Applied Mathematics and Computation 187 (2007) 250–265
www.elsevier.com/locate/amc
Kronecker operational matrices for fractional calculusand some applications
Adem Kilicman *, Zeyad Abdel Aziz Al Zhour
Department of Mathematics, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Institute of Advanced Technology, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract
The problems of systems identification, analysis and optimal control have been recently studied using orthogonal func-tions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haarand many other functions. In the present paper, several operational matrices for integration and differentiation are studied.we introduce the Kronecker convolution product and expanded to the Riemann–Liouville fractional integral of matrices.For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient becausesometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find theexact and approximate solutions of the general system matrix convolution differential equations, the way exists whichtransform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, sev-eral systems are solved by the new and other approaches and illustrative examples are also considered.� 2006 Elsevier Inc. All rights reserved.
Keywords: Kronecker product; Convolution product; Kronecker convolution product; Vector operator; Operational matrix; Laplacetransform
1. Introduction
One of the principle reasons is that matrices arise naturally in communication systems, economic planning,statistics, control theory and other fields of pure and applied mathematics [2,4,7]. Another is because an m · n
matrix can be valued as representing a linear map from an n-dimensional vector space to an m-dimensionalvector space, and conversely, all such maps can be represented as m · n matrices. The operational matricesfor integration and differentiation are studied by many authors [1,3,6,10]. For example, Mouroutsos and
0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2006.08.122
* Corresponding author. Address: Institute of Advanced Technology, University Putra Malaysia (UPM), 43400 Serdang, Selangor,Malaysia.
E-mail addresses: [email protected] (A. Kilicman), [email protected] (Z.A.A. Al Zhour).
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 251
Sparis [6] solved the control problem using Taylor series operational matrices for integration, Chen et al. [3]solved the distributed systems by using Walsh operational matrices and Wang [10] introduced the inversions ofrational and irrational transfer functions by using the generalized block pulse operational matrices for differ-entiation and integration. The Kronecker convolution product is an interesting area of current research and infact plays an important role in applications, for example, Sumita [9], established the matrix Laguerre trans-form to calculate matrix convolutions and evaluated a matrix renewal function.
In the present paper, several operational matrices for integration and differentiation are studied. we intro-duce the Kronecker convolution product and expanded to the Riemann–Liouville fractional integral of matri-ces. For some applications, it is often not necessary to compute exact solutions, approximate solutions aresufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Ourmethod is extended to find the exact and numerical solutions of the general system matrix convolution differ-ential equations, the way exists which transform the coupled matrix differential equations into forms for whichsolutions may be readily computed. Finally, several systems are solved by the new and other approaches andillustrative examples are also considered.
As usual, the notations A{�1}(t) and detA(t) are the inverse and determinant of matrix function A(t),respectively, with respect to convolution. The notations AT(t) and Vec A(t) are transpose and vector-operatorof matrix function A(t), respectively. The term ‘‘Vec A(t)’’ will be used to transform a matrix A(t) into a vectorby stacking its column one underneath.The notations d(t) and Qn(t) = Ind(t) are the Dirac delta function andDirac identity matrix, respectively, where In is the identity scalar matrix of order n · n. Finally, the notationsA(t) ~ B(t) and A(t) * B(t) are the convolution and Kronecker convolution products, respectively. Finally,A � B stands for the Kronecker product ; A � B = [aijB]ij.
2. Operational matrices for integration and differentiation
2.1. Taylor series operational matrix for integration
A function f(t) that is analytic function at the neighborhood of the point t0 can be expanded in the followingformula:
f ðtÞ ¼X1n¼1
anunðtÞ; ð1Þ
where an ¼ 1n!
dnf ð0Þdtn
� �and un(t) = tn.
To obtain an approximate expression of the analytic function f(t) we may truncate the series (1) up to the(r + 1)th term:
f ðtÞ ’Xr
n¼1
anunðtÞ: ð2Þ
By defining the coefficient vector cT as:
cT ¼ ða0; . . . ; arÞ ð3Þ
and the power series basis vector uT(t) as:uTðtÞ ¼ ðu0ðtÞ; . . . ;urðtÞÞ: ð4Þ
Eq. (2) can written in the following compact form:f ðtÞ ’ cTuðtÞ: ð5Þ
The basis functions un(t) satisfy the following relation:unðtÞ ¼ tun�1ðtÞ: ð6Þ
Now, one can easily show that Z t0
unðxÞdx ¼ tnþ1
nþ 1� t
nþ 1unðtÞ ¼
1
nþ 1unþ1ðtÞ: ð7Þ
252 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
On the basis of Eq. (2), the definite integral of analytic function f(t) in the interval (0, t) may be approxi-mated as:
Z t0
f ðxÞdx ’Xr
n¼1
an1
nþ 1unþ1ðtÞ ¼
Xr
n¼1
bnunþ1ðtÞ; ð8Þ
where b0 ¼ 0; b1 ¼ a0; b2 ¼ a1
2; . . . ; br ¼ ar�1
r .In this equation the term br+1ur+1 has been truncated. Eq. (8) can also written as:
Z t0
f ðxÞdx ’ cTPuðtÞ; ð9Þ
where P is called the Taylor series operational matrix for integration:
P ¼
0 1 0 0 0 � � � 0 0 0
0 0 12
0 0 � � � 0 0 0
0 0 0 13
0 � � � 0 0 0
..
. ... ..
. ... ..
. ... ..
. ... ..
.
