AN OPERATIONAL METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO DERIVATIVES

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1 ACTA MATHEMATCA VETNAMCA Volume 24, Number 2, 1999, pp AN OPERATONAL METHOD FOR SOLVNG FRACTONAL DFFERENTAL EQUATONS WTH THE CAPUTO DERVATVES YUR LUCHKO AND RUDOLF GORENFLO Abstract. n the present paper, we first develop the operational calculus of Mikusiński s type for the Caputo fractional differential operator. This calculus is used to obtain exact solutions of an initial value problem for linear fractional differential equations with constant coefficients and fractional derivatives in Caputo s sense. The initial conditions are given in terms of the field variable and its derivatives of integer order. The obtained solutions are expressed through Mittag-Leffler type functions. Special cases and integral representations of solutions are presented. 1. ntroduction Fractional differential equations have excited in recent years a considerable interest both in mathematics and in applications. They were used in modeling of many physical and chemical processes and in engineering (see, for example, [1] [5], [9], [1], [16], [17]). n its turn, mathematical aspects of fractional differential equations and methods of their solution were discussed by many authors: the iteration method in [24], the series method in [1], the Fourier transform technique in [3], [12], special methods for fractional differential equations of rational order or for equations of special type in [2], [13], [15], [18], [19], [21], the Laplace transform technique in [9], [1], [16], [17], [21], [22], the operational calculus method in [11], [15]. Let us note that in mathematical treatises on fractional differential equations the Riemann-Liouville approach to the notion of the fractional derivative of order ( ) is normally used: (1) (D f)(x) := ( ) m d (J m f)(x), m 1 < m, m N, x >, dx Received July 15, Mathematics Subject Classification. Primary 45J5, secondary 44A4, 26A33, 33E3. Key words and phrases. Fractional differential equations, operational calculus, Caputo fractional derivative, Mittag-Leffler type functions. This work was partially supported by the Research Commission of Free University of Berlin (Project Convolutions )

2 28 YUR LUCHKO AND RUDOLF GORENFLO where (2) (J f)(x) := 1 Γ() (x t) 1 f(t)dt, >, x >, (J f)(x) := f(x), x > is the Riemann-Liouville fractional integral of order. n this interpretation, the fractional derivative is left-inverse (and not right-inverse) to the corresponding fractional integral, which is the natural generalization of the Cauchy formula for the n-fold primitive of a function f(t). As to the initial value problem for fractional differential equations with the fractional derivatives in the Riemann-Liouville sense, there are some troubles with the initial conditions, see [11], [15], [21], [22], [23]. Namely, these initial conditions should be given as the (bounded) initial values of the fractional integral J m and of its integer derivatives of order k = 1, 2,..., m 1. On the other hand, in modeling of real processes the initial conditions are normally expressed in terms of a given number of bounded values assumed by the field variable and its derivatives of integer order. n order to meet this physical requirement, an alternative definition of fractional derivative was introduced by Caputo [4] and adopted by Caputo and Mainardi [5] in the framework of the theory of Linear Viscoelasticity: (3) (D f)(x) := (J m f (m) )(x) = 1 Γ(m ) m 1 < m, m N, x >. (x t) m 1 f (m) (t)dt, n a series of papers (see [9], [16], [17] and the references there) the Caputo fractional derivative was considered and some of the simplest linear fractional differential equations with constant coefficients and fractional derivatives in the Caputo sense were solved by using the Laplace transform technique. n the present paper we use the operational calculus method to solve an initial value problem for a general linear fractional differential equation with constant coefficients and with the Caputo fractional derivatives. We first develop a modification of the Mikusiński operational calculus for the Caputo fractional derivative, a modification that is more suitable for application to the solution of differential and integral equations of fractional order. Firstly, we consider the space C α of functions which can have an integrable singularity in the neighborhood of zero, instead of the

