Pergamon Mathl. Comput. Modelling Vol. 28, No. 1, pp. 65-76, 1998

@ 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: SO8957177(98)00082-X 08957177/98 $19.00 + 0.00

Exact Solution of Coupled Mixed Diffusion Problems with Coupled Boundary Conditions

J. CAMACHO, L. J~DAR AND E. NAVARRO Departamento de Matemkica Aplicada, Universidad PoWcnica

P.O. Box 22.012, Valencia 46071, Spain 1jodarQmat.upv.e~

(Received and accepted December 1997)

Abstract-In this paper, existence conditions and construction of an exsct series solution for coupled diffusion problems of the type ut - Ausr = G(z,t), u(O,t) = 0, Bzl(1, t) + C&(1, t) = o, u(qO) = f(z), 0 < x 5 1, t > 0 are treated. Here A is a positive stable matrix, matrix C-‘B has real eigenvalues, and no diagonalizable hypothesis on these matrices is assumed. @ 1998 Elsevier Science Ltd. All rights reserved.

Keywords-coupled diffusion problem, Coupled boundary conditions, Vector Sturm-Liouville problem, Eigenfunction method, Moore-Penrose pseudoinverse matrix.

1. INTRODUCTION

Coupled diffusion problems are frequent in many different fields [l-7]. Discrete methods for solving such problems are based on finite difference or finite element methods [3,8]. Analytic- numerical methods have been recently proposed in [9,10]. If apart from the coupling in the partial differential equations, the boundary value conditions are also coupled, both the analytical and numerical solutions are much more difficult to obtain [l l-131, disregarding uncoupling techniques due to their well-known drawbacks [14]. In a recent paper [lo], a first result in the study of coupled homogeneous differential systems with coupled boundary conditions has been addressed.

In this paper, we consider systems of the form

~(2, t) - AG& t) = C(T r), O<z<l, t>o, (I)

U(0, t) = 0, t > 0, (2)

B u(1, t) + C uz(1, t) = 0, t > 0, (3)

r+,O) = f(z), Olzll, (4)

where u = (~1, . . . , urn) T, G(z, t), and f(z) are vectors in Cm, and A, B, C are matrices in UFxm such that C is invertible. Conditions for the existence of a solution of problems (2)-(5), as well as its construction is the aim of this paper whose organization is as follows. Section 2 deals with an algebraic study of the vector boundary eigenvalue problem

x”+xX=o, O<x<l, x20,

X(0) = 0,

BAjX(1) + CAjX’(1) = 0, O<j<p,

This work has been partially supported by the Spanish D.G.I.C.Y.T. Grant PB93-381~C02-02 and the Genemlitat Vaknciana Grant GV-C-CN-1005796.

65

66

where p is the degree homogeneous problem

J. CAMACHO et al.

of the minimal polynomial of A. In Section 3, a series solution of the

~(2, t) - Au,&, t) = 0, O<z<l, t>o, (5)

together with conditions (3),(4) is obtained, under appropriate conditions on f(z), as well as to matrices A, B, C. In particular, we assume that

there exist real eigenvalues of the matrix C-lB (6)

and Re(z) > 0 for all eigenvalues of A. (7)

Finally, in Section 4, conditions for the existence and construction of a series solution of prob- lems (2)-(4) are given using an eigenfunction vector approach.

Throughout this paper, the set of all eigenvalues of a matrix C in (Cmxm is denoted by a(C). If D is a matrix in (CnXn, we denote by ]]D]]2 the 2-norm of D, defined in [14, p. 561,

11~112 = w$ f$Y

where for a vector z in @“, ]]z]]z = (z’z)~/~ is the usual norm of z. The number ]]D]]z coincides with the square root of the maximum of the set {[WI; w E a(DHD)} where DH is the conjugate transpose of the matrix D. By [15, p. 5561, it follows

m-1 Ilm ~~e”“J~2 5 eta@) C

1/2D11; tk

k! ' t 10,

k=O

where a(D) = max{Re(w); w E a(D)}. The matrix D is said to be positive stable if Re(w) > 0 for all w in a(D). If B is a matrix in UYxm, we denote by Bt its Moore-Penrose pseudoinverse. An account of properties, examples, and applications of this concept may be found in [16]. In particular, the kernel of B, denoted by Ker B coincides with Im(1 - BtB), the image of the matrix I - BtB. The Moore-Penrose pseudoinverse of a matrix can be efficiently computed with MATLAB package [17]. Finally, by [18], we recall that

J p(k)

P,(x)e”“dx = 5 $o(-l)k+,

where P,(z) is a polynomial in z of degree m and P?‘(z) is the kth derivative of P,(z) with respect to z. The set of all the real numbers will be denoted by W.