0 0 0 0 0 � � � 0 1r�1
0
0 0 0 0 0 � � � 0 0 1r
0 0 0 0 0 � � � 0 0 0
26666666666666664
37777777777777775ðrþ1Þ�ðrþ1Þ
: ð10Þ
The above expression indicates that the Taylor series operational matrix for integration is uniquely simple.If we integrate the function f(t) twice in the interval (0, t), we have
Z t0
Z t
0
f ðxÞðdxÞ2 ’Xr
n¼1
an1
nþ 1
� �1
nþ 2
� �unþ2ðtÞ ¼
Xr
n¼1
cnunþ1ðtÞ; ð11Þ
where c0 ¼ c1 ¼ 0; c2 ¼ a0; c3 ¼ a1
2; . . . ; cr ¼ ar�1
rðr�1Þ.Therefore, Eq. (11) can also written as:
Z t0
Z t
0
f ðxÞðdxÞ2 ’ cTQuðtÞ ð12Þ
and Q is called the operational matrix for double integration:
0 0 12
0 0 0 � � � 0 0 0
0 0 0 16
0 0 � � � 0 0 0
0 0 0 0 112
0 � � � 0 0 0
..
. ... ..
. ... ..
. ... ..
. ... ..
. ...
0 0 0 0 0 0 � � � 0 1ðr�1Þðr�2Þ 0
0 0 0 0 0 0 � � � 0 0 1rðr�1Þ
0 0 0 0 0 0 � � � 0 0 0
0 0 0 0 0 0 � � � 0 0 0
26666666666666666664
37777777777777777775
: ð13Þ
It easily to show that Q = P2. Therefore, Eq. (12) may now written as:
Z t0
Z t
0
f ðxÞðdxÞ2 ’ cTP 2uðtÞ: ð14Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 253
Now we can extend Eq. (14) to the case of k multiple integrations as:
Z t
0
� � �Z t
0
f ðxÞðdxÞk ’ aTP kuðtÞ: ð15Þ
2.2. Generalized block pulse operational matrix for integration
The block pulse functions form a complete set of orthogonal functions [10] which defined on the interval[0,b) by
wiðtÞ ¼1 i�1
m b 6 t < im b;
0 elsewhere . . .
�ð16Þ
for i = 1,2, . . . ,m. It is also known that for any absolutely integrable function f(t) on [0,b) can be expanded inblock pulse functions:
f ðtÞ ’ nT/mðtÞ; ð17Þ
where the mean-square error of approximation is minimized, nT = [f1, f2, . . . , fm] and /m(t) = [w1(t),w2(t), . . . ,wm(t)]. Here
fi ¼mb
Z b
0
f ðtÞwiðtÞdt ¼ mb
Z imð Þb
i�1mð Þb
f ðtÞwiðtÞdt: ð18Þ
There are many different starting points for the discussion of classical fractional calculus [8]. One beginswith a generalization of repeated integration. If f(t) is absolutely integrable on [0, b), it can be found that in[8,10]
Z t0
dtn
Z tn
0
dtn�1 � � �Z t3
0
dt2
Z t2
0
f ðt1Þdt1 ¼1
ðnþ 1Þ!
Z t
0
ðt � t1Þn�1f ðt1Þdt1 ¼1
ðnþ 1Þ! tn�1 � f ðtÞ; ð19Þ
where n = 1,2, . . . , and 0 6 t < b. On writing C(n) = (n � 1)!, an immediate generalization in the form of theoperation Ia defined for a > 0 is
ðIaf ÞðtÞ ¼ 1
CðaÞ
Z t
0
ðt � t1Þa�1f ðt1Þdt1 ¼1
CðaÞ ta�1 � f ðtÞ; 0 6 t < b; ð20Þ
where C(a) is the Gamma function and ta�1 � f ðtÞ ¼R t
0 f ðt � t1Þa�1ðt1Þdt1 is called the convolution product ofta�1 and f(t). Eq. (20) is called the Riemann–Liouville fractional integral of order a for the function f(t). Now iff(t) is expanded in block pulse functions, as shown in Eq. (17), the Riemann–Liouville fractional integralbecomes
ðIaf ÞðtÞ ¼ 1
CðaÞ ta�1 � f ðtÞ ’ nT 1
CðaÞ fta�1 � /mðtÞg: ð21Þ
Thus if ta�1* /m(t) can be integrated, then expanded in block pulse functions, the Riemann–Liouville frac-
tional integral is solved via the block pulse functions.Let f1(t) = ta�1, f2(t) = wi(t), a > 0, then
1
CðaÞ ta�1 � wiðtÞ ¼
1
CðaÞ f1ðtÞ � f2ðtÞ: ð22Þ
Taking the Laplace transform of Eq. (22) gives
£1
ta�1 � wiðtÞ� �
¼ £1
f1ðtÞ � f2ðtÞ� �
¼ 1F 1ðsÞF 2ðsÞ; ð23Þ
CðaÞ CðaÞ CðaÞ
254 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
where
F 1ðsÞ ¼ £fta�1g ¼ CðaÞsa
;
F 2ðsÞ ¼ £fwiðtÞg ¼1
s£
d
dtwiðtÞ
� �� ,
1
se�
i�1m½ �bs � e�
im½ �bs
n o:
ð24Þ
Since C(a + 1) = aC(a), thus Eq. (23) can be written as:
£1
CðaÞ ta�1 � wiðtÞ
� �¼ 1
Cðaþ 1ÞCðaþ 1Þ
saþ1e�
i�1m½ �bs � e�
im½ �bs
n o: ð25Þ
Taking the inverse Laplace transform of Eq. (25) yields
1
CðaÞ
Z t
0
ðt � t1Þa�1wiðt1Þdt1 ¼1
Cðaþ 1Þ t � i� 1
mb
� �a
u t � i� 1
mb
� �� t � i
mb
� �a
u t � im
b� �
; ð26Þ
where u(t) is the unit step function. Further, it can be derived from Eq. (18) that
