MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS. Igor Podlubny. Abstract

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1 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS Igor Podlubny Dedicated to Prof Rudolf Gorenflo, on the occasion of his 70-th birthday Abstract A matrix form representation of discrete analogues of various forms of fractional differentiation and fractional integration is suggested The approach, which is described in this paper, unifies the numerical differentiation of integer order and the n-fold integration, using the so-called triangular strip matrices Applied to numerical solution of differential equations, it also unifies the solution of ordinary integer- and fractional-order differential equations, and of fractional integral equations The suggested approach leads to significant simplification of the numerical solution of fractional integral and differential equations Mathematics Subject Classification: 26A33 (main), 65L12, 15A99, 39A70 Key Words and Phrases: fractional calculus, discrete approximations, fractional differential equations, fractional difference approximations, isoclinal matrices, triangular strip matrices 1Introduction There are several well-known approaches to unification of notions of differentiation and integration, and their extension to non-integer orders [13] The approach, which is described in this paper, unifies the numerical differentiation of integer order and the n-fold integration, using the so-called triangular Partially supported by grant 1/7098/20 of the Slovak Grant Agency for Science (VEGA)

2 360 I Podlubny strip matrices [1, 8, 14] Applied to numerical solution of differential equations, it also unifies solution of ordinary integer- and fractional-order differential equations, and of fractional integral equations Triangular strip matrices already appeared in some studies on fractional integralequations[2,3,6,7,9,10,11],butuntiltodaytheirusefulnessforapproximating fractional derivatives and solving fractional differential equations has not been recognized The structure of this paper is the following First of all, triangular strip matrices, and operations on them, are introduced Then discrete forms of the integer-order differentiation and of the n-fold integration are considered using triangular strip matrices, and a generalisation for the case of an arbitrary (noninteger) order of differentiation and integration is presented The advantages of the use of triangular strip matrices for numerical solution of fractional integral and differential equations of some important types are described and illustrated with four examples 2Triangular strip matrices In this paper we deal with matrices of a specific structure, which are called triangular strip matrices [14, p 20], and which have been mentioned in [1, 8] We will consider lower triangular strip matrices, ω ω 1 ω ω 2 ω 1 ω L =, (1) ω 1 ω2 ω 1 ω 0 0 ω ω 1 ω2 ω 1 ω 0 and upper triangular strip matrices, ω 0 ω 1 ω 2 ω 1 ω 0 ω 0 ω 1 ω 1 U = 0 0 ω 0 ω2 (2) ω1 ω 2 ω 0 ω ω 0 A lower (upper) triangular strip matrix is completely described by its first column (row) Because of this, it may be convenient in the future to use a compact notation of the form L = ω 0,ω 1,,ω T,

3 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 361 U = ω 0,ω 1,,ω, where T denotes matrix transposition However, in this paper we prefer to use full matrix notation for clarity Obviously, if matrices C and D are both lower (upper) triangular strip matrices, then they commute: CD = DC (3) Denoting Ω = L ω 0 E, Ψ = U ω 0 E, (4) where E is the unit matrix, we can write L = ω 0 E +Ω, U = ω 0 E +Ψ (5) We can also consider ( +1) ( + 1) matrices E p +, p =1,,withones on p-th diagonal above the main diagonal and zeroes elsewhere, and matrices Ep, p =1,,withonesonp-th diagonal below the main diagonal and zeroes elsewhere We will also denote E 0 ± E the unit matrix It can be shown that E ± p E± q = { E ± p+q, (p + q ), O, (p + q>), (6) from which follows that for integer k (E ± p )k = { E ± pk, (pk ), O, (pk > ), (E ± 1 )+1 = O (7) oting that Ω = ω k E k, Ψ = ω k E + k, (8) k=1 k=1 it can be shown that ( + 1)-th power of Ω and of Ψ gives the zero matrix: Ω +1 = O, Ψ +1 = O (9) Using (9) it is easy to check that the inverse matrices (L ) 1 and (U ) 1 are given by the following explicit expressions [5, p 62]: (L ) 1 = ω 1 0 E ω 2 0 Ω + ω 3 0 Ω2 + +( 1) ω 1 0 Ω, (10) (U ) 1 = ω0 1 E ω 2 0 Ψ + ω0 3 Ψ2 + +( 1) ω0 1 Ψ (11) There is a link between the matrix polynomials and the triangular strip matrices amely, if we introduce the polynomial ϱ (z), ϱ (z) =ω 0 + ω 1 z + ω 2 z ω z, (12)