3 AN OPERATONAL METHOD 29 space C[, ) of functions continuous on the half-axis used in Mikusiński s operational calculus. Since the kernel of the Riemann-Liouville integral operator J belongs to the space C α if 1 > α and α 1, this function space seems to be more suitable for the development of the operational calculus for the Caputo derivative. Secondly, the abovementioned property allows us to consider the Riemann-Liouville integral operator J as the convolution product with the power function x 1 /Γ() in the space C α if 1 > α, and this power function plays in our considerations the same role, as the function {1} in Mikusiński s operational calculus. As an application of this calculus we obtain solutions of linear fractional differential equations with constant coefficients and Caputo derivatives. 2. Some properties of the caputo fractional derivative Since both the Riemann-Liouville and the Caputo derivatives ((1) and (3), respectively) are defined through the Riemann-Liouville fractional integral (2) and this operator plays a very important role in the development of the corresponding operational calculus, there are some coinciding elements in the operational calculi for both fractional derivatives. The operational calculus for the Riemann-Liouville fractional derivative was already developed in [8], [11], [15] and we shall cite some theorems from there without proofs. We begin by defining the function space C α, α R, which was used by Dimovski [6] in his development of the operational calculus for the hyper-bessel differential operator. Definition 2.1. A real or complex-valued function f(x), x >, is said to be in the space C α, α R, if there exists a real number p, p > α, such that f(x) = x p f 1 (x) with a function f 1 (x) in C[, ). Clearly, C α is a vector space and the set of spaces C α is ordered by inclusion according to (4) C α C β α β. Theorem 2.1. The Riemann-Liouville fractional integral operator J,, is a linear map of the space C α, α 1, into itself, that is, J : C α C +α C α.

4 21 YUR LUCHKO AND RUDOLF GORENFLO Remark 2.1. n the case f C α for a value α 1 and for 1 we have J f C C[, ). t is important to note, that the operator J, >, has the following convolutional representation in the space C α, α 1: (5) (J f)(x) = (h f)(x), h (x) := x 1 /Γ(), f C α, where (g f)(x) = g(x t)f(t)dt, x > is the Laplace convolution. For the Laplace convolution itself the inclusion (6) g f C α1 +α 2 +1 C 1, f C α1, g C α2, α 1, α 2 1 holds true. The representation (5) and the commutativity of the Laplace convolution (see [6], [2]) lead to the following property of the Riemann- Liouville fractional integral: (J δ J η f)(x) = (J η J δ f)(x), f C α, α 1, δ, η. Next, using the associativity of the Laplace convolution and the Euler integral of the first kind for the evaluation of (h δ h η )(x), we obtain (7) (J δ J η f)(x) = (J δ+η f)(x), f C α, α 1, δ, η, that is also well known. n particular, it follows from (7) that (8) (J }. {{.. J } f)(x) = (J n f)(x), f C α, α 1,, n N. n t is obvious, that the Caputo derivative (3) is not defined on the whole space C α. Let us introduce a subspace of C α, which is suitable for dealing with the Caputo derivative. Definition 2.2. A function f(x), x >, is said to be in the space C m α, m N = N {}, iff f (m) C α. Remark 2.2. The space Cα m doesn t coincide with the space C α (m) = {f : there exist p > α, f C m [, ) such that f(x) = x p f(x)}, considered in

5 AN OPERATONAL METHOD 211 [14]. For example, if f(x) = cos(x)/ x, x >, then f C (1) 1, f C1 1 and for the function f(x) 1, x >, we have f C (1), f C1. We state five of the properties of the space C m α, which will be used in the further discussions. 1) C m α is a vector space. 2) C α C α. 3) f f C m α for a value α 1 and an index m 1, then f (k) (+) := lim x + f (k) (x) < +, k m 1, and the function is in C [, ). { f(x), x >, f(x) = f(+), x = Proof. f Cα m means f (m) := φ C α. Let us fix x = X >. Since f (m) = φ C[ξ, X], < ξ < X, we arrive at X ξ f (m) (t) dt = f () (X) f () (ξ), where the left- and right-hand sides as functions of ξ are continuous on (, X]. Furthermore, since we get X lim ξ + ξ f (m) (t) dt = X φ(t) dt < +, f () (+) := lim ξ + f () (ξ) = f () (X) X φ(t) dt < +, hence X f () (X) = φ(t) dt + f () (+).