2. ON A CLASS OF VECTOR BOUNDARY EIGENVALUE PROBLEMS

In this section, we seek eigenvalues and eigenfunctions of the boundary value problem

X” + x2x = 0, O<x<l, x10, (10)

X(0) = 0, (11)

BA_1’X(l) + CAjX’(1) = 0, OSj<p, (12)

where p is the degree of the minimal polynomial of matrix A. The general solution of (11) is

Xx(x) = sin(Xx)DA + cos(Xx)Ex,

Coupled Mixed Diffusion Problems 67

where DA, EA are UY vectors. Condition (12) gives

Xx(z) = sin(Xz)Dx,

from (12), there exists nonzero solutions of problem (11),(12) if there are nonzero vectors DA such that

[B sin(X) + XCcos(X)] AjDx = 0, OIj<p. (13)

Let HA be defined by HA = B sin(x) + X Ccos(X). (14)

Note that for X > 0, a necessary condition in order for HA to be singular is that sin(X) # 0. Also, for X > 0, HA is singular if and only if C-lB + IXcot X is singular. Hence, condition (13) for possible eigenvalues X > 0 can be written in the form

(C-‘B + IXcot X) AjDx = 0, OIj<p. (15)

Assume that ,u E (T (-C-lB) nlw,

and let X, be the unique solution of equation

06)

x, cot x, = p, (272 - 1,; I x, < (2n + l);, n 2 1. (17)

Let G, and cP be defined by

GP

G, = C-‘B + /.LI, FP = %A

(18)

G,,AP-I

Under the hypothesis rank cP < m, (19)

for the values of X = X, defined by (17), there are nonzero vector DA satisfying (15) and this equation can be written in the form

FPDx, = 0, n 1 1. (20)

By Theorem 2.3.2 of [19], the solution set of (20) takes the form

DA, = (I - $GT,) Sx,, SXn E cm. (21)

Hence, for X, > 0 satisfying (17), a set of solutions of problem (ll),( 12) is given by

XX,(Z) = sin&z) (I - Gl,+FP) Sx,, sx, E a?. (22)

Note that for X = 0, the solution set of (11),(12) takes the form

X0(z) = Doz, Do E P.

By imposing (12), one gets

(B + C)AjDo = 0, O<j<p. (23)

68 3. CAMACHO et al.

A necessary condition for the existence of nonzero vectors Do in @” satisfying (23) is the singu-

larity of matrix B + C or G1 = C-‘B + I. This means that,

/I = 1 E 0 (-C-‘B). (24)

In this case, one gets the eigenfunctions

X0(x) = x Do =x I-$5 ( >

so, so E a?, (25)

where 51 is given by (18).

THEOREM 2.1. Let p be a real eigenvalue of the matrix -C-l B and let A be a matrix in UZmx”,

where p is the degree of the minimal polynomial of A.

(i) If p = 1, X0 = 0 is an eigenvalue of problem (11),(12) if rank 51 < m, where G1 = C-‘B + I. Eigenfunctions X0(z) associated to X0 = 0 are given by (25), where SO is an arbitrary nonzero vector in U?.

(ii) Let p # 1, let G, and cp be defined by (18). If rank 6$ < m, problem (11),(12) has a

sequence of positive eigenvalues {Xn}n21, where X, is the unique solution of equation (17)

in [(2n - l)n/2, (27~ + 1)7r/2[. For each eigenvalue X ,,, one gets the eigenfunctions XA, (x)

defined by (22), h w ere SA, is an arbitrary nonzero vector in U?.

COROLLARY 2.1. With the hypotheses and notation of Theorem 2.1 for each eigenvalue X, of

problem (11),(12), the vector function

u (x, t, A,) = exp (-Az At) Xx, (xl (26)

is a solution of the boundary value problem (2), (3), and (5).

PROOF. Note that by (11) and (26), one satisfies

ut (x,t,X,J = -XiAexp (-XzAt) Xx,(x) = -&4u(x,t,&J,

uez (x, t, X,) = exp (-XiAt) Xim(x) = -Xiu (x, t, X,) .

Hence,

ut (2, t, A,) - Au,, (x, t, X,) = 0, O<X<l, t>O, 7x20.