4 362I Podlubny and take into account the relationship (7), then we can write: ϱ (E 1 )=ω 0E + ω 1 E 1 + ω 2(E 1 )2 + + ω (E 1 ) = L, (13) ϱ (E + 1 )=ω 0E + ω 1 E ω 2(E + 1 )2 + + ω (E + 1 ) = U, (14) where L and U defined by relationships (1) and (2) If we define the truncation operation, trunc ( ), which truncates (in a general case) the power series ϱ(z), ϱ(z) = ω k z k (15) to the polynomial ϱ (z), trunc (ϱ(z)) def = k=0 ω k z k = ϱ (z), (16) k=0 then we can consider the function ϱ(z) as a generating series for the set of lower (or upper) triangular matrices L (or U ), =1, 2, We will need the following properties of the truncation operation: trunc (γλ(z)) = γ trunc (λ(z)), (17) trunc (λ(z)+µ(z)) = trunc (λ(z)) + trunc (µ(z)), (18) ) trunc (λ(z)µ(z)) = trunc (trunc (λ(z)) trunc (µ(z)) (19) 3Operations with triangular strip matrices Due to the special structure of triangular strip matrices, the operations with them, such as addition, subtraction, multiplication, and inversion, can be expressed in the form of operations with their generating series (15) Let us consider two ( +1) ( + 1) lower triangular strip matrices: matrix A with elements a k, k =0, 1,, in its first column, and matrix B with elements b k, k = 0, 1,, in its first column Denoting λ(z) andµ(z) the generating series of A and B respectively, and using the representation (13), we can write: A = a k (E1 )k = λ (E1 ), B = b k (E1 )k = µ (E1 ), (20) k=0 k=0 where λ (z) = trunc (λ(z)), µ = trunc (µ(z)), and therefore A ± B = (a k ± b k )(E1 )k (21) k=0

5 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 363 In symbolic form, using the generating series λ(z) and µ(z) and the properties of truncation operation (17) and (18), this can be written as A ± B trunc (λ(z) ± µ(z)) = λ (z) ± µ (z) = = (a k ± b k )z k (a k ± b k )(E1 )k (22) k=0 This means that the coefficient on k-th diagonal of the sum of two lower triangular strip matrices is equal to the sum of k-th coefficients of the generating series of those matrices Therefore, summation of lower triangular strip matrices is equivalent to summation of their respective generating series with a subsequent truncation The multiplication by a constant γ is simple: γa trunc (γλ(z)) = γλ (z) = k=0 γa k z k k=0 γa k (E1 )k, (23) and it is equivalent to multiplication of the generating series by γ followed by truncation Taking into account the property (7) of the matrix E1, we obtain the product of A and B : ( )( ) A B = a k (E1 )k b k (E1 )k (24) k=0 = k=0 ( k k=0 k=0 k=0 ) a i b k i (E1 )k (25) i=0 Using the truncation operation, the product of the matrices A and B can also be expressed in terms of their generating series: ( ( k ) ) A B trunc (λ(z)µ(z)) = trunc a i b k i z k k=0 i=0 ( k ) ( k ) = a i b k i z k a i b k i (E1 )k (26) i=0 i=0 In other words, the product of two lower triangular matrices A and B is equivalent to the truncated product of their generating series The use of the generating series is especially convenient for inverting the lower triangular strip matrices If A is a lower triangular strip matrix with a generating function λ(z), then the generating function for the inverse matrix (A ) 1 is simply y(z) =λ 1 (z) Indeed, k=0

6 364 I Podlubny ) A (A ) 1 trunc (λ(z) λ 1 (z) =1 E (27) This means that the coefficients on the first column of the inverse matrix (A ) 1 are the coefficients of the polynomial ) y (z) = trunc (λ 1 (z), (28) which is the truncation of the generating series for the inverse matrix This method of inversion of triangular strip matrices is even simpler than the formulas (10) and (11) All the above rules involving generating functions can also be used for upper triangular strip matrices 4Integer-order differentiation Let us consider equidistant nodes with the step h: t k = kh, (k =0,,), in the interval [a, b], where t 0 = a and t = b 41 Backward differences For a function f(t), differentiable in [a, b], we can consider first-order approximation of its derivative f (t) atthepointst k, k =1,,, using first-order backward differences: f (t k ) 1 h f(t k)= 1 h (f k f k 1 ), k =1,, (29) All the formulas (29) can be written simultaneously in the matrix form: h 1 f f 0 h 1 f(t 1 ) f 1 h 1 f(t 2 ) = f 2 h (30) h 1 f(t 1 ) f 1 h 1 f(t ) f In the formula (30) the column vector of function values f k (k =0,,)is multiplied by the matrix B 1 = 1 h , (31)

7 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 365 and the result is the column vector of approximated values of f (t k ), k =1,,, with the exception of the first element, depending on the value of the function f(t) at the initial point, namely h 1 f 0 = h 1 f(a) We can look at the matrix B 1 as at a discrete analogue of first-order differentiation The generating function for the matrix B 1 is β 1 (z) =h 1 (1 z) (32) Similarly, we can consider the approximation of the second-order derivative using second-order backward differences: f (t k ) 1 h 2 2 f(t k )= 1 h 2 (f k 2f k 1 + f k 2 ), k =2,,, (33) which in the matrix form corresponds to the relationship h 2 f 0 h 2 ( 2f 0 + f 1 ) h 2 2 f(t 2 ) h 2 2 f(t 1 ) h 2 2 f(t ) = 1 h f 0 f 1 f 2 f 1 f (34) In the formula (34) the column vector of function values f k (k =0,,)is multiplied by the matrix B 2 = 1 h (35) and the result is the column vector of approximations of f (t k ), k =2, 3,,, with the exception of the first two elements, namely h 2 f 0 and h 2 ( 2f 0 +f 1 ) We can look at the matrix B 2 as at a discrete analogue of second-order differentiation The generating function for the matrix B 2 is β 2 (z) =h 2 (1 2z + z 2 )=h 2 (1 z) 2 (36) Further, we can consider a matrix B p,wherep is a positive integer:

