operational matrix

Walsh 0 Fractiona f

erational Matrices for Calculus and Their

Application to Distributed Systems

by c. F. CHEN and Y. T. TSAY

Department of Electrical Engineering, University of Houston, Houston, Texas

and T. T. wu

Engineering and Applied Physics Department, Harvard University, Cambridge,

Massachusetts

ABSTRACT: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to J/s, J(s’+ l), e-’ and eMJ” etc. are established. Several distributed parameter problems are solved by the new approach.

I. Introduction

Since Corrington (1) constructed Walsh tables for solving high-order differential equations and Chen and Hsiao (2) developed the Walsh operational matrix for solving state equations, the Walsh operational method has been successfully applied to various problems such as time domain synthesis (3) in system theory; piecewise gain determination in optimal control (4) and the direct method of solution of variational problems (5).

The unique feature of the operational matrix of Chen and Hsiao is that it performs like an integrator in the time domain or l/s in the Laplace domain. We naturally ask: can this operational matrix be generalized to fractional calculus? In other words, are there Walsh operational matrices corresponding to irrational functions such as Js, J(s’+ 1) or transcendental functions such as e --s , e+” etc.?

This paper will investigate this problem and develop several Walsh matrices for fractional calculus, then apply these matrices for solving distributed parameter systems.

II. Review of Operational Matrix for Znteger Integration

It is known that Walsh functions 4i(t) are a set of square waves which are orthonormal. Figure 1 shows the functions from & to &5 in the dyadic order.

For an arbitrary function f(t), we can expand it into Walsh series, if the

267

C. F. Chen, Y. T. Tsay and T. T. Wu

FIG. 1. Walsh functions.

function is absolutely integrable in [0, 1). We write:

f(t) = “EO G&(0

where

G = I

’ &(t)f(r) dr 0

(I)

(2)

are determined such that the following integral square error is minimized, i.e.

E = [f(t) - nfO GbW12 dt. (3)

For example, if we expand f(r) = t, the result is

f(t) = #Jo(t) -ad*(t) -Q&(f) -+#Q(f) + . . . (4)

Partial sums of (4) form stairwise waves that approximate the original function t, as shown in Fig. 2.

If we perform integration on the Walsh functions, various triangular waves are obtained. Evaluating the Walsh coefficients for these triangular waves,

268 Journal of The Franklin Institute

Walsh Operational Matrices for Fractional Calculus

1/2~o-1/4~, lzc- 1120, -lIG#, -1189,

I!!z!c-

Fro. 2. Decomposing a ramp into Walsh series.

gives the following formula: (Typically, with no loss of generality, only the first eight Walsh waves are considered.)

f _;i 1 -% l 0

$ 0 0 -Q Q 0 0 0

or

= 0 Q 0 0 ,__________________, ii 0 0 0

0 A 0 0

0 0 l iz 0

0 0 0 h

-4 0 0 0

0 4 0 0

0 0 --ii 0

0 0 0 4 ____________________-. 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(5)

The subscript means the dimension taken. It is preferable to take 2*, where s1 is an integer, as a dimension number. Note that the ith row of the square matrix shows the Walsh series coefficients of the function J4i-1 dt, i = 1, 2, . . . . For example, the first row of (5) gives the Walsh coefficients of Jc$~ dt. The eight coefficients may be found in the first eight terms of (4), of course.

The square matrix G performs as an integrator and is called the Walsh operational matrix for integer integration. It is interesting to note that if G is partitioned into four equal parts as shown; the upper left corner of GCBxsj is identical to GC4X4j; and the upper left part of Gc4x4j is Gc2x2j. Therefore, the regularity of the structure of the G matrix enables us to enlarge the G matrix.

Vol. 303, No. 3. March 1977 269


We write the general operational matrix for integration as follows:

1: i 2: i

___+___: 2 . . --

: . . : I mIm : 1I : i -; (3 :

____:____-::.____________~

G( ?nXrn) = 5 m m i O(‘B) / : -- 5 : 2m (‘4) (61 __________________________:__ __________:

f_I m (9) ; O(T) i __~~~~~~~~~~~~~~~~~~~~~~~~____________.________~~~~~

&I@) : %

The operational matrix (6) perform integration for integer calculus nicely. Therefore, it can be used to solve linear differential equations effectively. Chen and Hsiao demonstrate its power in solving state equations, in synthesizing transfer functions in the time domain, and in finding time-varying gains for optimal control.

We will generalize this matrix for fractional integration or non-integer calculus.

