Instabilities in the solution of a heat conduction problem using taylor series and alternative approaches

Instabilities in the Solution of a Heat Conduction Problem Using Taylor Series and Alternative Approaches byMOHSENRAZZAGH1

Department of Mathematics and Statistics, Mississippi State, MS 39762, U.S.A.

Mississippi

State University,

and MEHDIRAZZAGHI Department of Mathematics and Computer Bloomsburg, PA 17815, U.S.A.

Science, Bloomsburg

University,

ABSTRACT : A direct method for solving a heat conduction problem using the Taylor series is discussed. It is shown that the implementation of Taylor series for this problem involves the use of an ill-conditioned matrix. Alternative approaches using shifted Le.qendre and sh$ed Chebyshev polynomials with satisfactory results are given.

I. Introduction Orthogonal functions and polynomial series, often used to represent an arbitrary time function, have recently been used to solve various problems of dynamic systems. Typical examples are the Walsh function (l), Block-pulse functions (2), Laguerre polynomials (3), Legendre polynomials (4), Chebyshev series (5), Fourier series (6) and Taylor series (7). The direct methods of Ritz and Galerkin in solving variational problems are well known. Chen and Hsiao (1) introduced the Walsh series method to variational problems. Due to the nature of the Walsh functions, the solutions obtained were piecewise constant. Hwang and Shih (3) took Laguerre series approach, Chang and Wang (4) used shifted Legendre series approach, Horng and Chou (5) calculated Chebyshev series approach and Razzaghi and Razzaghi (7) utilized Taylor series. In this paper the polynomial series expansion is used to demonstrate the instabilities in the numerical solution of a heat conduction problem. The problem was introduced in (1) and piecewise continuous solution was derived via Walsh functions. Here, we first adopt an approach using the Taylor series. These series are not based on orthogonal functions, nevertheless, they possess the operational matrix of integration. It will be seen that to obtain the Taylor series coefficients we need to use a matrix commonly known as the Hilbert matrix. Hilbert matrices are known to be ill-conditioned and hence the Taylor series are not suitable for the solution of the heat conduction problem. To overcome this difficulty, we use the shifted Legendre and shifted Chebyshev polynomials by transforming them into

l'heFranklinInstitute001&0032189 $3.00+0.00

683

Mohsen

Razzaghi and Mehdi Razzaghi

Taylor series with satisfactory digital computation.

II. Properties

results.

The method

is simple and very suitable

for

of Taylor Sevies

A function f = f(t) that is analytic in the neighborhood of the point to = 0 may be expanded in a power series using MacLaurin formula as follows : -1 f(t) = 1 c T,(f) (1) 1-O where T; (t) = t’

If f(r)

is truncated

and

up to its first M terms, then (1) can be written ,n-~I f(r) = c C;T,(t) = CT(t) ;= 0

as

where c’

= [C”

c,

..’

T’(t) = [To(f) T,(f) . . The integration

of the vector

II

(2)

T,,, I(~

(3)

c,,,

T(t) defined in (3) can be approximated

by

f T(t’) dt’ z PT(t)

s0 where P is the operational

matrix

of integration

(4)

and is given by

OIOO... 0

0

4

0

0

000: .

P=

0 L

..

. .

. .

0

0

0

...

0

.

0

...

Ji

oooo...

0

III. Relation Between the TaJ?lov Series and Orthogonal

Polynomials

Since the elements of the basis vector of anv orthogonal combinations of the Taylor series, we have

polynomials

(5)

T,(t) = R, T(r) where 684

R, is a linear

transformation

relating

the basis

are linear

vector

7’,(t) of a given

Instabilities

in the Solution of a Heat Conduction

Problem

orthogonal set of polynomials to those of Taylor series. To obtain the elements rv of the matrix R,, we can use the corresponding recurrence formula of the series. The transformation matrices for shifted Legendre and shifted Chebyshev polynomials (first kind) can be obtained similar to (8). Also, the integral from t = 0 to t = 1 of the cross product of two basis vectors T,(t) defined in (5) can be obtained as

S’

T,(t)Tt‘(t)

dt = R,v

0

T(t)7”(t) [S’0

dt

1 R:

or H, = R,HR,T

(6)

with

H-7=

S’ 0

T,(t) T.:(t)dt

(7)

I

H=

T(t)F(t)

dt.

Equation (6) has very important practical implications and shows that once the transformation matrix R, is determined, the matrix H, can be obtained from H with r(t) defined in (3). If we denote by h,,, the (i, j)th entry of H, then hi, = _--

1

z+j+

1

i=O,l,...

,m-1

j=O,l,...,

m-l.

We note that the matrix H is indeed what is commonly known as the Hilbert matrix of order m. It is symmetric and non-singular whose inverse is explicitly known. For a discussion of Hilbert matrices which are classic examples for demonstrating round-off error difficulties, see Burden and Faires (9) and Knuth (10). The above relations are now used to solve a heat conduction problem discussed in (1).