8 366 I Podlubny B p = 1 h p ω ω 1 ω ω 2 ω 1 ω ω p ω p 1 ω ω p ω p 1 ω 0, (37) ( ) p ω j =( 1) j, j =0, 1, 2,,p (38) j The matrix B p is a discrete analogue of differentiation of p-th order, if backward differences of the p-th order are used The generating function for the matrix B p is β p (z) =h p (1 z) p (39) For the generating functions of the form β p (z) wehave: from which in view of (26) follows that β 2 (z) = β 1 (z)β 1 (z) β p (z) = β 1 (z) β 1 (z) }{{} p β p+q (z) = β p (z)β q (z) =β q (z)β p (z), B 2 = B 1 B, 1 (40) B p = B1 B 1 B 1, }{{} (41) p B p+q = B p Bq = Bq Bp, (42) where p and q are positive integers 42 Forward differences Similarly to the previous section, we obtain that the matrix F p,wherep is a

9 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 367 positive integer, ω 0 ω p 1 ω p ω 0 ω p 1 ω p F p = 1 h p 0, (43) ω 0 ω 1 ω ω 0 ω ω 0 ( ) p ω j =( 1) j, j j =0, 1, 2,,p (44) is a discrete analogue of differentiation of p-th order, namely of ( 1) p f (p) (t), if forward differences of the p-th order are used The generating function for F p is the same as for B p : β p(z) =h p (1 z) p Since the generating functions are the same as in case of the matrices B p,we have for F p the similar properties: F 2 = F 1 F 1, (45) F p = F 1 F 1 F 1, }{{} (46) p F p+q = F p F q = F q F p, (47) where p and q are positive integers It also should be noted that transposition of the matrix B p,representingthe backward difference operation, gives the matrix F p, which corresponds to forward differentiating: ( B p ) T = F p (F, p ) T = B p (48) 5 n-fold integration ow let us turn to the integration To deal with operations, which are inverse to the differentiation, we have to consider definite integrals with one limit fixed and another moving 51 Moving upper limit of integration Let us take a function f(t), integrable in [a, b], and consider integrals with fixed lower limit and moving upper limit: g 1 (t) = t a f(t)dt, (49)

10 368 I Podlubny for which we have g 1 (t) =f(t) in(a, b) Let us consider equidistant nodes with the step h: t k = kh, (k =0,,), in the interval [a, b], where t 0 = a and t = b We can use the left rectangular quadrature rule for approximating the integral (49) at the points t k, k =1,,: k 1 g 1 (t k ) h f i, k =1,, (50) i=0 All the formulas (50) can be written simultaneously in the matrix form: g 1 (t 1 ) f 0 g 1 (t 2 ) f 1 g 1 (t 3 ) f 2 = h (51) g 1 (t ) f 1 g 1 (t + h) f We see that the column vector of function values f k (k =0,,)ismultiplied by the matrix I 1 = h , (52) and the result is the column vector of approximated values of the integral (49), namely g 1 (t k ), k =1,,, with the exception of the last element, which corresponds to the node lying outside of the considered interval [a, b] We can look at the matrix I 1 as at a discrete analogue of left rectangular quadrature rule for evaluating the integral (49) The generating function for I 1 is ϕ 1 (z) =h(1 z) 1 (53) It must be noted here that the matrix I 1 is inverse to the matrix B1,which corresponds to backward difference approximation of the first derivative We have: B 1 I1 = I1 B1 trunc (β 1 (z) ϕ 1 (z)) = 1 E (54) Therefore, having one of these matrices, we can immediately obtain another by a matrix inversion

11 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 369 Similarly, we can consider the two-fold integral with a moving upper boundary: g 2 (t) = t a t dt a f(t)dt, (55) for which we have g 2 (t) =g 1 (t) =f(t) in(a, b) Using the left rectangular quadrature rule twice for approximating g 2 (t k )and taking into account that g 1 (t 0 )=0,wehave: k 1 g 2 (t k ) = h i=0 k 1 = h 2 k 1 g 1 (t i )=h i 1 i=1 j=0 i=1 k 1 g 1 (t i )=h i=1 k 2 f j = h 2 (k j 1) f j j=0 i 1 h f j j=0 = h 2( (k 1)f 0 +(k 2)f f k 3 + f k 2 ), (56) k =2, 3,, The equations (56) can be written simultaneously in the matrix form: g 2 (t 2 ) g 2 (t 3 ) g 2 (t ) g 2 (t + h) g 2 (t +2h) = h f 0 f 1 f 2 f 1 f (57) We see that the column vector of function values f k (k =0,,)ismultiplied by the matrix I 2 = h , (58) and the result is the column vector of approximated values of the integral (55), namely g 2 (t k ), k =2,,, with the exception of the last two elements, which correspond to the nodes lying outside of the considered interval [a, b] We can