6 212 YUR LUCHKO AND RUDOLF GORENFLO By liberating x = X and putting f () () := f () (+) we recognize f () as continuous on [, ) and obtain the representation (9) f(x) = (J m φ)(x) + φ(t) = f (m) (t), k= f (k) (+) xk k!, x, f (k) (+) = lim x + f (k) (x) < +, k m 1. 4) f f C m α for a value α 1, then f C m (, ) C [, ). ndeed, since f (m) C α, we have f (m) C(, ). The inclusion f C () [, ) follows from property 3). 5) For real α 1 and index m 1 the following equivalence holds: f C m α f(x) = (J m φ)(x) + k= c k x k k!, x, φ C α. The first statement ( ) is already proved (see (9)), the second one ( ) is checked directly. Let us note that in this case we have f (k) () = c k, k =, 1,..., m 1, f (m) = φ. On the basis of the properties 1)-5) of the function space Cα m we prove now some theorems, important for the development of the corresponding operational calculus. Theorem 2.2. Let f C m 1, m N. Then the Caputo fractional derivative D f, m, is well defined and the inclusion { D C 1, m 1 < m, f C k 1 [, ) C 1, m k 1 < m k, k = 1,..., m 1 holds true. Proof. n the case m 1 < m the inclusion under consideration follows from the definition of the Caputo derivative D and Theorem 2.1. The property 4) of the space Cα m, m 1, and the corresponding mapping properties of the Riemann-Liouville fractional integral (see [24]) give us the inclusion D f C k 1 [, ) for m k 1 < m k, k = 1,..., m 1. The inclusion C k 1 [, ) C[, ) C 1 follows from (4).

7 AN OPERATONAL METHOD 213 Theorem 2.3. Let f C m 1, m N and m 1 < m. Then the Riemann-Liouville and the Caputo fractional derivatives are connected by the relation: (1) (D f)(x) = (D f)(x) + k= f (k) (+) Γ(1 + k ) xk, x >. Proof. Making use of the representation (9), we get ( ) m d (D f)(x) = (J m f)(x) dx ( ) [ m d = (J m (J m f (m) )(t) + dx = (J m f (m) )(x) + = (D f)(x) + k= ( d dx k= ) [ m (J m ]) f (k) (+) tk (x) k! ]) k= f (k) (+) Γ(1 + k ) xk, x >. f (k) (+) tk k! n finding the last expression we have made use of the rules for fractional integration and differentiation of the power function p ν (x) = x ν, ν > 1, namely, (x) (11) (12) (J p ν )(x) = (D p ν )(x) = Γ(ν + 1) Γ(ν ) xν+,, x >, Γ(ν + 1) Γ(ν + 1 ) xν,, x >. Corollary 2.1. t follows from the representation (1) that the Riemann- Liouville fractional derivative (D f)(x) is not, in the general case, in the space C 1, if f C m 1. There are only three exceptional cases: 1) f = m N, then D f D f f (m) C 1. 2) f f (k) (+) =, k =,..., m 1, then D f D f C 1.

8 214 YUR LUCHKO AND RUDOLF GORENFLO 3) f < < 1, then D f C 1 because of (D f)(x) = (D f)(x) + f() Γ(1 ) x. Theorem 2.4. Let m 1 < m, m N, α 1 and f C m α. Then (13) (J D f)(x) = f(x) k= Proof. Using the relations (3), (7), and (9) we get f (k) (+) xk k!, x. (J D f)(x) = (J J m f (m) )(x) = (J m f (m) )(x) = f(x) k= f (k) (+) xk k!, x. Theorem 2.5. Let f C m 1, m N, f() = = f () () = and g C 1 1. Then the Laplace convolution h(x) = f(t)g(x t) dt is in the space C m+1 1 and h() = = h (m) () =. Proof. Let m =. The property 4) of the space C 1 1 gives us the inclusion g C[, ). n this case h(x) = x ε h 1 (x) for an ε >, h 1 C[, ), i.e., h C[, ) and h() =. We have also h (x) = f(t)g (x t) dt + f(x)g(), x >. Since g C 1 and f C 1 it follows from the inclusion (6) that x f(t)g (x t) dt C 1 and, consequently, h C 1, which proves our theorem in the case m =. For m = 1, it follows from the property 4) of

9 AN OPERATONAL METHOD 215 the space C 1 1, that f C[, ). Using the condition f() = and the representation obtained above for h we arrive at h () = and h (x) = f (x t)g (t) dt + f (x)g(), x >. Since g C 1 and f C 1, we get the inclusion h C 1. Repeating the same arguments m times, we arrive at h() = = h (m) () = and the representation h (m+1) (x) = which gives us the inclusion h C m+1 1. f (m) (x t)g (t) dt + f (m) (x)g(), x >, 3. Operational calculus for the caputo derivative For the sake of simplicity we shall consider in our further discussions the case of the space C 1, which turns out to be the most interesting one for applications. As in the case of Mikusiński s calculus, we have Theorem 3.1. The space C 1 with the operations of the Laplace convolution and ordinary addition becomes a commutative ring (C 1,, +) without divisors of zero. This ring can be extended to the quotient field M by following the lines of Mikusiński [2]: M := C 1 (C 1 \ {})/, where the equivalence relation ( ) is defined, as usual, by (f, g) (f 1, g 1 ) (f g 1 )(x) = (g f 1 )(x). Thus, we can consider the elements of the field M as convolution quotients f/g and define the operations in M as follows: f g + f 1 g 1 := f g 1 + g f 1 g g 1