IV. Application Consider

to a Heat Conduction Problem Using Taylor Series

the extremization .I =

s where g(x) is a known

o’[:y”

function

of --yg(x)]

dx =

’ F(x, y, y’) dx s0

satisfying

S’

g(x) dx = 1

0

with the boundary Vol. 326. No. 5, ,,p. 6X3-690. Printed in Great Brilam

(9)

(10)

conditions 1989

685

Mohsen

Razzaghi

and Mehdi Razzughi y’(0) = 0,

y’(1) = 0.

(11)

Schechter (11) gave a physical interpretation of this problem by noting an application in heat conduction. Reference (1) considered the case where g(x) is given

O
1 (12)

i<,Y
and gave an approximate solution using Walsh series. Here, we solve the same problem using Taylor series. Note that the exact solution is OdX<$

$X2

y(x) = ( To solve this problem

using Taylor y’(x)

integrating

-;:.u2+x-; ix’ -_x+ ;

;
(13)

; d x < 1.

series, we assume

II, I = 1 C,T,(x) = C’T(.x) r=O

(14)

both sides, and using (4) we obtain y(x) 2 CTPT(x)

+y(O).

(15)

In view of (12), we write (9) as I,‘2

.,=;[y”dx+~‘4ydx-3[~,“‘ydr-~‘4ydx]+[~yd

ydx

x-

s0

1 (16)

Substituting

(11) and (15) into (16) and using (4) and (8) we get J = ;CTHC+

where 2;(x) is defined as v(x) =

C’P[4v(!,) -4c(;)

\s

T(x’)dx’=

[x,;

+v(l)]

(17)

;y.

(18)

,...,

0

The boundary

conditions

(11) can be expressed C”T(0)

We now minimize Suppose

(17) subject

= 0,

CTT(l)

in terms of Taylor = 0.

(19)

to (19) using the Lagrange

J* = J+q,

series as

multiplier

technique.

CTT(0)f~2CT’T(l)

(20)

where II, and yI are the two multipliers. Differentiating setting the partial derivative equal to zero, we obtain

(20) with respect to C and

686

Journal

of the Franklin Pergamon

lnst~tulr Press plc

Instabilities

in the Solution of a Heat Conduction

Problem

TABLE I Taylor series

X

m=7

Shifted Chebyshev Poly. m = 7

Shifted Legendre Poly. m = 7

Walsh series m=8

0.0

0.00000

0.00000

0.00000

0.00000

0.2 0.4 0.6 0.8 1.0

-6.44431 -0.01213 - 7.58556 - 1.29599 -7.66917

0.02452 0.03228 - 0.04021 -0.10592 -0.12342

0.02319 0.03227 -0.04176 -0.10617 -0.12497

0.01172 0.01172 -0.03516 -0.11328 -0.12500

aJ*

mat = HC+P[~~(:)-~U(~)+U(~)]+~,T(O)+~~T(I)

= 0.

(21)

Equations (19) and (21) define a set of (m+2) simultaneous linear algebraic equations from which the coefficient vector C and the multipliers q, and q2 can be found. In fact, because of the occurrence of the matrix H in (21) the results obtained using Taylor series are not satisfactory. These results are presented in Tables I and II. The occurrence of the Hilbert matrix in the numerical solution can be avoided by choosing other polynomial series. Here the solution is considered using shifted Legendre and shifted Chebyshev (first kind) polynomials which will be discussed in the following section.

V. Alternative

Approach

1. Shifted Legendre polynomials The shifted Legendre polynomials recursively related by

L,(t) are defined

Li+ ,(t) = [(2i+ 1) (2twith L,,(t) = 1 and L,(t)

= 2t-

l)Li(t)-iL,_

on the interval

,(t)]/(i+

[0, l] and

1)

1. TABLE II

Taylor series

Shifted Chebyshev

Shifted Legendre

x

m = 12

Poly. m = 12

m= 12

Exact

0.0 0.2 0.4 0.6 0.8 1.0

0.00000 5.60087 2.4328 1 5.10634 3.41005 3.15450

0.00000 0.02060 0.03319 -0.04557 -0.10513 -0.12530

0.00000 0.02060 0.03319 - 0.04557 -0.10513 -0.12530

0.00000 0.02000 0.03500 - 0.04500 -0.10500 -0.12500

Vol. 326, No. 5, pp. 683-690, Prmtcd in Great Britain

1989

687

Mohsen

Razzaghi and Mehdi Razzaghi

If ,f(t) is any square integrable function, we can approximate number of terms of the shifted Legendre polynomials as f(t)

= 1 D,L,(t) i= 0

it by a finite

= PL(t)

(22)

where the shifted Legendre coefficient vector D and the vector similar to (2) and (3): The integration of the vector L(t) can be approximated by

L(t) are defined

, L(t’) dt’ = P,L(t)

(23)

s0 where P, is the operational

matrix

of integration ...

0

0

0

..

0

0

0

0 ...

0 I 5 .. .

. .

0

0

0

0

0

...

1100 -:

0

:

0

-4 ...

0

0

P, =

0 Using (5) and

0

given by

0

0

. .