12 370 I Podlubny look at the matrix I 2 as at a discrete analogue of left rectangular quadrature rule for evaluating the two-fold integral (55) The generating function for I 2 is ϕ 2 (z) =h 2 (1 z) 2 (59) It must be mentioned here that the matrix I 2 is inverse to the matrix B2, which corresponds to backward difference approximation of the second derivative We have: B 2 I2 = I2 B2 trunc (β 2 (z) ϕ 2 (z)) = 1 E (60) Therefore, having one of these matrices, we can immediately obtain another by a matrix inversion If we consider p-fold integration with a moving upper limit, g p (t) = t a dτ p τ p a τ 2 dτ p 1 f(τ 1 )dτ 1, (61) and apply the left rectangular quadrature rule p times, then we arrive at the following relationship in the matrix form: g p (t p ) γ f 0 g p (t p+1 ) γ 1 γ f 1 g p (t ) = h p γ 2 γ 1 γ 0 0 f p, g p (t + h) γ 1 γ 2 γ 1 γ 0 0 f 1 g p (t + ph) γ γ 1 γ 2 γ 1 γ 0 f (62) involving the lower triangular strip matrix I p with the generating function ϕ p (z) =h p (1 z) p, a γ γ 1 γ I p = hp γ 2 γ 1 γ 0 0, (63) γ 1 γ 2 γ 1 γ 0 0 γ γ 1 γ 2 γ 1 γ 0 which is inverse to the matrix B p, corresponding to the backward difference approximation of the p-th derivative: B p Ip = Ip Bp trunc (β p (z) ϕ p (z)) = 1 E (64)

13 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 371 In view of (26) it follows from the properties of the generating functions ϕ p (z) =h p (1 z) p that I 2 = I 1 I, 1 (65) I p = I1 I 1 I 1, }{{} (66) p I p+q = I p Iq = Iq Ip, (67) where p and q are positive integers Moreover, the matrices I p commute also with the matrices B p 52 Moving lower limit of integration If we consider p-fold integration with a moving lower limit, y p (t) = b t dτ p b τ p dτ p 1 b τ 2 f(τ 1 )dτ 1, (68) then its discrete analogue is represented by the upper triangular strip matrix J p with the generating function ϕ p (z) =h p (1 z) p : γ 0 γ 1 γ 2 γ 1 γ 0 γ 0 γ 1 γ 2 γ 1 J p = hp 0 0 γ 0 γ 1 γ 2 (69) γ 0 γ γ 0 The matrix J p is inverse to the matrix F p, corresponding to the backward difference approximation of the p-th derivative: F p J p = J p F p trunc (β p (z) ϕ p (z)) = 1 E (70) In view of (26) it follows from the properties of the generating functions ϕ p (z) =h p (1 z) p that J 2 = J 1 J 1, (71) J p = J 1 J 1 J1, }{{} (72) p J p+q = J p J q = J q J p, (73) where p and q are positive integers Moreover, the matrices J p with the matrices F p commute also

14 372I Podlubny It also should be noted that transposition of the matrix I p,representingthe integration with moving upper limit, gives the matrix J p, which corresponds to integration with moving lower limit: ( ) I p T ) = J p (J, p T = I p (74) 6Fractional differentiation The triangular strip matrices can also be used for fractional derivatives In this case, we arrive at lower (upper) triangular matrices, which have no zeros below (above) the main diagonal 61 Left-sided fractional derivatives Let us consider a function f(t), defined in [a, b], such that f(t) 0fort<a (Functions satisfying this condition are often called causal functions) We assume that the function f(t) is good enough for considering its left-sided fractional derivative of real order α (n 1 α<n), ad α t f(t) = ( ) 1 d n t Γ(n α) dt a f(τ)dτ, (a <t<b) (75) (t τ) α n+1 Let us take equidistant nodes with the step h: t k = kh (k =0, 1,,), in the interval [a, b], where t 0 = a and t = b Using the backward fractional difference approximation for the α-th derivative at the points t k, k =0, 1,,, we have: ad α t k f(t) α f(t k ) h α ( ) k α = h α ( 1) j f k j, k =0, 1,, (76) j j=0 All + 1 formulas (76) can be written simultaneously in the matrix form: h α α f(t 0 ) ω (α) f 0 h α α f(t 1 ) ω (α) 1 ω (α) f 1 h α α f(t 2 ) = 1 ω (α) 2 ω (α) 1 ω (α) f 2 h α, h α α f(t 1 ) ω (α) (α) 1 ω 2 ω (α) 1 ω (α) 0 0 f 1 h α α f(t ) ω (α) ω (α) (α) 1 ω 2 ω (α) 1 ω (α) f 0 ( ) (77) ω (α) α j =( 1) j, j =0, 1,, (78) j

15 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 373 In the formula (77) the column vector of function values f k (k =0,,)is multiplied by the matrix B α = 1 h α ω (α) ω (α) 1 ω (α) ω (α) 2 ω (α) 1 ω (α) ω (α) 1 ω (α) ω (α) ω (α) 1 2 ω (α) 1 ω (α) 0 0 (α) ω 2 ω (α) 1 ω (α) 0 and the result is the column vector of approximated values of the fractional derivative a Dt α k f(t), k =0, 1,, We can look at the matrix B α as at a discrete analogue of left-sided fractional differentiation of order α The generating function for the matrix B α is (79) β α (z) =h α (1 z) α (80) Since for lower triangular matrices B α and Bβ we always have B α B β = Bβ Bα = B α+β, we can consider such matrices as discrete analogues of the corresponding left-sided fractional derivatives a Dt α and a D β t,wheren 1 α<nand m 1 α<m, only if ( ) ( ) adt α ad β t f(t) = a D β t adt α f(t) = a D α+β t f(t), which holds if f (k) (a) =0, k =1, 2,,r 1, (81) where r =max{n, m} This means that if left-sided fractional derivatives of a function f(t) oforders less than some integer r are considered, than they can all be replaced with their corresponding discrete analogues, if the function f(t) satisfies the conditions (81) 62 Right-sided fractional derivatives Let us consider a function f(t), defined in [a, b], such that f(t) 0fort>b We assume that the function f(t) is good enough for considering its right-sided fractional derivative of real order α (n 1 α<n), td α b f(t) = ( ) ( 1)n d n b f(τ)dτ, (a <t<b) (82) Γ(n α) dt (τ t) α n+1 t