10 216 YUR LUCHKO AND RUDOLF GORENFLO and f g f1 g 1 := f f 1 g g 1. The proof of the fact that the set M is a commutative field with respect to the operations + and is based on Theorem 3.1. t is easily seen that the ring C 1 can be embedded in the field M by the map ( > ): f h f h, with, by (5), h (x) = x 1 /Γ(). Defining the operation of multiplication with a scalar λ from the field R (or C) by the relation λ f g := λf g, f g M, and remembering the fact, that the set C 1 is a vector space, we check that the set M is a vector space too. Since the constant function f(x) λ, x >, is in the space C 1, we should distinguish the operation of multiplication with a scalar in the vector space M and the operation of multiplication with a constant function in the field M. n this last case we shall write (14) {λ} f g = λh +1 h f g = {1} λf g. t is easy to check that the element = h h of the field M is the unity of this field with respect to the operation of multiplication. Let us prove that this element of the field M is not reduced to a function lying in the ring C 1 and, consequently, it can be regarded as a generalized function. ndeed, let the unity = h h be reduced to some function f C 1. Then we have (15) h h f h h h h = (f h ) h. Making use of the representations (5) and (8) we rewrite the last relation in the form (16) x 2 1 Γ(2) = (J 2 f)(x),

11 AN OPERATONAL METHOD 217 where (J 2 f)(x) is the Riemann-Liouville fractional integral (2). t is well known (see [15] for the case of the space C 1 ), that the Riemann-Liouville fractional derivative (1) is left-inverse to the Riemann-Liouville fractional integral. Applying the operator D 2 to the equality (16) and using the formula (12) for evaluation of the right-hand side, we obtain (17) f(x), x >. Relations (16) and (17) lead to the contradictory formula x 2 1 Γ(2), x >, which shows, that the relation (15) is contradictory for any f C 1 and, consequently, the unity element of the field M is not reduced to function in the ring C 1. Later we shall consider some other elements of the field M possessing this property, in particular, the element which will play an important role in the applications of operational calculus and is given by Definition 3.1. The algebraic inverse of the Riemann-Liouville fractional integral operator J is said to be the element S of the field M, which is reciprocal to the element h in the field M, that is, (18) S = h h h h h h 2, where (and in what follows) = h h denotes the identity element of the field M with respect to the operation of multiplication. As we have already seen, the Riemann-Liouville fractional integral J can be represented as a multiplication (convolution) in the ring C 1 (with the function h, see (5)). Since the ring C 1 is embedded into the field M of convolution quotients, this fact can be rewritten as follows: (19) (J f)(x) = S f. As to the Caputo fractional derivative, there exists no convolution representation in the ring C 1 for it, but it is reduced to the operator of multiplication in the field M.

12 218 YUR LUCHKO AND RUDOLF GORENFLO Theorem 3.2. Let f C m 1, m 1 < m, m N. Then the following relation holds true in the field M of convolution quotients: (2) D f = S f S f, f (x) := Proof. Theorem 2.4 gives us the representation (J D f)(x) = f(x) f (x), f (x) = k= k= f (k) (+) xk k!. f (k) (+) xk k!, x. Upon multiplying both sides of this relation by S, if we apply the relation (19), we shall obtain the assertion (2). We already know (see (8)), that for >, n N h n (x) := h... h = h n (x). }{{} n Let us extend this relation to an arbitrary positive real power exponent: (21) h λ (x) := h λ (x), λ >. We have for any λ > the inclusion h λ C 1, and the following relations can be easily checked (α >, β > ): (22) h α h β = h α h β = h (α+β) = h α+β, (23) h α 1 = h β 2 1 α = 2 β. Then we define a power function of the element S with an arbitrary real power exponent λ: (24) S λ = h λ, λ <,, λ =,, λ >. h λ

