r) 6 ( ) h ( ) 11987 6 h + a(1) w(ti ; r) − y(ti ; r) + a(1) w(ti ; 1) − y(ti ; 1) − h MP5 + O(h7 ) 6 6 12960

w(ti+1 ; r) − y(ti+1 ; r)

= w(ti ; r) − y(ti ; r) +

So, w(ti+1 ; r) − y(ti+1 ; r)

( ) h h ≤ 1 + |a(1)| w(ti ; r) − y(ti ; r) + |a(1)| w(ti ; 1) − y(ti ; 1) 6 3 11987 6 + h MP5 + O(h7 ) 12960

By assumption 2 |a(1)| w(ti ; 1) − y(ti ; 1) = ε r w(ti ; r) − y(ti ; r) , (εr ≥ 0), We have

( ) w(ti+1 ; r) − y(ti+1 ; r) ≤ 1 + h |a| + h εr w(ti ; r) − y(ti ; r) + 11987 h6 MP5 + O(h7 ) 6 6 12960

Similarly, the following inequality can be obtained. ) h h 11987 6 |w(ti+1 ; r) − y(ti+1 ; r)| ≤ 1 + |a| + εr |w(ti ; r) − y(ti ; r)| + h MP5 + O(h7 ), (ε ≥ 0) . 6 6 12960 (

Denote Ui = w(ti ; r) − y(ti ; r), Vi = w(ti ; r) − y(ti ; r). Then, using Lemma(4.1) |Ui |

|Vi |

)i ( h h ≤ 1 + |a(1)| + εr |U0 | 6 6 )i ( )( 1 + h6 |a(1)| + 6h ε r − 1 11987 5 7 + MP + O(h ) , h 12960 6 (|a(1)| + ε r ) )i ( h h ≤ 1 + |a(1)| + εr |V0 | 6 6 )i ( )( 1 + h6 |a(1)| + h6 ε r − 1 11987 5 7 MP + O(h ) + , h 12960 6 (|a(1)| + ε r )

In particular, )N h h |U0 | ≤ 1 + |a(1)| + εr 6 6 )T )( ( 1 + h6 |a(1)| + h6 ε r h − 1 11987 5 7 MP + O(h ) + , h 12960 6 (|a(1)| + ε r ) (

|Ui |



15

( )N h h |V0 | ≤ 1 + |a(1)| + εr 6 6 )T ( )( 1 + h6 |a(1)| + h6 ε r h − 1 11987 5 7 + MP + O(h ) , h 12960 6 (|a(1)| + ε r )

|Vi |

Here, U0 = V0 = 0. If h → 0 we get UN → 0, VN → 0 and the proof is complete. 6 Examples In this section, some examples are given to illustrate our method. Moreover, we plot the obtained solutions and approximate them based on the r-cut representation at each case. Example 6.1. Consider the following FFDE[13] { ′ y (t) = (1 + r, 3 − r) ⊗ y(t), y(0) = (2 + r, 4 − r), t ∈ I = [0, +∞)

(6.17)

Since in this example a = (1 + r, 3 − r) ≥ 0, then (I): If y(t) is (i)-differentiable, the exact solution is: y(t; r) = (r − 3r + 3rt + 2)e2t , y(t; r) = −(r − 3r + 3rt − 4)e2t . Now, { } I1 = t ∈ I|y(t, 0) − y(t, 0) ≥ 0 = [0, +∞), { } ′ ′ I2 = t ∈ I|y (t, 0) − y (t, 0) ≥ 0 = [0, +∞) ∩

′

are intervals that on them, y and y are valid(are fuzzy numbers). Therefore, on I1 I2 = [0, +∞), y is (i)-differentiable solution. Approximatioe solutions w, w can be found by solving ODEs (5.9),(5.10)(see Figs.1 and 2). The numerical values are given in tables (see 1 and 2) and error ∆w is shown at t=0.1. Table 1. r 0.1 0.3 0.5 0.7 1 ∆w at t=0.1

RK-5 w(ti ; r) w(ti ; r) 2.2377786505 5.0958366444 2.5543164436 4.7557080388 2.8675421525 4.4589121526 3.1881567077 4.1408374746 3.6642082743 3.6642082743 3.0000e-010

RK-4 w(ti ; r) w(ti ; r) 2.2400431008 5.0727947823 2.5538293849 4.7570758880 2.8649750260 4.4416003754 3.1821217993 4.1263677266 3.6642082735 3.6642082735 5.0000e-010

Exact Solution y(ti ; r) y(ti ; r) 2.235167047 5.093249501 2.552731764 4.775684784 2.870296481 4.458120067 3.187861198 4.140555350 3.664208274 3.664208274