16 374 I Podlubny Similarly to the previous section, we can obtain the discrete analogue of the right-sided fractional differentiation on the net of equidistant nodes with the step h: t k = kh (k =0, 1,,), in the interval [a, b], where t 0 = a and t = b, which is represented by the matrix F α = 1 h α ω (α) 0 ω (α) (α) 1 ω 1 0 ω (α) 0 ω (α) (α) 1 ω 0 0 ω (α) 0 ω (α) ω (α) 0 ω (α) ω (α) 0 ω (α) 1 (83) The generating function for the matrix F α is the same as for the matrix Bα : β α (z) =h α (1 z) α It should also be mentioned that transposition of the matrix B α, corresponding to left-sided fractional differentiation, gives the matrix F p, which corresponds to right-sided differentiation: ( ) B α T ( ) = F α, F α T = B α (84) Similarly to the previous section, if right-sided fractional derivatives of a function f(t) of orders less than some integer r are considered, then they can all be replaced with their corresponding discrete analogues, if the function f(t) satisfies the conditions f (k) (b) =0, k =1, 2,,r 1 (85) 63 Sequential fractional derivatives For left-sided sequential fractional derivatives, in which all derivatives in the sequence can be arbitrarily interchanged, adt α f(t) = ad α 1 t ad α 2 t a D αn t f(t), (86) α =(α 1,α 2,, α n ), (and the same equidistant nodes as above) the discrete analogue B α has the form of the product of matrices B α k, corresponding to operators ad α k t, k =1, 2,,n: B α = B α n 1 Bα 2 Bαn = B α k (87) k=1 Similarly, for right-sided fractional derivatives, in which all derivatives in the sequence can be arbitrarily interchanged, tdb α f(t) = t D α 1 b td α 2 b t D αn b f(t), (88)

17 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 375 the discrete analogue F α is F α = F α 1 F α n 2 Fαn = F α k (89) k=1 7Fractional integration Discrete analogues of left- and right-sided fractional integrals can be obtained by inversion of the discrete analogues of the corresponding fractional derivatives 71 Left-sided fractional integration To obtain the matrix I α, corresponding to the discrete analogue of the leftsided fractional integration (α >0), adt α f(t) = 1 t (t τ) α 1 f(τ)dτ, (a <t<b), (90) Γ(α) a we simply invert the matrix B α, corresponding to the left-sided fractional differentiation: ) 1 I (B α = α (91) If ϕ α (z) denotes the generation function for I α and β α(z) is the generating function for B α, then taking into account the rule (28), we can write: ( ) I α ϕ (z) = trunc βα 1 (z) ( = trunc h α (1 z) α) Therefore, the matrix I α has the following form: ω ( α) ω ( α) 1 ω ( α) ω ( α) I α = hα 2 ω ( α) 1 ω ( α) ω ( α) ( α) 1 ω 2 ω ( α) 1 ω ( α) 0 0 ( α) ω 2 ω ( α) 1 ω ( α) 0 ω ( α) ω ( α) 1 (92) 72 Right-sided fractional integration Similarly, inversion of the matrix F α, corresponding to the right-sided fractional differentiation, gives the matrix J α,

18 376 I Podlubny J α = h α ω ( α) ω ( α) 0 ω ( α) 1 0 ω ( α) 0 ω ( α) ω ( α) 0 ω ( α) 1 1 ω ( α) 1 0 ω ( α) 1 ω ( α) ω ( α) ω ( α) 0 which is the discrete analogue of the right-sided fractional integration (α >0): (93) tdb α f(t) = 1 b (τ t) α 1 f(τ)dτ, (a <t<b) (94) Γ(α) t 73 Feller and Riesz potentials on a finite interval Let us consider the Riesz potential R α f(t) and the modified Riesz potential M α f(t) on a finite interval [13, Chap 3]: R α 1 b ϕ(τ)dτ f(t) =, (a <t<b), (95) 2Γ(α)cos(απ/2) t τ 1 α a M α f(t) = b 1 sign(t τ)ϕ(τ)dτ 2Γ(α)sin(απ/2) t τ 1 α, (a <t<b) (96) a Obviously, both these operators are linear combinations of the left-sided and right-sided fractional integrals: R α 1 ( ) f(t) = adt α f(t)+ t Db α f(t), (97) 2cos (απ/2) M α 1 ( ) f(t) = adt α f(t) t Db α f(t) (98) 2sin (απ/2) The matrices R α and Mα of their discrete analogues are linear combinations of the matrix I α, corresponding to the left-sided fractional integral, and the matrix J α, corresponding to the right-sided fractional integral: R α = 1 2cos (απ/2) ( Iα + J α ), (99) M α 1 = 2sin (απ/2) ( Iα J) α (100) The Feller potential operator Φ α f(t) is also a linear combination of left- and right-sided fractional integrals, but with general constant coefficients u, v [13, Ch3]:

19 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 377 Φ α f(t) =u a Dt α and the matrix of its discrete analogue is + v t D α b f(t), (101) Φ α = ui α + vj α (102) umerical inversion of the Riesz and Feller potential operators (95), (96), and (101), reduces to inversion of the corresponding square matrices R α, Mα,and Φ α Example 1 Let us consider the fractional integral equation with the Riesz kernel: 1 Γ(1 α) 1 1 y(τ) dτ =1, ( 1 <t<1), (103) t τ α which has the solution [12, eq 6116] ( ) απ y(t) =π 1 Γ(1 α)cos (1 t 2 ) (α 1)/2 (104) 2 Writing the equation (103) in the form 1D (1 α) t y(t)+ t D (1 α) 1 y(t) =1, and replacing the fractional derivatives with their discrete analogues, we obtain the system of linear algebraic equations ( B (1 α) + F (1 α) ) Y = F, (105) ( T ( T where F = 1, 1,,1), for determination of Y = y(t 0 ),y(t 1 ),,y(t )), which represents the approximate solution of the equations (103) The numerical solution of the equation (103) for α =08 isshowninfig1 In this figure, and also in all subsequent figures, only a subset of the points of the obtained numerical solution is shown; otherwise it would be difficult to depict the analytical solution and the numerical one, which are very close each other 8umerical solution of fractional differential equations The use of triangular strip matrices significantly simplifies numerical solution of fractional differential equations Instead of writing unwieldy recurrence relationships for determination of the values of the unknown function in equidistant discretization nodes, one can immediately write a system of algebraic equations for those values For convenience, let us introduce a certain type of matrices, which are obtained from the unit matrix E by keeping only some of its rows and omitting all other rows: S 1 is obtained by omitting only the first row of E; S 2 is obtained by

20 378 I Podlubny analytical solution numerical solution (h=0005) Figure 1: Solution of equation 1 1 Γ(1 α) 1 t τ α y(τ) dτ =1 omitting only the second row; S 1,2 is obtained by omitting only the first and the second row of E; and, in general, S r1,r 2,,r k is obtained by omitting the rows with the numbers r 1,r 2,,r k If A is a square matrix, then the product S r1,r 2,,r k A contains only rows of A with the numbers different from r 1,r 2,,r k Similarly, the product ASr T 1,r 2,,r k contains only columns of A with the numbers different from r 1,r 2,,r k Because of this property, the matrix S r1,r 2,,r k is called an eliminator In case of infinite matrices, similar matrices appeared in [4] The following simple example illustrates the main property of eliminators: A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 AS T 1 = ; S 1 = a 12 a 13 a 22 a 23 a 32 a 33 [ ; S 1 AS T 1 = ] [ ] a21 a ; S 1 A = 22 a 23 ; a 31 a 32 a 33 [ a22 a 23 a 32 a 33 Considering ( +1) ( + 1) lower triangular matrices L (1) and upper triangular matrices U (2), and numbering rows and columns from 0 to, we have the following useful relationships: ]

21 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 379 { } { } L S 0 S0 T L 1 =, (106) U U { } { 1 } L S S T U = L 1, (107) U { } { 1 } L S 0,1,,k S0,1,,k T L k 1 =, (108) U U k 1 { } { } L S k, k+1,, S k, k+1,, T L k 1 = (109) U U k 1 In other words, simultaneous multiplication of a triangular strip matrix by the eliminator S 0,1,,k (or by S k, k+1,, )ontheleftands0,1,,k T (respectively, by S k, k+1,, T ) on the right preserves the type and the structure of the triangular strip matrix, and only reduces its size by k +1 rows and k +1 columns 81 Initial value problems for FDEs The general procedure of numerical solution of fractional differential equations consists of two steps First, initial conditions are used to reduce a given initial-value problem to a problem with zero initial conditions At this stage, instead of a given equation a modified equation, incorporating the initial values, is obtained Then the system of algebraic equations is obtained by replacing all derivatives (of fractional and integer orders) in the obtained modified equation by the corresponding matrices (B α for left-sided derivatives, F α for right-sided derivatives) for their discrete analogues We will consider an m-term linear fractional differential equation with nonconstant coefficients of the following form: m p k (t)d α k y(t) =f(t), (110) k=1 0 α 1 <α 2 <<α m, n 1 <α m <n, where D α k denotes either Riemann-Liouville or Caputo left-sided fractional derivative of order α k Let us denote P (k) ) (p =diag k (t 0 ),p k (t 1 ),,p k (t ) = p k (t 0 ) p k (t 1 ) 0 0, p k (t ) (111)