13 AN OPERATONAL METHOD 219 Using this definition and the relations (22) and (23), we get (α, β R): (25) S α S β = S α+β, (26) S α 1 = S β 2 1 α = 2 β. For many applications it is important to know the functions of S in M which can be represented by means of the elements of the ring C 1. One useful class of such functions is given by the following theorem. Theorem 3.3. Let the power series i 1,...,i n = a i1,...,i n x i 1 1 x i n n, x = (x 1,..., x n ) R n, a i1,...,i n R, be convergent at a point x = (x 1,..., x n ) with all x k, k = 1,..., n, and β >, α i >, i = 1,..., n. Then the function of S S β i 1,...,i n = a i1,...,i n (S α 1 ) i 1 (S α n ) i n = i 1,...,i n = a i1,...,i n h (β+α1 i 1 + +α n i n )(x), where h (x) is given by (5), defines an element of the ring C 1. For the proof of this theorem we refer to [11]. We give here some operational relations, which will be used in the further discussions. For more operational relations we refer to [8], [11], and [15]. (27) S ρ = x 1 E, (ρx ), where E α,β (z) is the generalized Mittag-Leffler function defined by E α,β (z) := k= z k, α, β >, z <. Γ(αk + β)

14 22 YUR LUCHKO AND RUDOLF GORENFLO The relation (27) can formally be obtained as a geometric series: S ρ = ρ h = k= = h ρh = ρ k x (k+1) 1 Γ(k + ) k= ρ k h k+1 = x 1 E, (ρx ). The m-fold convolution of the right-hand side of the relation (27) gives us the operational relation: (28) (S ρ) m = x E m,m(ρx ), m N, where E m α,β(z) := i= (m) i z i i 1 i!γ(αi + β), α, β >, z <, (m) i = (m + k). k= Let β >, α i >, i = 1,..., n. We then have the operational relation (29) n S β λ i S α i = x β 1 E (α1,...,α n ),β(λ 1 x α 1,..., λ n x α n ) with the multivariate Mittag-Leffler function E (a1,...,a n ),b(z 1,..., z n ) := k= and the multinomial coefficients l 1 + +ln=k l 1,...,ln (k; l 1,..., l n ) := n z l i i (k; l 1,..., l n ) Γ ( b + n ) a i l i k! l 1! l n! For the general properties of the Mittag-Leffler functions we refer to [7].

15 AN OPERATONAL METHOD Fractional differential equations We first apply the developed operational calculus to some simple fractional differential equations, which have already been considered by using the Laplace transform technique (see [9] and references there). We begin with the initial value problem ( > ) { (D y)(x) λy(x) = g(x), (3) y (k) () = c k R, k =,..., m 1, m 1 < m, λ R. The function g is assumed to lie in C 1 if N, in C 1 1 if N, and the unknown function y(x) is to be determined in the space C m 1. Making use of the relation (2), the initial value problem (3) can be reduced to the following algebraic equation in the convolution field M: S y λy = S y + g, y (x) = k= whose unique solution in the field M has the form y = y g + y h = c k x k k!, m 1 < m, S λ g + S S λ y. t turns out that the right-hand part of this relation can be interpreted as a function lying in the space C m 1, i.e., as a classical solution of the initial value problem (3). We shall prove this fact later for more general equations, here we shall only demonstrate formulas for this solution. t follows from the operational relation (27) and the embedding of the ring C 1 into the field M, that the first term of this relation, y g (solution of the inhomogeneous fractional differential equation (3) with zero initial conditions), is represented in the form (31) y g (x) = t 1 E, (λt )g(x t) dt. As to the second term, y h, it is a solution of the homogeneous fractional differential equation (3) (g(x) replaced by ) with the given initial conditions, and we have (32) y h (x) = k= c k u k (x), u k (x) = S S λ {xk k! }.

16 222 YUR LUCHKO AND RUDOLF GORENFLO Making use of the relation (33) x k k! = h k+1(x) = h (k+1)/ (x) = S (k+1)/, the formula (25), and the operational relation (29), we get the representation of the functions u k (x), k =,..., m 1, in terms of the generalized Mittag-Leffler function: u k (x) = S S λ {xk k! } = S (k+1)/ λs 1 = x k E,k+1 (λx ). Furthermore, due to the representation (5) of the Riemann-Liouville fractional integral, we have u k (x) = (J k u )(x), u (x) = E,1 (λx ) := E (λx ), z k where E α (z) = is the Mittag-Leffler function. Using the k= Γ(αk + 1) last representation we arrive at the relations u (l) k () = δ kl, k, l =,..., m 1, and therefore the m functions u k (x), k =,..., m 1, represent the general solution of the homogeneous fractional differential equation (3). Summarizing the obtained results, we get the solution of the initial value problem (3) in the form y(x) = t 1 E, (λt )g(x t) dt + k= c k x k E,k+1 (λx ), which can be rewritten in the case λ in terms of the Mittag-Leffler function: y(x) = 1 λ The next equation d dt (E (λt )) g(x t) dt + c k (J k E (λt ))(x). k= (34) { y (x) λ 1 (D y)(x) λ 2 y(x) = g(x), y() = c R, < < 1, λ 1, λ 2 R