tauðtÞ ’ ½c1; c2; . . . ; cm�/mðtÞ ¼ CT/mðtÞ; ð27Þ
whereci ¼bm
� �a iaþ1 � ði� 1Þaþ1
aþ 1: ð28Þ
Moreover, from the disjoint property of the block pulse functions [5]
wiðtÞwjðtÞ ¼wiðtÞ i ¼ j;
0 i 6¼ j:
�ð29Þ
It is obvious that
t � i� 1
mb
� �a
u t � i� 1
mb
� �’ ½0; 0; . . . ; 0; c1; c2; . . . ; cm�iþ1�/mðtÞ; ð30Þ
t � im
b� �a
u t � im
b� �
’ ½0; 0; . . . ; 0; c1; c2; . . . ; cm�1�/mðtÞ: ð31Þ
Thus Eq. (26) becomes
1
CðaÞ
Z t
0
ðt � t1Þa�1wiðt1Þdt1 ’1
Cðaþ 1Þ ½0; 0; . . . ; 0; c1; c2 � c1; . . . ; cm�iþ1 � cm�i�/mðtÞ ð32Þ
for i = 1,2, . . . ,m. Here,
cr � cr�1 ¼bm
� �a1
aþ 1½raþ1 � 2ðr � 1Þaþ1 þ ðr � 2Þaþ1� ð33Þ
for r = 2,3, . . . ,m � i + 1 and c1 ¼ bm
�a 1aþ1
.Now we can also write Eq. (32) as:
1
CðaÞ
Z t
0
ðt � t1Þa�1wiðt1Þdt1 ’bm
� �a1
Cðaþ 2Þ ½0; 0; . . . ; 0; n1; n2; . . . ; nm�iþ1�/mðtÞ; ð34Þ
where
n1 ¼ 1; np ¼ paþ1 � 2ðp � 1Þaþ1 þ ðp � 2Þaþ1 ðp ¼ 2; 3; . . . ;m� iþ 1Þ: ð35Þ
Finally, for i = 1,2, . . . ,m, Eq. (34) can be written as:
1
CðaÞ
Z t
0
ðt � t1Þa�1/mðt1Þdt1 ’ F a/mðtÞ; ð36Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 255
where
F a ¼bm
� �a1
Cðaþ 2Þ
1 n2 n3 � � � nm
0 1 n2 � � � nm�1
0 0 1 � � � nm�2
0 0 0 . .. ..
.
0 0 0 0 1
266666664
377777775
ð37Þ
is called the generalized block pulse operational matrix for integration.For example, if a = b = 1, we then have n1 = 1, np = p2 � 2 (2 � 1)2 + (p � 2)2 = 2, for all p = 2,3, . . . ,m
and
Z t
0
/mðt1Þdt1 ’ F 1/mðtÞ ¼1
2m
1 2 2 � � � 2
0 1 2 � � � 2
0 0 1 � � � 2
0 0 0 . .. ..
.
0 0 0 0 1
26666664
37777775/mðtÞ:
The generalized block pulse operational matrices Da for differentiation can be derived simply by inverting theFa matrices. Let
Da ¼ F �1a ¼
mb
� �aCðaþ 2Þ
1 n2 n3 � � � nm
0 1 n2 � � � nm�1
0 0 1 � � � nm�2
0 0 0 . .. ..
.
0 0 0 0 1
266666664
377777775
�1
¼ mb
� �aCðaþ 2Þ
d1 d2 d1 � � � dm
0 d1 d2 � � � dm�1
0 0 d1 � � � dm�2
0 0 0 . .. ..
.
0 0 0 0 d1
266666664
377777775:
ð38Þ
Thus
1 n2 n3 � � � nm
0 1 n2 � � � nm�1
0 0 1 � � � nm�2
0 0 0 . .. ..
.
0 0 0 0 1
266666664
377777775:
d1 d2 d3 � � � dm
0 d1 d2 � � � dm�1
0 0 d1 � � � dm�2
0 0 0 . .. ..
.
0 0 0 0 d1
266666664
377777775¼ Im; ð39Þ
where Im is the identity matrix of order m · m. After equating both sides of Eq. (39), we get
d1 ¼ 1; d2 ¼ �n2d1; . . . ; dm ¼ �Xm
i¼2
nidm�iþ1: ð40Þ
Note that if a = 0, then F0 = D0 = Im which corresponds to s0 = 1 in the Laplace domain. It is obvious thatthe Fa and Da matrices perform as s�a and sa in the Laplace domain and as fractional (operational) integratorsand differentiators in the time domain.
For example, let a = 0.5, m = 4, b = 1, the operational matrices F0.5 and D0.5 which correspond to s�0.5 ands0.5 are computed below:
n1 = 1, n2 ¼ 2ffiffiffi2p� 2 ¼ 0:8284, n3 ¼ 3
ffiffiffi3p� 4
ffiffiffi2pþ 1 ¼ 0:5393, n4 ¼ 8� 6
ffiffiffi3pþ 2
ffiffiffi2p¼ 0:4631, d1 ¼ 1,
d2 = �0.8284, d3 = 0.1470, n4 = �0.1111, Cð2:5Þ ¼ 0:75Cð0:5Þ ¼ 0:75ffiffiffipp¼ 1:3293 and
256 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
F 0:5 ¼
0:3761 0:3116 0:2028 0:1640
0 0:3761 0:3116 0:2028
0 0 0:3761 0:3116
0 0 0 0:3761
266664
377775
D0:5 ¼
2:6857 �2:2025 0:2028 �0:2954
0 2:6857 �2:2025 0:3098
0 0 2:6857 �2:2025
0 0 0 2:6857
266664
377775:
3. Convolution product and Riemann–Liouville fractional integral of matrices
Definition 1. Let A(t) = [fij(t)] and B(t) = [gij(t)] be m · m absolutely integrable matrices on [0, b). Theconvolution and Kronecker convolution product s are matrix functions defined by
(i) Convolution product:
AðtÞ � BðtÞ ¼ ½hijðtÞ�; hijðtÞ ¼Xm
k¼1
Z t
0
fikðt � t1Þgkjðt1Þdt1 ¼Xm
k¼1
fikðtÞ � gkjðtÞ: ð41Þ
(ii) Kronecker convolution product
AðtÞ ~ BðtÞ ¼ ½fijðtÞ � BðtÞ�ij: ð42Þ
Here, fij(t) * B(t) is the ijth submatrix of order m · m and A(t) ~ B(t) is of order m2 · m2.