16

Figure 1: The approximate solution to the FFDE at t = 0.1 and h = 0.01

Figure 2: The approximate solution to the FFDE (II): If ye(t) is (ii)-differentiable, the exact solution is ( ) ( ) 15 3r 2t −7 7r −2t y(t; r) = − e + + e , 4 4 4 4 ( ) ( ) 9 3r 2t 7 7r −2t y(t; r) = + e + − e . 4 4 4 4 Now, } { 1 log7 I1 = t ∈ I|y(t, 0) − y(t, 0) ≥ 0 = [0, ], 4 3 { } ′ ′ I2 = t ∈ I|y (t, 0) − y (t, 0) ≥ 0 = [0, +∞) ′

∩

are intervals that on them, y and y are valid(are fuzzy numbers). Therefore, on I1 I2 = [0, 14 log7 3 ], y is (ii)-differentiable solution. Approximatioe solutions w, w can be found by solving ODEs (5.13),(5.14)(see Figs. 3 and 4)



17


Table 2. r 0.1 0.3 0.5 0.7 1 ∆w at t=0.1

RK-5 w(ti ; r) w(ti ; r) 3.2093342357 4.1234608052 3.3100370286 4.0210282304 3.4109574986 3.9188104302 3.5120952283 3.8168078165 3.6642082743 3.6642082743 3.0000e-010

RK-4 w(ti ; r) w(ti ; r) 3.2093522329 4.1234429780 3.3100510117 4.0210143497 3.4109674758 3.9188005045 3.5121012079 3.8168018543 3.6642082735 3.6642082735 5.0000e-010

Exact Solution y(ti ; r) y(ti ; r) 3.199154200 4.129262348 3.302499549 4.025916998 3.405844899 3.922571649 3.509190249 3.819226299 3.664208274 3.664208274

Figure 4: The approximate solution to the FFDE In the next example, a numerical method to solve the first-order fuzzy differential equations (FDE) (i.e., coefficient is crisp) is proposed in[28]. This example is a special case of FFDE which we solve it with our method and show that the numerical results of the two methods are the same. Example 6.2. Let us consider the nuclear decay equation ′

y (t) = −λ ⊗ y(t), y(0) = y0 ,



18

where y(t) is the number of radionuclides present in a given radioactive material, λ is the decay constant and y0 is the inital number of radionuclides. In the model, uncertain is introduced if we have uncertain information on the inital number y0 of radionuclides present in the material. Note that the phenomenon of nuclear disintegration is consider a stochastic process, uncertainty being introduced by the lack of information on the radioactive material under study. To take into account the uncertainty, we consider y0 to be a fuzzy number. Let λ = 1, I = [0, .1] and y0 = (r − 1, 1 − r). (I): If y(t) is (i)-differentiable, the exact solution is: y(t; r) = (r − 1)et , y(t; r) = (1 − r)et . Now, { } I1 = t ∈ I|y(t, 0) − y(t, 0) ≥ 0 = [0, +∞) { } ′ ′ I2 = t ∈ I|y (t, 0) − y (t, 0) ≥ 0 = [0, +∞) ∩

′

are intervals that on them, y and y are valid(are fuzzy numbers). Therefore, on I1 I2 = [0, +∞), y is (i)-differentiable solution. Since in this example −λ < 0, Approximatioe solutions w, w can be found by solving ODEs (5.11),(5.12)(see Figs.5 and 6) The numerical values are given in tables (see 3 and 4) and error ∆w is shown at t=0.09. Table 3. r 0.1 0.3 0.5 0.7 0.9 ∆w at t=0.09

RK-5 w(ti ; r) -0.994653 -0.773619 -0.552585 -0.331551 -0.110517

w(ti ; r) 0.994653 0.773619 0.552585 0.331551 0.110517 4.5682e-010

Euler w(ti ; r) -0.994159 -0.773235 -0.552311 -0.331386 -0.110462

w(ti ; r) 0.994159 0.773235 0.552311 0.331386 0.110462 0.054879e-3

Exact Solution y(ti ; r) y(ti ; r) -0.994653 0.994653 -0.773619 0.773619 -0.552585 0.552585 -0.331551 0.331551 -0.110517 0.110517




19

Figure 6: The approximate solution to the FFDE (II): If ye(t) is (ii)-differentiable, the exact solution is y(t; r) = (r − 1)e−t , y(t; r) = (1 − r)e−t . Now, { } I1 = t ∈ I|y(t, 0) − y(t, 0) ≥ 0 = [0, +∞) { } ′ ′ I2 = t ∈ I|y (t, 0) − y (t, 0) ≥ 0 = [0, +∞) ′

∩

are intervals that on them, y and y are valid(are fuzzy numbers). Therefore, on I1 I2 = [0, +∞), y is (ii)-differentiable solution. Approximatioe solutions w, w can be found by solving ODEs (5.15),(5.16)(see Figs.7 and 8). It is notable that all the results obtained in the our numerical method is similar to the results of Nieto et al. in[28]. Therefore, solving this examples shows that our numerical method is capable of solving FDE and FFDE.