17 AN OPERATONAL METHOD 223 with = 1/2 and λ 1 <, λ 2 < corresponds to the Basset problem, a classical problem in fluid dynamics (see [9], [17]). We treat the general problem (34). The function g C 1 is given, and the unknown function y(x) is to be determined in the space C 1 1. With the help of the relation (2) the problem under consideration can be reduced to the algebraic equation in the field M: S 1 y λ 1 S y λ 2 y = g + S 1 y 1 λ 1 S y, y 1 (x) y (x) c. Applying the relation (26) we represent a unique solution of this equation in the field M in the form y = y g + y h = S 1 λ 1 S 1 λ 2 g + S 1 λ 1 S 1 S 1 λ 1 S 1 λ 2 y 1, y 1 (x) c. Using now the relations (25) and (29) we arrive at the representation S 1 λ 1 S 1 λ 2 = S1 1 λ 1 S (1 ) 1 λ 2 S1 1 = E (1,1),1 (λ 1 x 1, λ 2 x) with the multivariate Mittag-Leffler function. same technique and (14), (33), the relation y h (x) = S 1 λ 1 S 1 S 1 λ 1 S 1 λ 2 [ = c S 1 + λ 2 {c } = S 2 1 [ + λ 1 S 1 1 λ 2 S 1 1 = c [ 1 + λ2 xe (1,1),2 (λ 1 x 1, λ 2 x) ]. We also have, using the λ 2 S 1 λ 1 S 1 λ 2 ] ] c S 1 The unique solution of the initial value problem (34) has then the form y(x) = E (1,1),1 (λ 1 t 1, λ 2 t)g(x t) dt + c [ 1 + λ2 xe (1,1),2 (λ 1 x 1, λ 2 x) ]. The inclusion y C 1 1 will be shown in the next theorem for a more general situation.

18 224 YUR LUCHKO AND RUDOLF GORENFLO We consider now the general linear differential equation with constant coefficients and the Caputo derivatives. Theorem 4.1. Let > 1 > > n, m i 1 < i m i, m i N = N {}, λ i R, i = 1,..., n. The initial value problem (35) (D y)(x) n λ i (D i y)(x) = g(x), y (k) () = c k R, k =,..., m 1, m 1 < m, where the function g is assumed to lie in C 1 if N, in C 1 1 if N, and the unknown function y(x) is to be determined in the space C m 1, has a solution, unique in the space C m 1, of the form (36) y(x) = y g (x) + Here (37) y g (x) = k= c k u k (x), x. t 1 E ( ), (t)g(x t)dt is a solution of the problem (35) with zero initial conditions, and the system of functions (38) u k (x) = xk k! + n i=l k +1 λ i x k+ i E ( ),k+1+ i (x), k =,..., m 1, fulfills the initial conditions u (l) k () = δ kl, k, l =,...,. The function (39) E ( ),β (x) = E ( 1,..., n ),β(λ 1 x 1,..., λ n x n ) is a particular case of the multivariate Mittag-Leffler function (29) and the natural numbers l k, k =,..., m 1, are determined from the condition (4) { mlk k + 1, m lk +1 k.

19 AN OPERATONAL METHOD 225 n the case m i k, i =,..., m 1, we set l k :=, and if m i k + 1, i =,..., m 1, then l k := n. Proof. Since y C m 1, the initial value problem (35) can be reduced to the following algebraic equation in the field M by using the formula (2): (41) S y S y y (x) = k= n λ i (S i y S i y i ) = g, x k m c k k!, y i 1 i (x) = k= x k c k, i = 1,..., n. k! Equation (41) has a unique solution in the field M, which in view of the relation (26) is given by (42) y = y g + y h = S n λ i S i/ g + S y n S n λ i S i/ y i λ i S i/. Using the operational relation (29), Theorems 3.1 and 3.3, we can interpret the element y g of the field M as the function in the ring C 1 : y g (x) = t 1 E ( ), (t)g(x t)dt, where the function E ( ), (x) is given by (39). n the case N the function g(x) is in the space C 1 1 and Theorem 2.5 gives us the inclusion y g C 1. 1 Let us show that the function y g (x) is in the space C 1. m Multiplying the relation y g = S n λ i S i/ g by (S n λ i S i/ ) and then by h = S 1, we readily get (43) y g (x) = (J g)(x) + n λ i (J i y g )(x).