Definition 2. Let AðtÞ ¼ 1CðaijÞ t
aij�1h i
and B(t) = [gij(t)] be m · m absolutely integrable matrices on [0,b). TheRiemann–Liouville Fractional Integral of A(t) and B(t) is a matrix functions defined by
ðIaij BÞðtÞ ¼ AðtÞ � BðtÞ; aij > 0: ð43Þ
Definition 3. Let A(t) = [fij(t)] be m · m absolutely integrable matrices on [0, b). The determinant, inverse andk-power of A(t) respect to the convolution are defined by
(i) Determinant:
det AðtÞ ¼Xm
j¼1
ð�1Þjþ1f1jðtÞ � D1j; ð44Þ
where Dij the determinant of the (m � 1) · (m � 1) matrix function obtained from A(t) by deleting row i
and column j of A(t). We call Dij(t) the minor A(t) of corresponding to the entry of fij(t) of A(t).For exam-ple, if A(t) = [fij(t)] be 2 · 2 absolutely integrable matrices on [0, b), then
det AðtÞ ¼ f11ðtÞ � f22ðtÞ � f12ðtÞ � f21ðtÞ:
(ii) Inversion:Af�1gðtÞ ¼ ½qijðtÞ�; qijðtÞ ¼ ðdet AðtÞÞf�1g � adjAðtÞ: ð45Þ
Here, (det A(t)){�1} exists, (detA(t)){�1}* detA(t) = d(t) and note that A{�1}(t) * A(t) = A(t) * A{�1}(t) =
Qm(t), where Qm(t) = Imd(t) is the Dirac identity matrix and d(t) is the Dirac identity function.
(iii) k- power convolution product:AfkgðtÞ ¼ ½f fmgij ðtÞ�; f fmgij ðtÞ ¼Xm
r¼1
f fm�1gir ðtÞ � frjðtÞ: ð46Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 257
Theorem 4. Let A(t), B(t) and C(t) be n · n absolutely integrable matrices on [0,b).Then for any scalars a and b
ðiÞ ½aAðtÞ þ bBðtÞ� � CðtÞ ¼ aAðtÞ � CðtÞ þ bBðtÞ � CðtÞ; ð47ÞðiiÞ ðAðtÞ � BðtÞÞ � CðtÞ ¼ AðtÞ � ðBðtÞ � CðtÞÞ; ð48ÞðiiiÞ AðtÞ � QnðtÞ ¼ QnðtÞ � QnðtÞ � AðtÞ ¼ AðtÞ; QnðtÞ ¼ IndðtÞ; ð49ÞðivÞ ½AðtÞ � BðtÞ�T ¼ BTðtÞ � ATðtÞ; ð50ÞðvÞ ½AðtÞ � BðtÞ�f�1g ¼ Bf�1gðtÞ � Af�1gðtÞ; if Af�1gðtÞ and Bf�1gðtÞ are exist: ð51Þ
Proof. Straightforward by the definition of the convolution product of matrices. h
Lemma 5. Let A(t) = [fij(t)] and B(t) = [gij(t)] be m · m Semi-Markov Kernel matrices on [0,b). Then for
fixed t
ðAðtÞ � BðtÞÞði; jÞ 6 ðAðtÞBðtÞÞði; jÞ: ð52Þ
Proof. Since t 2 [0,b) (fixed), then
ðAðtÞ � BðtÞÞði; jÞ ¼Xm
k¼1
fikðtÞ � gkjðtÞ 6Xm
k¼1
fikðtÞgkjðtÞ ¼ ðAðtÞBðtÞÞði; jÞ: �
Theorem 6. Let A(t) = [fij(t)] and B(t) = [gij(t)] be m · m Semi-Markov Kernel matrices on [0,b). Then for any
matrix norm kÆk and positive integer k
kAðtÞ � BðtÞk 6 kAðtÞkkBðtÞk; ð53ÞkAðtÞfkgk 6 kAðtÞkk
: ð54Þ
Proof. Since kÆk is matrix norm, then by Eq. (52), we have
kAðtÞ � BðtÞk 6 kAðtÞBðtÞk 6 kAðtÞkkBðtÞk:
The proof of other inequality is by induction on k and Eq. (53), we have
kAfkgðtÞk ¼ kAfk�1gðtÞ � AðtÞk 6 kAfk�1gðtÞkkAðtÞk 6 � � � 6 kAðtÞkk: �
Theorem 7. Let A(t) = [fij(t)], B(t) = [gij(t)] be m · m Laplace transformable matrices. Then
ðiÞ £fAðtÞ � BðtÞg ¼ £fAðtÞg£fBðtÞg; ð55ÞðiiÞ £fAðtÞ ~ BðtÞg ¼ £fAðtÞg � £fBðtÞg: ð56Þ
Proof. (i) Straightforward by the convolution Theorem.
(ii) Suppose that £{fij(t)} = Fij(s) and £ {B(t)} = G(s), then by part (i), we have
£fAðtÞ ~ BðtÞg ¼ £½fijðtÞ � BðtÞ�ij ¼ ½F ijðsÞGðsÞ� ¼ £fAðtÞg � £fBðtÞg: �
Theorem 8. Let A(t), B(t) and C(t) be n · n absolutely integrable matrices on [0,b). Then
½AðtÞ ~ BðtÞ� � ½CðtÞ ~ DðtÞ� ¼ ½AðtÞ � CðtÞ� ~ ½BðtÞ � DðtÞ�: ð57Þ
258 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
Proof. The (i, j)th block of [A(t) ~ B(t)]*[C(t) ~ D(t)] is obtained by taking the convolution product of ith rowblock of A(t) ~ B(t) and the jth column block of C(t) ~ D(t),i.e.,
½fi1ðtÞ � BðtÞ � � � finðtÞ � BðtÞ� �
h1jðtÞ � DðtÞ
..
