142 ZAMM . Z. antzew. Math. Mech. 74 (1994) 2

erreicht man (wegen A k , o 2 m/2K) nach endlich vielen Schritten xk.0 2 b und

@(x, y ) 2 + min @ ( x ~ , ~ , y) = m,* ( 2 4 2 ) .

Damit ist ( 5 ) auf s( 5 x 5 p, y = y bewiesen. Um anschlieljend den Beweis im Rechteck cc 5 x 5 p, y 5 y I, A b mit

A ; = mg/2L (Lm/4L)

zu erbringen, wird Qy jeweils langs einer Strecke parallel zur y-Achse integriert. Diese Integration liefert namlich zu jedem festen x

k

@(x, y) 2 m; - L . A h = m y 2 ( 2 4 4 ) .

Nun wird obiger Algorithmus wieder auf der Strecke r 5 x 6 p, y = y + A b durchgefiihrt und ergibt mit gewissen Stellen xk.l

@(x,y + A ; ) 2 +min@(x,,,y + A ; ) = m: ( ~ m / 2 ) .

Dann wird das sich nach oben anschlieRende Rechteck der Hohe

A‘, = m7/2L (2_m/4L)

entsprechend abgearbeitet, usw. Da wir mit jedem Rechteck minde- stens um m/4L in y hoher kommen, ist nach endlich vielen Schritten die obere Begrenzungsstrecke des Rechtecks a 6 x 5 j, y 6 y 5 6 iiberschritten und ( 5 ) damit bewiesen.

Man beachte: In den einzelnen Rechenschritten werden (nebst K und L) nur Funktionswerte bei Q benutzt und nicht das ja unbekannte m.

Herrn Prof. Dr. D. GAIER danke ich fur eine kritische Durchsicht des Manuskriptes.

L i te ra tur

k

1 ANUERSON, G. D : VAMANAMURTHY, M. K.; VUORINEN. M.: Functional inequalities for complete elliptic integrals and their ratios SIAM J. Math. Anal. 21 (1990), 536-549.

2 ANDERSON, G . D.; VAMANAMURTHY, M. K.: VUORINEN, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. (im Druck).

3 DWIGHT, H. B.: Tables of integrals and other mathematical data 4. Aufl.; Macmillan Comp, New York 1961.

4 TRICOMI. F.: Elliptische Funktionen. Geest & Portig, Leipzig 1948.

Eingegangen: 13. September 1990, revidiert 22. Mai 1991 Anschrift: Dozent Dr. REINER KUHNAU, Fachbereich Ma- thematik und Informatik der Martin-Luther-Univ. Halle- Wittenberg, Universitiitsplatz 6, D-06108 Halle an der Saale, Deutschland

ZAMM . Z. angew. Math. Mech. 74 (1994) 2, 142-143 Akademie Verlag

GROETSCH, C. W.

A Numerical Method for the Surface Temperature of a Sphere under Nonlinear Boundary Conditions

MSC (1980): 45K05, 45D05, 65N35, 73830

The surface temperature of a homogeneous solid sphere that is being heated under the influence of a nonlinear radiation law is shown to satisfy a certain nonlinear Volterra integro-differential equation. We give a proof of the uniform convergence of a numerical method, based on spline approximation and collocation, for this equation.

1. In t roduct ion

In [l] the author presented a numerical method for a model, developed by MANN and WOLF [4], for the surface temperature of a half space under a nonlinear boundary condition. A similar method for a slab of finite thickness was investigated in [2]. In this note we model the surface temperature of a solid homogeneous sphere, initially at temperature zero, that is being heated under the

influence of a time varying ambient temperature f ( t ) , under the assumption that the surface heat flux is determined by a certain nonlinear function of the ambient and surface temperatures.

Suppose that the body is the unit ball and let r represent the radial distance from the origin. Then, under suitable normalizations, the temperature u(r, t ) satisfies the following initial-boundary value problem:

r 1 (1.1)

2 u t = A u = u , , + - u u , , O < r < l , t > O ,

u(r ,O) = 0 , 0 s r 5 1 . J We assume a nonlinear transfer law at the surface of the form

U A L t ) = R( f ( t ) ) - R(u(1, t ) ) 1 (1.2) wherefis a given positive bounded function representing the ambient temperature and R is a given increasing function satisfying R(0) = 0. For example, Newton’s law of heating is modeled by a linear function R, while a Stefan radiation law is modeled by a function R of the form R(z) = mz4. Other power functions have been used to model convective transfer laws [5]. Our aim is to give a uniform convergence proof of a numerical method for the surface temperature history y(t) := u( l , t).

2. An integro-different ia l equat ion

Denote the Laplace transform (with respect to t ) of u by U . After transformation the governing equations become

2 r

U ” + - L ” S U = O , s > O , U ’ ( l ) = G ,

where G is the Laplace transform of g( t ) := R(f ( t ) ) - R(y(t)) . The general form of a bounded solution of this equation is

u = C(e”7r - e-“Fp)/r, (2.1) where C is some functional of s alone. The boundary condition then gives

C = G(s)

(fi - 1)e“ + (fi + 1)e-v”‘

Substituting this into (2.1) and allowing r + 1 -, we find that

where @(s) is the Laplace transform of q(t) = u(1, t). Upon re- arrangement this gives

Hence by the convolution theorem we have

m

k ( t ) = exp (- n Z / t ) / f i - 1 . (2.4) * = - I T

In the next section we prove the convergence, on appropriate time intervals, of a simple numerical method for (2.3).