20 226 YUR LUCHKO AND RUDOLF GORENFLO t follows from this representation, that { (44) y g (x) = (J C 1 1, N, ψ 1 )(x), ψ 1 C 1, 1 N. Combining now the relations (44) and (43) and repeating the same arguments p times (p = [/( 1 )] + 1) we arrive at the representation { (45) y g (x) = (J C 1, N, ψ p )(x), ψ p C 1, 1 N. n the case = m N it follows from (45) that y g (m) = ψ p C 1 and y g () = = y g () () =. f N, m 1 < < m then the function h (x) = x Γ() C and we have h () = = h (m 2) () =. Using now the representation (45) and Theorem 2.5 we get the inclusion y g C 1 m and the relations y g () = = y g () () =. The last considerations prove the part of Theorem 4.1 concerning the solution of the problem (35) with zero initial conditions. Let us consider the element y h of the representation (42). Using the expressions for the functions y (x), y i (x), i = 1,..., n, from (41), we have y h = k= c k u k (x), u k = S l k S n λ i S i/ λ i S i/ { x k k! where the natural numbers l k, k =,..., m 1, are defined by (4). Applying the relation (33) for the function xk, some elementary transformations and then the operational relation (29), we find from the previous k! representation (k =,..., m 1): }, u k = S (k+1)/ S l k S n λ i S i/ λ i S i/ =

21 AN OPERATONAL METHOD 227 = S (k+1)/ = S (k+1)/ + = xk k! + n i=l k +1 + n i=l k +1 n i=l k +1 S n λ i λ i S i/ λ i S i/ S (k+1+ i)/ n λ i S ( i)/ λ i x k+ i E ( ),k+1+ i (x), where the function E ( ),β (x) is given by (39). According to the definition of the numbers l k, we have m i k for i = l k + 1,..., n. t follows then, that k + i, i = l k + 1,..., n. Using this inequality, we readily get the inclusion u k C m 1, k =,..., m 1 and the relations u (l) k () = δ kl, k, l =,..., m 1, and therefore the m functions u k (x), k =,..., m 1, represent the general solution of the homogeneous fractional differential equation (35). Remark 4.1. The results of Theorem 4.1 can be used in some cases for the initial value problem (35) with the Riemann-Liouville fractional derivatives instead of the Caputo fractional derivatives. n particular, as we have seen in Corollary 2.1, (D y)(x) (D y)(x) if = m N or y (k) () =, k =,..., m 1, m 1 < m. n the case < < 1 we can use the relation (D y)(x) = (D y)(x) + y() Γ(1 ) x to reduce the initial value problem with the Riemann-Liouville fractional derivatives to the initial value problem of the type (35). Remark 4.2. The developed theory can be rewritten without any difficulties for the case of the Caputo derivative of the complex order. n particular, Theorem 4.1 gives us then the solution of the initial value problem (35) for C if R() = m N, then = m), R() > R( 1 ) > > R( n ) if R( i ) = m i N, then i = m i ), m i 1 < R( i ) m i, m i N = N {}, i, λ i C, i = 1,..., n, y (k) () = c k C, k =,..., m 1, m 1 < R() m.