III. A Set of New Functions

Since Gc,,,,,,,, corresponds to l/s, if we want to study 4s for example, what we have to effect is an extension of the operational matrix idea: (1) to invert the operational matrix in order to find a matrix corresponding to the differentiator; (2) to take the square root of the differentiator matrix thus obtained for finding a new matrix equivalent to Js. However, difficulties arise: The inversion of a large matrix is not a trivial matter if m is large! Secondly, for taking the square root of a matrix, we naturally think of diagonalization of the matrix in question. It is easy to find that all the eigenvalues of the operational matrix are clustered together. There are m repeated roots and the value is 1/2m. There- fore, diagonalization of this problem is not a routine matter.

For overcoming these difficulties, we develop a new set of orthogonal functions. There is no loss of generality if we use GcaX4) for deriving the new set. Consider:

Or, in general,

[I 1 &,,, dt = G(,~,,&“+

(7)

(8)

Journal of The Franklin Institute 270


We define a set of new functions which relates the set of Walsh functions by the following formula:

or, in general,

&I, = W(?nXl?&(?n, (9a)

where the $i are called the block pulse functions with unity height and l/m width. They are shown in Fig. 3 for the first four functions. The square matrix in (9), denoted by WC4X4j, is called the Walsh matrix. The relation between the matrix and Walsh wave forms is shown in Fig. 4. From the figure we see that it is very easy to construct the Walsh matrix because the elements of the matrix correspond directly to the magnitude of the wave forms.

One of the properties of the W matrix is that

W?LXrnj = mL, (10) or

W;nXrn) =1_ WCrnXrn). (11) m

Substituting (9a) into (8), yields

I

,

WC mxm,kn, dt = &w,,,W~mxm,~w. 0

FIG. 3. Block pulse functions.

(12)

Vol. 303. No. 3, March 1977 271


1 1 1

1 -1 -1

-1 1 -1

-1 -1 1

FIG. 4. Walsh functions and Walsh matrix.

Therefore,

I

f

+%n) df = W;~,,,G~,,,,w~,,,,r~r(,,. 0

We define the following:

W;~xrn~G~rnxrn~W~rnxrn~ p H(,,,,.

Combining it with (11) gives

H( mxm) =;WGW.

Evaluation of the similarity transformation yields

J& 1

mxm) =- m

4 1 1 1 . . . 1

0 3 1 1 . . . 1

0 0 $ 1 . . . 1 . . . . . . . . . . . . . . . .

0 ;, ;, (j ;-_: 1

(13)

(14)

(15)

(16)

Matrix H is the operational matrix as expressed in the block pulse function domain. Of course, this matrix can be directly obtained from integration of Fig. 3, by reading the magnitudes as shown in Fig. 5.

It is seen that the set of the block pulse functions is more fundamental than the Walsh functions: The operational matrix for integration shown in (16) is in

272 Journal of The Frankfin Institute


r

+ 1 1

1 7 1

3

0

1

1

1

1 2 _

FIG. 5. Block pulse functions and their integrations.

terms of block pulse functions and is simpler than the operational matrix derived from Walsh.

Inspecting the H matrix, we make the following decomposition:

H( WlXm) =;t;(&, +Q~mx,,+Q~~!mxm,+...+Q~~~,> (17)

where

Qonxm)=

0 1 0 0 . . . 0

0 0 1 0 . . . 0

0 0 0 1 . . . 0 ....... . ..... ..*.

d d d d ::: i

0;

: : 1(,-l) . I

0;

,__;____________

o/o . . . 0

and it is easy to see the following important property:

Qt 0 : I _’

mxm) = E 3 ____:__rn_? Oi: 0

for i<m

and

Q; mxm) =qm,

Using (19) and (20), we write (17) as

for izm.

=; (--%,, + (I, -Qcmx,J1>

=& CL +Q(ntxn,,K -Qwen,)-‘.

(18)

(19)

(20)

(21)

273 Vol. 303, No. 3. March 1977


We rewrite (17) by expressing Hc,,,) as a polynomial of Qcmx,,,,

where H( mxm) = hn(Qmxm>

h&)=~($+x+r’+. . . F-l).

If q is an eigenvalue of Q~mxmj, it is known (6) that eigenvalue for Ht,,,, is

h=h,(q)=-l-%. 2m l-q

(24)

(22)

(23)

the corresponding

However, the eigenvalues of Qcmx,,,) in (18) are all zeros. From (24), we obtain

h=h,,,(O)=&

Therefore, we state that the eigenvalues of H(,,,,,) is 1/2m with multiplicity m. Equations (21) through (24) will be used repeatedly to derive various

generalized operational matrices.