22 380 I Podlubny Y = ( T ( T y(t 0 ),y(t 1 ),,y(t )), F = f(t 0 ),f(t 1 ),,f(t )) (112) Using these notations and taking into account that the discrete analogue of the left-sided fractional derivative D α k is B α k, we can write a discrete analogue of the fractional differential equation (110): m P (k) Bα k Y = F (113) k=1 82 Zero initial conditions If n 1 <α m <n, then the Riemann-Liouville and the Caputo formulations of the equation (110) are equivalent under the assumption of zero initial values of the function y(t) and its(n 1) derivatives [12]: y (k) (t 0 )=0, k =0, 1,,n 1 (114) Approximating the derivatives in the initial conditions (114) by backward differences, we immediately obtain: y(t 0 )=y(t 1 )= = y(t n 1 )=0 (115) The linear algebraic system for determination of y n,,y is obtained from the system (113) by omitting its first n rows and substituting the zero starting values (115) into the remaining equations This can be symbolically written with the help of eliminator: { { m S 0,1,,n 1 k=1 } } P (k) Bα k S0,1,,n 1 T {S 0,1,,n 1 Y } = S 0,1,,n 1 F (116) The solution of the linear algebraic system (116) along with the starting values (115) gives the numerical solution of the fractional differential equation (110) under zero initial conditions (114) If the coefficients p k (t) are constant, ie p k (t) p k, then the system (116) takes on the simplest form: m p k B α k n {S 0,1,,n 1Y } = S 0,1,,n 1 F (117) k=1 Example 2 Let us consider the following two-term fractional differential equation under zero initial conditions: y (α) (t)+y(t) =1, (118) y(0) = 0, y (0) = 0, (119)

23 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS analytical solution numerical solution (h=001) Figure 2: Solution of the problem y (18) (t)+y(t) =1,y(0) = 0,y (0) = 0 which has the analytical solution y(t) =t α E α,α+1 ( t α ) (120) The numerical solution of the problem (118) (119) can be found from the system (117), where we have m =2,α 1 = α, α 2 =0,n =2,p 1 = p 2 =1, B α 1 n = Bα 2, Bα 2 n = E 2, F =(1, 1,,1) T For these values, the system }{{} of algebraic equations for determining y k, k =2, 3,, takes on the form: { B α 2 + E 2 } {S0,1 Y } = S 0,1 F (121) It should be also added that from the initial conditions we have y 0 = y 1 =0 The numerical solution of the problem (118) (119) for α = 18 isshownin Fig2 83 Initial conditions in terms of integer-order derivatives If the fractional derivatives in the equation (110), where n 1 <α m <n,are Caputo derivatives, then the initial conditions are expressed in terms of classical integer-order derivatives and can be non-zero: y (k) (t 0 )=c k, k =0, 1,,n 1 (122)

24 382I Podlubny The solution of the initial-value problem (110) (122) can be written in the form y(t) =y (t)+z(t), (123) where y (t) is some known function, satisfying the conditions y (k) (t 0 )=c k, k = 0, 1,,n 1, and z(t) is a new unknown function Substituting (123) into the equation (110) and the initial conditions (122), we obtain for the function z(t) an initial-value problem with zero initial conditions, which can be solved as described in Section 82 Example 3 Let us consider the following two-term fractional differential equation under non-zero initial conditions: y (α) (t)+y(t) =1, (124) y(0) = c 0, y (0) = c 1 (125) The analytical solution, obtained with the help of the Laplace transform of the Caputo fractional derivatives [12], is given by the expression y(t) =c 0 E α,1 ( t α )+c 1 te α,2 ( t α )+t α E α,α+1 ( t α ) (126) To obtain numerical solution, we have first to transform the problem (124) (125) to the problem with zero initial conditions For this, let us introduce an auxiliary function z(t), such that y(t) =c 0 + c 1 t + z(t) Substituting this expression into the equation (124) and in the initial conditions (125), we obtain the problem for finding z(t): z (α) (t)+z(t) =1 c 0 c 1 t, (127) z(0) = 0, z (0) = 0 (128) The numerical solution of this problem can be found as described in Section 82, and the numerical solution y(t) of the problem (124) (125) is obtained using the relationship y(t) =c 0 + c 1 t + z(t) The numerical solution of the problem (124) (125) for α =18, c 0 =1,c 1 = 1 is shown in Fig3 84 Initial conditions in terms of R-L fractional derivatives Initial value problems for fractional differential equations with non-zero initial conditions in terms of Riemann-Liouville (R-L) derivatives, namely ad α k 1 t y(t) = c k, k =0, 1,,n 1, (129) t a can be also transformed to initial-value problems with zero initial condition Such a transformation allows us to circumvent the difficulty consisting in the fact that there is still no known approximation for such initial conditions

25 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS analytical solution numerical solution (h=001) Figure 3: Solution of the problem y (18) (t)+y(t) =1,y(0) = 1,y (0) = 1 Example 4 Let us consider the following two-term fractional differential equation under non-zero initial conditions: y (α) (t)+y(t) =1, (130) y (α 1) (0) = c 0, y (α 2) (0) = c 1 (131) The analytical solution, obtained with the help of the Laplace transform of the Riemann Liouville fractional derivative [12], is given by the expression y(t) =c 0 t α 1 E α,α ( t α )+c 1 t α 2 E α,α 1 ( t α )+t α E α,α+1 ( t α ) (132) To obtain numerical solution, we have first to transform the problem (130) (131) to the problem with zero initial conditions For this, let us introduce an auxiliary function z(t), such that y(t) =c 0 t α 1 + c 1 t α 2 + z(t) Substituting this expression into the equation (130) and into the initial conditions (131), we obtain the problem for finding z(t): z (α) (t)+z(t) =1 c 0 t α 1 c 1 t α 2, (133) z(0) = 0, z (0) = 0 (134)