1 2m-3 0

1

0

2m-3

1

0

2m-1

(61,we get

s I

H, =

L(t)LT(t)

dt = RLHR;

0

where RL is the transformation matrix for shifted Legendre polynomials and H is given by (8). Clearly, since the weight function in the orthogonality property of the shifted Legendre polynomials is unity we can also obtain H, using this property. Indeed H, is given by a diagonal matrix with the ith diagonal element obtained from (2i+ I))‘, i = 0, 1,. . . , (m- 1). To solve the heat conduction problem using shifted Legendre polynomials, we assume y’(x)

=

1

D,L,(x)

= DTL(x)

,=O

and hence analogous J=

where VJx) 688

to (17), we have $DTH,D+DTP,[4VL(:)-4VVL(+)+

V,(l)]

(24)

is defined as Journal of the Franklin Institute Pergarnon Press plc

instabilities in the Sorption of a Heat Conduction Problem

s x

V=(x) =

L(x’) dx’ = R,

T(Y) dx’

0

R’[x,~,...;~]T*

= The boundary polynomials as

We now minimize technique. Suppose

conditions

(11) can be expressed

in terms of shifted

BTL(0)

= 0

or

DTRLT(0)

= 0,

(25)

DrL(I)

= 0

or

DrR,T(I)

= 0,

(26)

(24) subject

to (25) and (26) using

the Lagrange

J* = J+g,DTRLT(0)+~qf)TRLT(l) where setting linear q3 and

Legendre

multiplier

(27)

yapand q4 are the two multipliers. Differentiating (27) with respect to L) and the partial derivative equal to zero, we obtain a set of (m+2) simuhaneous algebraic equations from which the coefficient vector D and the multipliers q4 can be found. The results are presented in Tables I and II.

2. Shifted Chebyshev polynomials (first kind) The shifted Chebyshev recursively related by

polynomials

&+,(t)

&(t) are defined

on the interval

[0, I] and

= (2-4t)S,(t)-S+,(t)

with S,(t)

= 1

and

S,(t)

= l-2t.

The solution of the heat conduction problem using shifted Chebyshev polynomials can be obtained similar to the previous case. The results for m = 7 and m = 12 using Taylor series and shifted Legendre and shifted Chebyshev polynomials are presented in Tables I and II. The exact solution and Walsh series approximation with m - 8 are also given for comparison. For m = 12 the shifted Legendre and shifted Chebyshev polynomials approximation of J’(X) are nearly the same as the exact solution and clearly better than the solution obtained using Walsh series.

VI. ConcCusion Polynomial series and the associated matrix of integration (operational matrix) are applied to solve a heat conduction problem. The present approach provides solutions in the form of smooth continuous time functions instead of piecewise constant approximations using Walsh series method. The accuracy of the computational results depends on the kind of poIynomia1 used. It is shown that the application of Taylor series for this problem is not satisfactory because of the Vol. 326, No. 5, pp. 683-690, Printed in Great Britam

1989

689

Mohsen

Razzaghi and Mehdi Razzaghi

occurrence of the Hilbert matrix. Alternative approaches with very satisfactory results using shifted Legendre and shifted Chebyshev polynomials are given by transforming the properties of the Taylor series to the above orthogonal polynomials.

(1) C. F. Chen and C. H. Hsiao, “A Walsh Series direct method for solving variational problems”,

J. Franklin Inst., Vol. 300, No. 4, pp. 2655280, 1975.

(2) N. S. Hsu and B. Cheng, “Analysis and optimal control of time-varying (3) (4) (5) (6) (7) (8) (9) (10) (11)

690

linear systems via block-pulse functions”, ht. J. Control, Vol. 33, pp. 1107-l 122, 1981. C. Hwang and Y. P. Shih, “Laguerre Series direct method for variational problems”, J. Opt. Ttieor_v Applic., Vol. 39, No. 1, pp. 143-149, 1983. R. Y. Chang and M. L. Wang, “Shifted Legendre direct method for variational problems”, J. Opt. Theor_y Applic., Vol. 39, pp. 2999307, 1983. I. R. Horng and J. H. Chou, “Shifted Chebyshev direct method for solving variational problems”, Irzt. J. S~vtems Sci., Vol. 16, pp. 855861, 1985. M. Razzaghi and M. Razzaghi, “Fourier series direct method for variational problems”, ht. J. Control, Vol. 48, pp. 887-895, 1988. M. Razzaghi and M. Razzaghi, “Taylor series direct method for variational problems”, J. Franklin Inst., Vol. 325, pp. 125-131, 1988. P. D. Sparis and S. G. Mouroutsos, “The operational matrix of polynomial series transformation”, ht. J. Systems Sci., Vol. 16, pp. 117331184, 1985. R. L. Burden and J. D. Faires, “Numerical Analysis”, 3rd Edn, Prindle, Webber and Schmidt, Boston, MA, 1985. D. E. Knuth, “Fundamental Algorithms”, Addison-Wesley Pub. Co., Reading, MA, 2nd Edn, 1973. R. S. Schechter, “The Variation Method in Engineering”, McGraw-Hill, New York, 1967.

Journal

of the

Frankhn Pergamon

,nst,tute Press plc