IV. Operational Matrix for Differentiation

Theoretically, it should not be very difficult to find the operational matrix for differentiation, when that for integration is known. However, it is much simpler to find the inversion in the block pulse domain.

Let us denote the operational matrix for differentiation as Bc,xmj. We have

B (mxm)=HTAxm)=2m(L +Q(mxm&,-Q(mxm))

= 2m(I, -2Q~,.,,+2Q&,,+. . . i-lYelQK&,,,

(-l)iQ&,X,, . (26)

For m = 4, we have

(27)

Similarly transformating back to the Walsh domain yields the operational matrix for differentiation, denoted by Dcmxmj:



For m = 4, we have 0 01 0

0 00 1

D (4X4)

= 8 1 ’ -1 0 4 -2

0 -12 0

I In general, the formula is

.________:______________~_______________,__________ ,

: rn’

;_____-mt__ _____:

;. I ‘. 1

-% /

‘. : . . DC,,,) = 2m

: ..I ; ‘_______I_______.~___________7 i -2I,p,

-I(?) i 41,?:, / % i

‘_______________‘______________:___________

(29)

(30)

From (26) the eigenvalue, h-‘, of matrix H&,) can be expressed as the eigenvalue, q, of Qcmx,,,)

m-l

b =4m ++ 2 (+.ji i-l >

(31)

or

b=2m ‘3 i+q' (32)

We could use (30) directly to evaluate the operator of differentiation or use (26) and (28) indirectly to deal with it. When the dimension number m is higher, the former is not as simple as the latter.

V. Operational Matrices for Fractional Calculus

Now we try to find the operational matrix for fractional differentiation. The general form of (32) could be written as follows:

b’l’ = (&?)l” (33)

where 1 is an integer. Equation (33) can be expanded into polynomial of q and terminated at qm-‘. As the result, Eq. (33) becomes

b”’ = (2m)‘“p,,,(q) (34)

Vol. 303, No. 3, March 1977 275

C. F. Chen, Y. T. Tsay and T. T, Wu

where pl,,, is the polynomial of order m- 1 for l/1 differentiation. Thus the operational matrix for l/1 differentiation from (26) is given by

B”’ (mxm) = CW"h,dQ~mxm,). (35)

In the Walsh domain, the corresponding l/1 differentiation operational matrix is

111 D( mxm) = (2m)“’ W;ntxm~~,,m(Q~rnxrn))W~rnxrn). (36)

To illustrate the procedure, we try to find the operational matrix of Js. By choosing m = 4, we have

l-q f &n(q) = p2,4(q) = - ( 1 1+q

= l-q++q2-;q3+. . . (37)

Then (35) becomes

&x4) = &? x 4)(I (4x4r-Q~4x4)+tQf4x4)-iQ:4x4)+. . .I. (38)

However,

Therefore, we have Qi=O for ir4.

Bhx4j = J@)[I, 4x4) - Q(4x4) ++Q:4x4) -$Qf4x4d

In the Walsh domain, using (36) gives

3 1 3 1

(40) 3 -1 17 -5

For finding the operational matrix of fractional integration, we can use the reasoning similar to the fractional differentiation case.

Generalizing (24), yields

(41)

where n.,,, is the polynomial of order m - 1 for l/1 integration. The operational matrix for l/l-integration in terms of the block pulse



function is given by

l/I Wmxm,

1 =1/1 w,m(Q~mxm,) @ml (43)

and the corresponding l/l-integration operational matrix in the Walsh domain is easily found as

111 GM,,,, =& W;~,,,y,,(Q~,xrn))W~,xrn) (44)

1

=m(2m)“’ WC ,xm,7/1,m(Q(mxm,)W(,,m) (45)

Illustratively, we try to find the operational matrix for l/ Js, assuming m = 4, i.e.

Then X,,(g) = Y2.4k.I) = 1+ 4 +ts’ +tq3.

: &,x4) = Js +I (4x4) + Q~4x4) +fQ;4x4) +4Q?4x4,)

(46)

1 1 1 t

1 I 0 1 1 1

=J80011

0 0 0 1

From (45), we have

1 1 G;4x4) = -

I

4.B WC4x4jH;4x41Wt4x4j

I

17 -5 -3 -1

1 5 7 1 -5

=- 1612 3 I 3 1

(47)

-1 5 -1

I

5

Of course, this result can be obtained directly from inverting the matrix of the first example.