20

Table 4. r 0.1 0.3 0.5 0.7 0.9 ∆w at t=0.09

RK-5 w(ti ; r) -0.813531 -0.632746 -0.451961 -0.271177 -0.090392

w(ti ; r) 0.813531 0.632746 0.451961 0.271177 0.090392 9.1352e-005

Euler w(ti ; r) -0.813943 -0.633067 -0.452191 -0.271314 -0.090438

w(ti ; r) 0.813943 0.633067 0.452191 0.271314 0.090438 0.045534e-3

Exact Solution y(ti ; r) y(ti ; r) -0.814353 0.814353 -0.633386 0.633386 -0.452418 0.452418 -0.271451 0.271451 -0.090483 0.090483

Figure 8: The approximate solution to the FFDE

7 Conclusion Fifth order Runge-Kutta method for solving first-order fully fuzzy differential equations (FFDE) under strongly generalized H-differentiability was studied. We showed that FFDE could be divided in four differential equations under H-differentiability and each case satisfies the Lipschitz condition and have a unique solution. The convergence of this method is discussed and using an algorithm for fifth order Runge-Kutta method, the solution is approximated in each case. In the end, we give some examples to illustrate the theory. Higher order Runge-Kutta methods will be considered in our future research. Acknowledgements The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from editor too. References [1] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by taylor method, Comput. Methods Appl. Math, 2 (2002) 113-124. http://dx.doi.org/10.2478/cmam-2002-0006 [2] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by Runge-kutta method, Comput. Methods Appl. Math, 2 (2001) 1-13.



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[3] S. Abbasbandy, T. Allahviranloo, Lopez-pouso oscar, J.J. Nieto, Numerical solutions of fuzzy differential equation inclusions, Comput. Methods Appl. Math, 48 (2004) 1633-1641. http://dx.doi.org/10.1016/j.camwa.2004.03.009 [4] S. Abbasbandy, N. Ahmady, E. Ahmady, Numerical solutions of fuzzy differential equations by predictor corrector method, Inf. Sci, 177 (2007) 1633-1647. http://dx.doi.org/10.1016/j.ins.2006.09.015 [5] S. Abbasbandy, T. Kiani NA, M. Barkhordari, Toward the existence and uniqueness of solution of second-order fuzzy differential equations, Inf. Sci, 179 (2009) 1207-1215. http://dx.doi.org/10.1016/j.ins.2008.11.004 [6] A. Ban, B. Bede, Properties of the cross product of fuzzy numbers, Fuzzy Math, 14 (2006) 513-531. [7] B. Bede, IJ. Rudas, Al. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inf. Sci, 177 (2007) 1648-1662. http://dx.doi.org/10.1016/j.ins.2006.08.021 [8] B. Bede, SG. Gal, Generalizations of differentiability of fuzzy number valued fuction with application to fuzzy differential equations, Fuzzy Sets Syst, 151 (2005) 581-599. http://dx.doi.org/10.1016/j.fss.2004.08.001 [9] B. Bede, J. Fodor, Product type operations between fuzzy numbers and their applications in geology, Acta Polytec Hung, 3 (2006) 123-139. [10] J. J. Buckley, T. Feuring, Fuzzy differential equation, Fuzzy Sets, 110 (2000) 43-54. http://dx.doi.org/10.1016/S0165-0114(98)00141-9 [11] J. J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets Syst, 105 (1999) 241248. http://dx.doi.org/10.1016/S0165-0114(98)00323-6 [12] W. Congxin, S. Shiji, Existence theorem to the Cauchy problem of fuzzy differential equation under compatnesstype conditions, Info. Sci, 108 (1998) 123-134. http://dx.doi.org/10.1016/S0020-0255(97)10064-0 [13] P. Darabi, S. Moloudzadeh, H. Khandani, A numerical method for solving first-order fully fuzzy differential equation under strongly generalized H-differentiability, Metho. and Appl, to appeared (2015). http://dx.doi.org/10.1007/s00500-015-1743-0 [14] D. Dubois, H. Prade, Towards fuzzy differential calculus : Part 3, differentation, Fuzzy Sets Syst, 8 (1982) 225233. http://dx.doi.org/10.1016/S0165-0114(82)80001-8 [15] M. Friedman, M. Ming, A. Kandel, Numerical solution of fuzzy differential and integral equations, Fuzzy Sets Systems, 106 (1999) 35-48. http://dx.doi.org/10.1016/S0165-0114(98)00355-8 [16] SG. Gal, Approximation theory in fuzzy setting. In: Anastassious GA (ed) Hand book of analytic-computational methods in applied mahematics, Chapman Hall C R C Press, New york, 617-666. http://dx.doi.org/10.1201/9781420036053.ch13 [17] D. N. Georgiou, J. J. Nieto, R. Rodriguez, Initial value problem for higher-order fuzzy differential equations, Nonlinear Anal, 63 (2005) 587-600. http://dx.doi.org/10.1016/j.na.2005.05.020



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Fifth order Runge-Kutta method for solving first-order

Fifth order Runge-Kutta method for solving first-order

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