.
hnjðtÞ � DðtÞ
26664
37775 ¼
Xn
k¼1
ðfikðtÞ � hkjðtÞ � BðtÞ � DðtÞÞ
and the (i, j)th block of the right hand side [A(t) * C(t)] ~ [B(t) * D(t)] obtained by the definition of the Kro-necker convolution product xij(t)*(B(t) * D(t)) where xij is the (i, j)th element in A(t) * C(t). But the rule of con-volution product
xijðtÞ ¼Xn
k¼1
ðfikðtÞ � hkjðtÞÞ: �
Lemma 9. Let u(t) and v(t) be m-vectors. Then
VecðuðtÞ � vTðtÞÞ ¼ vðtÞ ~ uðtÞ: ð58Þ
Proof. Straightforward by the definitions of convolution and Kronecker convolution products. h
Theorem 10. Let A(t), X(t), and B(t) be m · m absolutely integrable matrices on [0,b).Then
ðiÞ Vec½AðtÞ � X ðtÞ � BðtÞ� ¼ ½BTðtÞ ~ AðtÞ� � Vec X ðtÞ; ð59ÞðiiÞ Vec½AðtÞX ðtÞ � BðtÞ� ¼ ½BTðtÞ � AðtÞ� � Vec X ðtÞ: ð60Þ
Proof. Let X(t) be contains x1(t), . . . ,xm(t) vectors each is of order m · 1 and let ej(t) be m · 1a vector of zerosexcept for a d(t) in the jth position, for each j = 1,2 . . . ,m, then X ðtÞ ¼
Pmj¼1xjðtÞ � eT
j ðtÞ. Now, since A(t) * xj(t)and BT(t) * ej(t) are vectors of order m · 1, then we have
VecfAðtÞ � X ðtÞ � BðtÞg ¼ VecXm
j¼1
AðtÞ � xjðtÞ � eTj ðtÞ � BðtÞ
( )¼Xm
j¼1
VecfðAðtÞ � xjðtÞÞ � ðBTðtÞ � ejðtÞÞTg
¼Xm
j¼1
f½BTðtÞ � ejðtÞ� ~ ðAðtÞ � xjðtÞÞg ¼Xm
j¼1
f½BTðtÞ ~ AðtÞ� � ðejðtÞ ~ xjðtÞÞg
¼ ½BTðtÞ ~ AðtÞ� �Xm
j¼1
VecðxjðtÞ ~ eTj ðtÞÞ ¼ ðBTðtÞ ~ AðtÞÞ � Vec
Xm
j¼1
ðxjðtÞ ~ eTj ðtÞÞ
¼ ðBTðtÞ ~ AðtÞÞ � Vec X ðtÞ:
Similarly, we can proof (ii). h
4. Some applications
4.1. Solution of the state–space equations and optimal control problem using Taylor series and Kronecker product
The state–space equation is given by
x0ðtÞ ¼ AxðtÞ þ BuðtÞ; xð0Þ ¼ x0; 0 6 t 6 b; ð61Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 259
where xðtÞ 2 Rn, uðtÞ 2 Rm analytic and A, B are known constant matrices. The input vector u(t) may be ex-panded in Taylor series as follows:
uðtÞ ¼
u1ðtÞu2ðtÞ
..
.
umðtÞ
2666664
3777775 ¼
h10 h11 h12 � � � h1;r�1
h20 h21 h22 � � � h2;r�1
..
. ... ..
. ... ..
.
hm0 hm1 hm2 � � � hm;r�1
2666664
3777775 �
u0ðtÞu1ðtÞ
..
.
ur�1ðtÞ
2666664
3777775 ¼ H � uðtÞ; ð62Þ
where H is a known constant matrix.Similarly, the state vector x(t) may also be expanded in Taylor series as follows:
xðtÞ ¼
x1ðtÞx2ðtÞ
..
.
xnðtÞ
2666664
3777775 ¼
f10 f11 f12 � � � f1;r�1
f20 f21 f22 � � � f2;r�1
..
. ... ..
. ... ..
.
fn0 fn1 fn2 � � � fn;r�1
2666664
3777775:
u0ðtÞu1ðtÞ
..
.
ur�1ðtÞ
2666664
3777775 ¼ ½f0; f1; f2; . . . ; fr�1�:uðtÞ ¼ F � uðtÞ; ð63Þ
where F contains the unknown coefficients of the Taylor expansion of the state vector x(t).Next by integrating the space–state equation in Eq. (61), we have
xðtÞ � xð0Þ ¼ AZ t
0
xðrÞdrþ BZ t
0
uðrÞdr: ð64Þ
Introducing Eqs. (62), (63) in (64) and using the integration property of the operational matrix P expressedby (15), we obtain
ðF � AFP ÞuðtÞ ¼ ðBHP þ EÞuðtÞ; ð65Þ
where E = [x0 0 0 . . . 0].In Eq. (65), the matrix BHP + E is known; therefore, we have the following system for the unknown matrixF:
AFP � D; D ¼ �BHP � E: ð66Þ
This system may be written in a simpler form by using the Kronecker product, actually we haveMf ¼ ðA� P T � InrÞf ¼ d; ð67Þ
where M = PT � A � Inr is a nr · nr matrix andf ¼ Vec F ¼
f0
f1
..
.
fr�1
266664
377775; d ¼ Vec D ¼
d0
d1
..
.
dr�1
266664
377775;
i.e., fi and di are the ith columns of F and D, respectively. Here,
M ¼ P T � A� Inr ¼
pA pA � � � pA
pA pA � � � pA
..
. ... ..
. ...
pA pA � � � pA
266664
377775� Inr
is called the Kronecker operational matrix, where Inr is an identity matrix of order nr · nr.The solution of Eq. (67) is f = M�1d = (PT � A � Inr)
�1d.