VI. Application to Distributed Systems

For many engineering systems described by partial differential equations or time-varying differential equations, when evaluated in the Laplace domain, fractional functions or transcendental functions of s result. For example, problems in thermal processes, in hole diffusion of transistors, in electromagne- tic devices, in transmission lines and in percolation processes often have mathematical models involving J(s), J(s2+ l), eeJ” etc. To find their inverse

Vol. 303. No. 3. March 1977 277

C. F. Chen, Y. T, Tsay and T. T. Wu

Laplace transform is not a trivial matter. There is a long history in attacking these types of problems. To mention a few: Pade’s approximation (6); Carlson and Halijak’s approach (7) using regular Newton’s process; Lerner’s work on partial analogue approximation (8); Kiloneitreva and Netushil’s technique (9) developing special functions; Chen and Chiu’s use the Fast Fourier transform

(10). The operational matrix method for fractional calculus developed in this

paper is a general and powerful approach to the distributed problem. We now apply the method to various distributed systems.

Bessel functions

One of the well known functions

F(s) = $+ 1) (48)

can be constructed as the Laplace transform of the solution of a time-varying system. We would like to find its inverse Laplace transform as the first application. Rewrite

F(s)=J(s”l+l)=J(s:+l) -f&F,(s)+.

Suppose we are interested in the solution for

OStc8.

Time scaling must be used. Note that the general rule of scaling is based on the following

for t-+cwl

where a! is the scaling factor. For this case, (Y = 8

where

Fl _s. = 0 48 8 xW8)*+ 11 *

For m = 8, substituting (32) into (Sl), yields

20 - 4) =J(S-6q+5q2)’

278

(49)

(50)

(52)

(53)

Journal of The Franklin Institute


Taylor series expansion at q = 0 gives

F,(q) = 0.8944 - 0.3578q - 0.5009q2- 0.3578q3

-0.0429q4+0.2284q5+0.2905q6+0.1346q7+. . . (54)

Replacing q by Q gives the operational matrix of this problem in the IJ domain. Note that l/s in (51) should be expressed by the JI functions, i.e.

$11 1 1 1 1 1 1 l]&@. (55)

Consequently, (48), becomes

z-1J(s’: 1) s =z-‘!F, 2

0 8

=[l 1 1 1 1 1 1 l]

il.8944

0

-0.3578 . . . 0.1346

-0.3578 $cg)

0.8944

= 0.8944+,+ 0.5367q1 + 0.03584& - 0.3220J/3

-0.3650~4-0.1365J/,+0.1721~6+0.3067~7. (56)

The exact solution and (56) are compared in Fig. 6. If we use the W matrix (9a>, we can transform (56) in terms of the Walsh

functions. Of course, the two approaches yield the same answer.

Generalized exponential function

Consider the following function

s-; F(s) =-

J(s)+1 (57)

We rewrite it as

F(s) = L__.lo

1+L s

F,(s) - i .

JS Assume m = 4, we are interested in the interval 0 5 t < 2.4. The scaling factor is s + s/2.4. Then we have

Vol. 303, No. 3, March 1977 279

C. F. Chen, Y. T. Tsay and T. 1. Wu

FIG. 6. Solution of Bessel function

0.25

t

FIG. 7. Solution of generalized exponential function.



Using (32), gives

1 F,(q) = 1+o.5477(1+q+aqz+;q3+. . .)

= 0.6461- 0.2287q - O.O3341q*- 0.09602q3

Then replacing q by Q, gives

F 1(4X4) = 0.64611w, -0.2287Qc4,4,-0.02341Q;Lw

- O.O9602Q&,,.

(58)

Finally, we obtain

-0.2287 -0.03341 -0.09602

Y-IF(s) -0.2287

=[l 1 1 l] 0.6461

0 0.6461 --“d”;‘;,“, (4) (59)

.

0 0 0.6461

=[0.6461, 0.41074, 0.3840, 0.2880]&+.

Figure 7 shows the exact solution and the approximation calculated by the new method.

Heat conduction

For the heat conduction problem, we easily arrive at the following function:

F(s) = 5 e+“.

Find the inverse Laplace transform for 0 5 t C 8. By using the scale s+s/8 and letting m =4, we have

F(s) = - si8+iexp (-\ii)=iexp(-dz)kiF,(q). (60)

Taylor expanding and evaluating the coefficients, give

(1+ q)-=( 1 - q)-“2

10 + q)-3(l - q)-l + (-1+2q)(1+ q)-yl- q)-3’21

a3F1(q) T=exp (-V/~)[(l+q)12(1-q)1’2+(2q-l)(l+q)-4(l-q)-2

- 3(1+ qY4(1 - q)-l + (1 - q)-2(l + q)-3 - 2(1+ q)-5’2(1 _ q)-3/2

+ (29 - I)(-5/2)(1+ q)-7’2(1 - q)-3’2 + (2q - 1)(1 + q)-s’2(3/2)(1 _ q)-5/2].