26 384 I Podlubny analytical solution numerical solution (h=001) Figure 4: Solution of the problem y (18) (t)+y(t) =1;y (08) (0) = 1; y ( 02) (0) = 1 The numerical solution of this problem can be found as described in Section 82, and the numerical solution y(t) of the problem (130) (131) is obtained using the relationship y(t) =c 0 t α 1 + c 1 t α 2 + z(t) The numerical solution of the problem (130) (131) for α =18, c 0 =1,c 1 = 1 is shown in Fig4 85 onlinear FDEs The triangular strip matrices can be useful also for solving fractional differential equations of a general form Let us write, for example, an equation with left-sided fractional derivatives y (αi) (t) = a D α i t y(t) : y (α1) (t) =f(t, y (α2) (t), y (α3) (t),,y (αk) (t)), (135) (0 <α 1 <α 2 <<α k n) assuming that the initial conditions are already transformed to zero initial conditions Replacing all fractional derivatives in the equation (135) with their discrete analogues and utilizing zero initial conditions, we obtain a nonlinear algebraic system B α 1 Y = f(et,b α 2 Y,B α 3 Y,,B α k Y ), (136) y j =0, j =1, 2,,n 1,

27 MATRIX APPROACH TO DISCRETE FRACTIOAL CALCULUS 385 where Y =(y 0,y 1,,y ) T, t =(t 0,t 1,,t ) T, y j = y(t j ), t j = jh, (j = 0, 1,,), and E is ( +1) ( + 1) unit matrix 9Conclusion The suggested approach, using the triangular strip matrices, provides: a uniform approach to discretization of derivatives of arbitrary real order, including the classical integer-order derivatives, and various types of fractional derivatives, including the left- and right-sided derivatives, and the sequential fractional derivatives; a uniform approach to numerical solution of differential equations of integer order and of fractional order; a convenient language for discretization of differential equations of arbitrary real order; a method for numerical solution of initial value problems and boundary value problems for ordinary differential equations of arbitrary real order; a possible method for numerical solution of non-linear differential equations of arbitrary real order The triangular matrix approach can also be used for obtaining new quadrature formulas for fractional integrals For this, any approximation of fractional derivatives should be written in the form of a triangular strip matrix, inversion of which gives the corresponding quadrature formula for fractional integrals Similarly, new approximations of fractional derivatives can be obtained by inverting the triangular strip matrices, corresponding to quadrature formulas for fractional integrals Acknowledgment The idea of the approach, described in this paper, appeared due to inspiring discussions with Prof Rudolf Gorenflo, Dr Yurii Luchko, and Prof Ralf Kornhuber, during the short visit of the author to the Free University of Berlin in December 1999, sponsored by the Free University References [1] B V Bulgakov, Kolebaniya (Vibrations) Gostekhizdat, Moscow (1954) (In Russian) [2] R F C a m e r o n and S M c K e e, High accuracy convergent product integration methods for the generalized Abel equation Journal of Integral Equations 7 (1984),

28 386 I Podlubny [3] R F C a m e r o n and S M c K e e, The analysis of product integration methods for Abel s equation using discrete fractional differentiation IMA Journal of umerical Analysis 5 (1985), [4] R G C o o k e, Infinite Matrices and Sequence Spaces MacMilan and Co, London (1950) (Russian transl: Beskonechnye matrisy i prostranstva posledovatel nostei, Fizmatgiz, Moscow (1960)) [5] BP Demidovich, Lektsii po matematicheskoy teorii ustoychivosti (Lectures on the Mathematical Theory of Stability) auka, Moscow (1967) (In Russian) [6] L J D e r r, Triangular matrices with the isoclinal property Pacific Journal of Mathematics 37, o 1 (1971), [7] P P B E g g e r m o n t, A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations Journal of Integral Equations 3 (1981), [8] FR Gantmakher, Teoriya matrits (Theory of Matrices) auka, Moscow (1988) (In Russian) [9] P A W H o l y h e a d and S M c K e e, Stability and convergence of multistep methods for linear Volterra integral equations of the first kind SIAM J umer Anal 13, o 2 (April 1976), [10] P A W H o l y h e a d, S M c K e e, and P J T a y l o r, Multistep methods for solving linear Volterra integral equations of the first kind SIAM J umer Anal 12, o 5 (October 1975), [11] S M c K e e, Discretization methods and block isoclinal matrices IMA Journal of umerical Analysis 3 (1983), [12] I Podlubny, Fractional Differential Equations Academic Press, San Diego (1999) [13] S G S a m k o, A A K i l b a s, and O I M a r i c h e v, Integraly i proizvodnye drobnogo poryadka i nekotorye ich prilozheniya (Integrals and Derivatives of the Fractional Order and Some of Their Applications) auka i Tekhnika, Minsk (1987) (In Russian; English transl: Fractional Integrals and Derivatives, Gordon and Breach, Amsterdam (1993)) [14] DA Suprunenko and RI Tyshkevich, Perestanovochnye matritsy auka i Tekhnika, Minsk, 1966 (English transl: Commutative Matrices, Academic Press, ew York (1968)) Dept of Informatics and Control Engineering Faculty of BERG Received: June 15, 2000 Technical University of Kosice Kosice, SLOVAK Republic podlbn@tukesk