260 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
The main advantage of this approach is that matrix M = PT � A � Inr due to the form of the operationalmatrix P, can be easily proven lower triangular. Therefore, one does not have to inverse the matrix M to findthe solution.
Example 11. Consider the following state–space equation
�xðtÞ ¼0 1
�2 �3
� xðtÞ þ
0
1
� uðtÞ;
where xð0Þ ¼ 01
� , u(t) = 2t + 3, 0 6 t 6 b.
If we choose r = 4 terms, then h = [3,2,0,0]T, u(t) = [u0(t),u1(t),u2(t),u3(t)] = [1, t, t2, t3].Now
D ¼ �BHP � E ¼ 1 0 0 0
�1 �3 �1 0
� d ¼ Vec D ¼ ½1;�1; 0;�3; 0;�1; 0; 0�T
M ¼ P T � A� I8 ¼
�I2 0 0 0
A �I2 0 0
0 A2�I2 0
0 0 A3�I2
26664
37775 ¼
�1 0 0 0 0 0 0 0
0 �1 0 0 0 0 0 0
0 1 �1 0 0 0 0 0
�2 �3 0 �1 0 0 0 0
0 0 0 1=2 �1 0 0 0
0 0 �1 �3=2 0 �1 0 0
0 0 0 0 0 1=3 �1 0
0 0 0 0 �2=3 �1 0 �1
26666666666664
37777777777775:
Now, f = M�1d = [0, 1,1,0,0,0,0,0]T. Therefore,
xðtÞ ¼x1ðtÞx2ðtÞ
� ¼ ½f0; f1; f2; f3�:uðtÞ ¼
0 1 0 0
1 0 0 0
� uðtÞ ¼
t
1
� :
The optimal control problem for the system in Eq. (61) with the performance index
Y ¼ 1
2
Z b
0
½xTQxþ uTRu�dr ð68Þ
has the well known solution
u� ¼ R�1BTkðtÞ; ð69Þ
where k(t) satisfies the following canonical equation:x0ðtÞk0ðtÞ
� ¼ A BR�1BT
Q �AT
" #xðtÞkðtÞ
� ; ð70Þ
with the boundary conditions x(0) = x0, and k(b) = 0.To simply the solution of the system (70), we change the independent variable t as s = b � t.We now have
x0ðsÞk0ðsÞ
� ¼ A BR�1BT
Q �AT
" #xðsÞkðsÞ
� ¼ �N
xðsÞkðsÞ
� ; ð71Þ
with the boundary conditions x(t = 0) = x(s = b) = x0 and k(t = b) = k(s = 0) = 0.Integration of Eq. (71) from 0 to s yields:
xðsÞkðsÞ
� �
xð0Þkð0Þ
� ¼ �N
R s0
xðrÞdrR s0
kðrÞdr
" #: ð72Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 261
If we assume that the functions x(s) and k(s) are analytic, then we may expand them in Taylor series asfollows:
xðsÞkðsÞ
� ¼ W uðsÞ: ð73Þ
The matrix W is the 2n · r Taylor coefficient matrix that must be determined. With the use of Eqs. (73), (72)can be written as:
W uðsÞ þ NWPuðsÞ ¼ SuðsÞ; ð74Þ
where P is the operational matrix expressed by (15). By equating the corresponding terms of the Taylor series,we obtain the following matrix equation:W þ NWP ¼ S: ð75Þ
Applying the Vec-notation and Kronecker product, we haveVecW þ VecðNWP Þ ¼ Vec W þ ðP T � NÞVec W ¼ Vec S: ð76Þ
Now, Eq. (75) may be written as:Tw ¼ s where T ¼ I þ P T � N ; ð77Þ
where T is a 2nr · 2nr square matrix. Also,S ¼ xðs ¼ 0Þ 0 0 � � � 0
0 0 0 � � � 0
�
and
w ¼
w0
w1
..
.
wr�1
26664
37775; s ¼
s0
s1
..
.
sr�1
26664
37775
are the vectors formed by the vector elements wi, si that are the columns of matrices W and S, respectively.Due to the structure of operational matrix of integration P, the matrix T formed by the kronecker product
of PT and N, may be easily proved lower triangular. This fact simplifies immensely the solution of the systemin Eq. (77) that generally has 2nr equations.
4.2. Solution of the general coupled convolution matrix differential equations using Kronecker Products
The General Coupled Convolution Matrix Differential Equations is given by
X 01ðtÞ ¼ A11ðtÞ � X 1ðtÞ � B11ðtÞ þ A12ðtÞ � X 2ðtÞ � B12ðtÞ þ � � � þ A1pðtÞ � X pðtÞ � B1pðtÞX 02ðtÞ ¼ A21ðtÞ � X 1ðtÞ � B21ðtÞ þ A22ðtÞ � X 2ðtÞ � B22ðtÞ þ � � � þ A2pðtÞ � X pðtÞ � B2pðtÞ
..
.
X 0pðtÞ ¼ Ap1ðtÞ � X 1ðtÞ � Bp1ðtÞ þ Ap2ðtÞ � X 2ðtÞ � Bp2ðtÞ þ � � � þ AppðtÞ � X pðtÞ � BppðtÞ
; ð78Þ
where Aij(t), Bij(t) are given m · m matrices with entries in are a restricted set of functions on [0, b), which isclosed under convolution and isomorphic via the Laplace transforms to the proper rational functions withnatural multiplication, Xi(t) unknown matrix functions to be solved and Xi(0) = Ci (i, j = 1,2, . . . ,p).
If we use the Vec- notation and the Kronecker convolution product as in Eq. (59) to the system in Eq. (78),we have
Vec X 01ðtÞVec X 02ðtÞ
..
.