Vol. 303, No. 3, March lY77 281


Then Taylor series

1 a2 I, = F,(o) +$ F,(O) - q +- 2 F,(O) - q2 +$ $3 FIKO * q3

2!a q

F,(q)= (l+q+$q3)e-l.

Replacing q by Q, we finally have

e -1

e -1 e-’ 0 -c 1 0 e-l e-’ 0 &4)

0 0 e-’ e-’

.O 0 0 e-’ I

(61)

=0.3679~o+0.7358~l+0.73584~2+0.79714~3.

Figure 8 shows the exact solution and the approximation of the heat conduction problem.

Percolation problem

It is well known that the percolation problem is difficult to solve. We use the same procedure as above.

Consider the function

F,(s).

If m = 4 and 01 t < 8, we substitute s by s/8 for scaling. Then we obtain

=e ,+~+km2+m3+ - - 2 2! 3! ..* 1

=e-~[1+0.5q+0.125q2+0.0185q3+. . .I.

Consequently, we have

rO.6065 0.3033 0.07582 0.011231

1 =[l 1 1 l] I o 0 0.6065 0 0.3033 0.6065 h4)

0 0 0

0.07582 0.3033 I

0.6065

= 0.6065%+ 0.90984+, + 0.9856$2 + 0.9969*+ (63)



I -6 se

* 2 4 6 8

t

FIG. 8. Solution of heat conduction problem.

* -zi fe

02.

* 2 4 6 8

1

FIG. 9. Solution of percolation problem.

Vol 303, Yo. 3, March 1977 283


The comparison between the exact solution and the approximation is shown in Fig. 9.

VIZ. Conclusion

Fractional functions in general, and the functions Js, J(s” + l), e-’ and eeJ” in particular, are inverted by the Walsh operational matrix approach. A new set

of orthogonal functions &i(t), which is closely related to the Walsh set 4i(t), is derived. Various operational matrices for the typical functions are established, both in the $ domain and the + domain. The application of these operational matrices for the solution of distributed-parameter systems is demonstrated.

Acknowledgements

The first author wishes to acknowledge the support of the National Science Foundation grant HES75-20759. To one of the reviewers the authors wish to express their gratitude for valuable suggestions.

References

(1) M. S. Corrington, “Solution of differential and integral equations with Walsh functions”, ZEEE Trans. on Circuit Theory, Vol. CT-20, No. 5, pp. 470-476, Sept. 1973.

(2) C. F. Chen and C. H. Hsiao, “A state space approach to Walsh series solution of linear systems”, Znt. J. Systems Sci. Vol. 6, No. 9, pp. 833-858, 1975.

(3) C. F. Chen and C. H. Hsiao, “Time-domain synthesis via Walsh functions”, ZEE Proc. Vol. 122, pp. 565-570, May 1975.

(4) C. F. Chen and C. H. Hsiao, “Design of piecewise constant gains for optimal control via Walsh functions”, IEEE Trans. on Automatic control, Vol. AC-20, No. 5, pp. 596-603, Oct. 1975.

(5) C. F. Chen and C. H. Hsiao, “Walsh series direct method for solving variational problems” J. Franklin Inst., Vol. 300, No. 4, pp. 265-280, Oct. 1975.

(6) 3. L. Stewart, “Generalized Pade approximation”, IRE Proc. pp. 2003-2008, 1960.

(7) G. E. Carlson and C. A. Halijak, “Approximation of fractional Capacitors (l/S)“” by the regular Newton Process”, IEEE Trans. on Circuit Theory, Vol. m-11, pp. 210-213, 1964.

(8) R. M. Lerner, “The design of a constant angle or power-law magnitude impe- dance”, ZEEE Trans. on Circuit Theory, Vol. CT-IO, pp. 98-107, 1963.

(9) M. B. Kilomeitreva and A. V. Netushil, “Transients in an automatic control systems with irrational transfer functions”. Automation and Remote Control. Vol. 203. pp. 359-364, 1965.

(10) C. F. Chen and R. F. Chiu, “Evaluation of irrational and transcendental transfer functions via the fast Fourier transform”, Znt. J. Electron. Vol. 35, No. 2, pp. 267-278, 1973.


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