3. A spl ine-col locat ion method

Suppose T > 0 and for a given positive integer N , let h = T / N , ti = j h for j = 0, 1, ..., N . Let 1, be the piecewise linear ‘.hat” spline satisfying l j ( t i ) = 0 for i += j , l j ( t j ) = 1. We form a piecewise linear approximation yh to y of the form

143 ~

Short Communications

where the coeffcients qj are determined by substituting qh into (2.3) and collocating at the points t i , i = 1, ..., N. This results in the following system of equations for the coefficients of the approximate solution cph:

i

h ( R ( f ( f , ) ) - R(cp,)) = 1 aijcpj, i = 1, ..., N , (3.1) j = 1

where

(3.2)

I

qm = h j k ( ( m + S ) h)d.u.

L 2 e - i t - n ) 2 i = 1 + 2 f e - n z + r c o s 2 n n v

yat ,I- - r " = 1

(3.3) n

The rcmarkable identity

._

of JACOB[ 13, p. 991 shows that the kernel k(t) is positive and decreasing. It therefore follows from (3.3) that the sequence {q,,,} is also positive and decreasing. From (3.2) we then see that atj = q0 > 0 and aij < 0 for j < i.

Note that any solution cpi of (3.1) is a fixed point of the function

From the properties of the matrix entries aij and the function R, it follows easily that for each i the function gi has a unique fixed point cpt E (0, M), where 0 < f ( t ) < M. Therefore the nonlinear system of equations (3.1) which determines the approximation qh has a unique solution for each h > 0.

Our convergence analysis will require a bound for the matrix norm ~ A ; ' , I , . This is provided by the following lemma.

Lemma 3.1 :

iiA;'Il, 5 ( k ( T ) h ) - ' .

part of A,. Therefore Proof We have A, = yo[ - Sh, where -Sh is the subdiagonal

A; = qOl(1 - S,,/q0)-l .

But

For notational convenience, let

(Kcp) ( t ) := t

k( t - T) cp'(T) dr . 0

Lemma 3.2: lfcp E C2[0, TI, then

max IK(cp - Lhv) (t,)l = O(h'. '). l < i g N

Proof: By (2.4) we have 1,

K(P - Lhq) (ti) = J L(ti - T) (CP - Lh'?)' (7) dT > 0

where

i ( t ) = f exp ( - n 2 / t ) / f i

Hence for some constant C ( T )

n = - m

The remaining quantity is a half-order derivative of the spline error and it follows from the techniques of [6] (see also [2]) that

lK(q - L h q ) (till = o(h' '1. We may now present a convergence theorem. Theorem 3.3: Suppose cp E C2[0, TI and that R sut is fks u

Lipschifz condition with constant L on [0, MI. If L < k(T) , then I l q h - dlo = 0(h'.5).

Proof: Let

D = [q(tt), ...) q( t& E IRN ,

h [ ~ ( ~ h ) - ~ ( 4 1 = A , ( ~ -

Oh = [PI, ..., Cp"]' E R" . We then have from (2.3), (3.1) and lemma 3.2,

+ o ( h 2 . 5 ) .

Therefore

u - oh = hA,'[R(u*) - R(u)] + A; 'O(h2 ,5 )

and hence, by lemma 3.1,

(1 - ~ I I A ; I I ~ ~ , L ) 1 1 ~ - uqx = o(h1.5).

Therefore

References

I GROBTSCA, C. W.: Convergence of a numerical algorithm for a nonlinear heat transfer

2 GROETSCH, C. W.: A simple numerical model for nonlinear warming of a slab. Proc.

3 MAGNUS. W.: OnEKHernNcER, F.: Formeln und Sitze fur die speziellen Funktionen

4 MANN, W. R.: WOLF, F.: Heat transfer between solids and gases under nonlinear

problem Z. angew. Math. Mesh. 65 (1985). 645.

Internat. Sympos. Cornputat. Math., Matsuyama, Japan, to appear.

der Mathematischen Physik. Sprinzer-Verlag, Berlin 1943.

h boundary conditions. Quart Appl. Math. 9 (1951), 163- 184.

natural convection. Z. angeu. Math. Mech. 68 (1988). 58-59. 4,- I = 1 k ( ( N - 1) h + X) dx 2 hk(Nh) = hk(T) . 0 5 SALINIKOV. V . ; PETROVIC. S . Heating problem of a horizontal semi-infinite solid by

6 SCHULTZ, M. H.: Spline analysis. Prentice-Hall. Englewood Cliffs. N . J . 1973 0

Another needed ingredient in the convergence proof is a bound for the error incurred when cp in the right hand side of (2.3) is re- placed by its linear spline interpolant

Received May 13. 1991, revised November IS, 1991

Address: Prof. Dr. CHARLES W. GROETSCH, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A.