Vec X 0pðtÞ
266664
377775 ¼
BT11ðtÞ ~ A11ðtÞ BT
12ðtÞ ~ A12ðtÞ � � � BT1pðtÞ ~ A1pðtÞ
BT21ðtÞ ~ A21ðtÞ BT
22ðtÞ ~ A22ðtÞ � � � BT2pðtÞ ~ A2pðtÞ
..
. ... ..
. ...
BTp1ðtÞ ~ Ap1ðtÞ BT
p2ðtÞ ~ Ap2ðtÞ � � � BTppðtÞ ~ AppðtÞ
266664
377775
Vec X 1ðtÞVec X 2ðtÞ
..
.
Vec X pðtÞ
26664
37775: ð79Þ
262 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
This system may be written in a simpler form by
x0ðtÞ ¼ HðtÞ � xðtÞ; xðtÞ ¼ c; ð80Þ
wherex0ðtÞ ¼
Vec X 01ðtÞ
Vec X 02ðtÞ
..
.
Vec X 0pðtÞ
26666664
37777775; xðtÞ ¼
Vec X 1ðtÞ
Vec X 2ðtÞ
..
.
VecX pðtÞ
26666664
37777775
ð81Þ
HðtÞ ¼
BT11ðtÞ ~ A11ðtÞ BT
12ðtÞ ~ A12ðtÞ � � � BT1pðtÞ ~ A1pðtÞ
BT21ðtÞ ~ A21ðtÞ BT
22ðtÞ ~ A22ðtÞ � � � BT2pðtÞ ~ A2pðtÞ
..
. ... ..
. ...
BTp1ðtÞ ~ Ap1ðtÞ BT
p2ðtÞ ~ Ap2ðtÞ � � � BTppðtÞ ~ AppðtÞ
266666664
377777775: ð82Þ
Now if we use Laplace transform to Eq. (80), we have
£fx0ðtÞg ¼ £fHðtÞ � xðtÞg: ð83Þ
Let Y(s) = £{x(t)}, G(s) = £{H(t)}, we haveY ðsÞ ¼ fsI � GðsÞgf�1gc; sI > GðsÞ: ð84Þ
Taking the inverse Laplace transform of Eq. (84) yieldsxðtÞ ¼ £�1 s I � GðsÞs
� �� �f�1g( )
c ¼ £�1 1
sI � GðsÞ
s
� �f�1g !( )
c ¼ I � ðI � I � HðtÞÞf�1gc; ð85Þ
where I is an identity scalar matrix.Suppose that Q(t) = I * H(t), then (I � I * H(t)){�1} = (I � Q(t)){�1} can be obtained by two ways, either by
truncated series development or by explicit inversion within the above described convolution algebra. In thefirst case, we have
ðI � QðtÞÞf�1g ¼ I þXn
k¼1
QfkgðtÞ þ RnðtÞ: ð86Þ
Here,
Qfnþ1gij ðtÞ ¼
Xk2N
Z t
0
QikðuÞQfngkj ðt � uÞdu ð87Þ
and Q{1}(t) = Q(t).The total error is given by
RnðtÞ ¼X1
k¼nþ1
QfkgðtÞ ¼ Qfnþ1gðtÞ � ðI þ QðtÞ þ Qf2gðtÞ þ � � �Þ ¼ Qfnþ1gðtÞ � ðI � QðtÞÞf�1g: ð88Þ
Now for any matrix norm satisfying:
kQfngðtÞk 6 kQðtÞkn; ð89Þ
then the total error due to the above truncation is also bounded by (89), if kQ(t)k < 1, then
kRnðtÞk 6kQðtÞknþ1
1� kQðtÞk : ð90Þ
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 263
Thus for given maximum error, say e we have
nþ 1 6lnfeð1� kQðtÞkÞg
ln kQðtÞk : ð91Þ
In the direct inversion method, we calculate (I � Q(t)){�1} directly:
ðI � QðtÞÞf�1g ¼ ðdetðI � QðtÞÞÞf�1g � adjðI � QðtÞÞ: ð92Þ
Example 12. The following coupled matrix convolution equations:
AðtÞ � X ðtÞ þ Y ðtÞ � BðtÞ ¼ CðtÞ;DðtÞ � X ðtÞ þ Y ðtÞ � EðtÞ ¼ F ðtÞ
has a unique solution if and only if the matrixH ¼ QnðtÞ ~ AðtÞ BTðtÞ ~ QmðtÞQnðtÞ ~ DðtÞ ETðtÞ ~ QmðtÞ
" #2 MI
2mn
is non-singular; in this case the unique solution is given by
Vec X ðtÞVec Y ðtÞ
� ¼ H f�1gðtÞ �
VecCðtÞVec F ðtÞ
�
and the corresponding homogeneous matrix convolution equationsAðtÞ � X ðtÞ þ Y ðtÞ � BðtÞ ¼ 0;DðtÞ � X ðtÞ þ Y ðtÞ � EðtÞ ¼ 0
has a unique solution: X(t) = Y(t) = 0.Because if we use the Vec-notation and Kronecker product, we have
QnðtÞ ~ AðtÞ BTðtÞ ~ QmðtÞQnðtÞ ~ DðtÞ ETðtÞ ~ QmðtÞ
" #�
VecX ðtÞVec Y ðtÞ
� ¼
Vec CðtÞVec F ðtÞ
� :
This system has a unique solution if and only if the matrix H is non-singular.
4.3. Invert the rational and irrational transfer functions
We will use generalized block pulse operational matrices Fa and Da that defined in Eqs. (37) and (38) to getthe time functions of rational and irrational transfer functions. The irrational transfer functions are derivedfrom distributed systems. It can be found in [3].