22 228 YUR LUCHKO AND RUDOLF GORENFLO The solution (36)-(38) of the initial value problem (35) was obtained in terms of the Mittag-Leffler type function E ( ),β (x), which is given by its series representation (46) E ( ),β (x) = k= l 1 + +ln=k l 1,...,ln n (λ i x i ) l i (k; l 1,..., l n ) Γ(β + n ( i )l i ) Let us find an integral representation of this function. We shall use the Hankel integral representation of the reciprocal Gamma-function 1 Γ(z) = 1 2πi Ha(+) e ζ ζ z dζ, z C, where Ha(ε+) is the Hankel path, a loop which starts from along the lower side of the negative real axis, encircles the circular disc ζ = ζ > ε > in the positive sense and ends at along the upper side of the negative real axis. Then we substitute this representation into (46) and get E ( ),β (x) = k= 1 2πi l 1 + +ln=k l 1,...,ln Ha(+) (k; l 1,..., l n ) n (λ i x i ) l i e ζ ζ β n ( i)l i dζ. Changing the order of integration and summation, using the multinomial formula with exponent 1 to get a closed form for a sum in integrand and substituting ζ = sx, we finally get (47) E ( ),β (x) = x 1 β 1 e sx s β ds 2πi Ha(λ+) s n, λ i s i ( n ) 1/( 1 ) λ = max 1, λ i.

23 AN OPERATONAL METHOD 229 Corollary 4.1. Applying the representation (47) to the formulas (37) and (38), we rewrite the solution (36) of the initial value problem (35) in the form (48) y(x) = where u δ (t)g(x t) + k= c k u k (x), x, (49) u δ (x) = 1 2πi Ha(λ+) e sx ds s n, λ i s i (5) u k (x) = xk k! + 1 2πi = 1 2πi Ha(λ+) Ha(λ+) e sx n i=l k +1 λ i s i s n λ i s i e sx [ s l k λ i s i s n λ i s i ] ds s k+1 ds, k =,..., m 1. sk+1 n particular, if n =, λ n, then l = n 1 and we have [ ] u (x) = 1 e sx s n 1 λ i s i ds 2πi s n 1 s λ i s i λn Ha(λ+) n this situation we have the relation f n >, we get u (x) 1. u δ (x) = 1 λ n u (x). Remark 4.3. The initial value problem (35) for the three cases: 1) n = 1, 1 =, λ 1 = 1, 2) n = 2, = 1, λ 2 = 1, 2 =, and 3) n = 2,

24 23 YUR LUCHKO AND RUDOLF GORENFLO = 2, λ 2 = 1, 2 = was considered in [9] by using the Laplace transform method. n this research and survey paper the form (48) - (5) of the solution was obtained and used, by evaluating the contribution of poles of the integrand by the residue theorem and transforming the Hankel path Ha(λ+) into the Ha(+), to represent it as a sum of oscillatory and monotone parts. n addition, asymptotic expansions, plots and interesting particular cases are given there. General results concerning the methods of evaluation of the poles of the integrand in the integral representations of the type (49), (5), asymptotic expansions of such representations as well as a lot of interesting particular cases can be found in the paper [12]. Finally, we consider some examples. Example 1. Let the right-hand part of the fractional differential equation (35) be a power function: g(x) = x α, α > 1, if N, α, if N. Γ(α + 1) Since x α Γ(α + 1) = h α+1(x) = h (α+1)/(x) = S (α+1)/, we get by using (42), (25), (29), and (47) the following representations of the part y g (x) of the solution (36): y g = = S n n = 1 2πi λ i S i/ S (+α+1)/ Ha(λ+) g = λ i S ( i)/ e sx s α 1 ds s n λ i s i S n λ i S i/ = x +α E ( ),+α+1 (x) S (α+1)/ Example 2. We consider now the equation (35) with i = (n i)α, i = 1,..., n, = nα, q 1 < q, q N. Then the solution (36) can be represented in terms of the generalized Mittag-Leffler function Eα,β m (x), see (28). ndeed, using the relation (26) and representing the corresponding

25 AN OPERATONAL METHOD 231 rational function as a sum of partial fractions, we get y g = S n p = n j j=1 m=1 λ i S i/ c jm g = S n α n λ i S n i α g (S α β j ) m g, n n p = n. The operational relation (28) gives us the representation with u δ (x) = y g (x) = p n j j=1 m=1 We have also (k =,..., q 1) q 1 y h (x) = c k u k (x), k= u k = xk k! + v k (x) = = n i=l k +1 S n α n p k n { } x k i=l k +1 k! Sα n n n jk j=1 m=1 λ i S n i α λ i S n i α = u δ (t)g(x t) dt c jm x α E m α,mα(β j x α ). p k λ i S n i α λ i S n i α n jk j=1 m=1 = xk k! + (J k+1 v k )(x), c jmk (S α β jk ) m c jmk x α Eα,mα(β m jk x α ), p k j=1 n jk = n. n the case α Q the generalized Mittag-Leffler function E m α,mα(x) can be represented in terms of special functions of the hypergeometric type (see [9]).

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