Consider any transfer function G(s) whose time function is denoted by g(t). Let
GðsÞ ¼ 1
sZðsÞ; ð93Þ
where 1s is the Laplace transform of unit step function. The meaning of (93) is that g(t) can be derived by apply-
ing a unit step function to a new system whose transfer function is Z(s) = sG(s).Given the rational transfer function G(s),
GðsÞ ¼ b1sn�1 þ b2sn�2 þ � � � þ bn�1sþ bn
sn þ a1sn�1 þ � � � þ an�1sþ an: ð94Þ
Then
ZðsÞ ¼ b1sn þ b2sn�1 þ � � � þ bn�1s2 þ bnssn þ a1sn�1 þ � � � þ an�1sþ an
¼ g1 þ g2s�1 þ g3s�2 þ � � � þ gkþ1s�k þ � � � ; ð95Þ
where gi can be derived by cross multiplications of (95) to yield the following linear recursive equation:
g1 ¼ b1; g2 ¼ b2 � a1g1; gn ¼ bn �Xn�1
i¼1
an�igi;
gnþ1 ¼ �Xn
i¼1
anþ1�igi; gnþr ¼ �Xnþr�1
i¼1
anþr�igi:
ð96Þ
264 A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265
Eq. (95) is the Maclaurin series of G(s). For irrational transfer functions G(s), Z(s) can always be expandedin Maclaurin series. This can be seen from the following examples. After associating each sa with F�a(a < 0) orDa(a > 0) operational matrices, the approximation of g(t) via the block pulse functions can be immediatelyderived and found in [3,10].
Example 13. Given the rational transfer function G(s),
GðsÞ ¼ s4 þ 3s3 þ 2s2 þ 5sþ 1
s5 þ 5s4 þ 3s3 þ 7s2 þ sþ 1¼ 1
sð1� 2s�1 þ 9s�2 � 41s�3 þ 192s�4 � � � �Þ;
) gðtÞ ’ ½1; 1; . . . ; 1�ðI � 2F 1 þ 9F 2 � 41F 3 þ 192F 4 � � � �ÞumðtÞ:
For b = 5, m = 10,
gðtÞ ’ ½0:7325; 0:5878; 0:6187; 0:6847; 0:7374; 0:7396; 0:6679; 0:5185; 0:3081; 0:0681�u10ðtÞ:
Example 14. The Laplace transform of the Bessel function of the first kind and zero order J0(t) is
GðsÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffis2 þ 1p ¼ 1
s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ s�2p ¼ 1
s1� 1
2s�2 þ 3
8s�4 � 5
16s�6 þ 35
128s�8 � � � �
� �;
) gðtÞ ’ ½1; 1; . . . ; 1� I � 1
2F 2 þ
3
8F 4 �
5
16F 6 þ
35
128F 4 � � � �
� �umðtÞ:
For b = 8, m = 8,
gðtÞ ’ ½0:9197; 0:5060;�0:0382;�0:3628;�0:3094;�0:0090; 0:2484; 0:2561�u8ðtÞ:
Example 15. The Laplace transform of the generalized exponential function is
GðsÞ ¼1ffiffispffiffiffiffiffiffiffiffiffiffiffisþ 1p ¼ 1
sð1þ s�0:5Þ�1 ¼ 1
sð1� s�0:5 þ s�1 � s�1:5 þ s�2 � s�2:5 þ � � �Þ;
) gðtÞ ’ ½1; 1; . . . ; 1�ðI � F 0:5 þ F 1 � F 1:5 þ F 2 � F 2:5 þ � � �ÞumðtÞ:
For b = 2.4, m = 4,
gðtÞ ’ ½0:6201; 0:4448; 0:3742; 0:3306�u4ðtÞ:
Example 16. The Laplace transform of the following irrational transfer function is
GðsÞ ¼ 1ffiffisp e�
ffiffisp¼ 1
s
ffiffisp
e�ffiffisp¼ 1
ss0:5 � sþ 1
2s1:5 � 1
6s2 þ 1
24s2:5 � 1
120s3 þ � � �
� �;
) gðtÞ ’ ½1; 1; . . . ; 1� D0:5 � D1 þ1
2D1:5 �
1
6D2 þ
1
24D2:5 �
1
120D3 þ � � �
� �umðtÞ:
For b = 5, m = 5,
gðtÞ ’ ½0:3782; 0:3425; 0:1824; 0:0840; 0:1413�u5ðtÞ:
Acknowledgements
The present research has been partially supported by University Putra Malaysia (UPM) under the grantIRPA09-02-04-0259-EA001.
References
[1] Al Zhour Zeyad, Kilicman Adem, Some applications of the convolution and Kronecker products of matrices, in: Proceeding of theInternational Conference on Math. 2005, UUM, Kedah, Malaysia, 2005, pp. 551–562.
A. Kilicman, Z.A.A. Al Zhour / Applied Mathematics and Computation 187 (2007) 250–265 265
[2] T. Chen, B.A. Francis, Optimal Sampled-Data Control Systems, Springer, London, 1995.[3] C.F. Chen, Y.T. Tsay, T.T. Wu, Walsh operational matrices for fractional calculus and their applications to distributed systems, J.
Frankin Inst. 303 (3) (1977) 267–284.[4] A.E. Gilmour, Circulant Matrix Methods for the Numerical solutions of Partial Differential Equations by FFT Convolutions, New
Zealand, 1987.[5] K. Maleknejad, M. Shaherezaee, H. Khatami, Numerical solution of integral equations systems of second kind by Block Pulse
Functions, Appl. Math. Comput. 166 (2005) 15–24.[6] S.G. Mouroutsos, P.D. Sparis, Taylor Series approach to system identification, analysis and optimal control, J. Frankin Inst. 319 (3)
(1985) 359–371.[7] L. Nikolaos, Dependability analysis of Semi-Markov system, Reliab. Eng. Syst. Safety 55 (1997) 203–207.[8] B. Ross, Fractional Calculus and its Applications, Springer-Verlag, Berlin, 1975.[9] H. Sumita, The Matrix Laguerre Transform, Appl. Math. Comput. 15 (1984) 1–28.
[10] C.-H. Wang, On the generalization of Block Pulse Operational matrices for fractional and operational calculus, J. Frankin Inst. 315(2) (1983) 91–102.