By condition (ll), one gets ~(0, t, X,) = 0. Note also that

Bu(l,t,X,)+C~,(l,t,X,)=Bexp(X~At)Xx~(l)+Cexp(-X~At)X~~(l). (27)

For each t, the matrix exponential exp(-XiAt) can be expressed as a polynomial of the same

degree as the minimal polynomial of A, [20, p. 557)

em (--A?&) = ho(t) + bl(t)A + . . . + bp_l(t)~~-l, (28)

where bj(t) for 0 5 j < p are scalars. By condition (12) and (27),(28), one gets

P-1

Bu(l,t,Xn) fcu, (i,t,X,) = Cbj(t) {~WX~,(I) +cL@x;~(I)} = 0, t > 0. (29) I jr0

Coupled Mixed Diffusion Problems

3. EXACT SERIES SOLUTION OF THE HOMOGENEOUS PROBLEM

69

In this section, we construct a solution of problems (2)-(5) by superposition of the solutions

~(2, t, X,) of problems (2), (3), and (5) obtained in Section 2. Appropriate conditions on f(z) and A must be found apart from those imposed in Theorem 2.1. Suppose that the matrix -C-lB

has k distinct real eigenvalues

let Gpj = C-'B + CLjI ad

hi = G,, G,, - . . Gpi_l Gpi+l . . . G,, , R=G,&, l<i<k. (31)

Let S be the vector subspace defined by

S=KerR. (32)

By the decomposition theorem [21, p. 5361, one gets

S = KerG,, @KerG,, @...@KerG,,. (33)

Note that if we consider the polynomial of degree k - 1, defined by

then polynomials {Qi(z)}ei are coprime and by Bezout’s theorem [21, p. 5381, there exists

complex numbers {ai}ti such that

I = 2 aiQi(z) = Q(x), (35) i=l

where

15iIk, (36)

that is, Q(z) is the Lagrange interpolating polynomial taking the value 1 at each z = pi, 1 5 i 5 k,

see [22]. By the matrix functional calculus, one gets

I= (-l)k-l~oiR~i = (-l)*-‘&aiQi(-C-‘B). i=l i=l

(37)

Let gi(z) be the projection of f(z) on the subspace Ker G,,;, defined by

gi(z> = (-1)“-‘4’&~.f(~)v (38)

and note that gi(z) + - -. + gk(z) = f(z) for 0 5 z 5 1. Assume that f(z) lies in S, that is

Rfbc) = 0, for all z with 0 5 z 5 1. (39)

Then by (31), (32), and (38), it follows that

G,igg(z) = (-l)k-laiG~iR~if(~) = (-l)k-‘oiRf(z) = 0, O<z<l. (40)

70

Let us assume that

J. CAMACHO et al.

and

f is twice continuously differentiable in (0, l] with f (0) = 0

then

@if(l) - &f’(l) = 0, l<i<k,

gi is twice continuously differentiable in [0, l] and gi(0) = 0,

and by (38)) one gets

/Jigi - gi’(1) = (-I)“-’ % [Pi&f (1) - &f’(l)] = 0, llilk.

Let us consider the problem

ut(~, t) - A~,,(~, t) = 0, 0<2<1, t>o,

U(0, t) = 0, t > 0,

Bu(l,t) + CU,(l,t) = 0, t > 0,

U(T 0) = 9i(Z), o<z:1.

If pi # 1, by Section 2, the eigenvalues of the scalar regular Sturm-Liouville problem

y(2) + /@y = 0, O<z<l

Y(0) = 0, /.&Y(l) - Y’(l) = 0

are {X&l, defined by

x~&x; = /Li, (2n - 1); 5 Xi < (2n + l):, n2 1, lsilk.

(41)

(42)

(43)

(44)

(pi)

(45)

(46)

By the expansion theorem in the series of eigenfunctions of the scalar regular Sturm-Liouville problem (45), see [23, p. 901 or [24], ss gi(s) satisfies (43) and (44), working componentwise, one gets

(47)

and series (47) is uniformly convergent to gi(z) on [0, 11. Assume the condition

Ker GPi is an invariant subspace of A.

By (38), (40), and (48), it follows that

(48)

GPiA$J(z)S = 0, 05z51, OIj<p, (49)

or

G,,Ajg&) = 0, 05z11, OIj<p. (50)

Note that (49) or (50) means that vector ci defined by (47) lies in Ker GLi and that equation

is compatible (see (20) and (21)). S ummarizing under conditions (41), (42), and (49), the series

u~(z, t) = c exp (-XAAt) sin (-A$) 4 (51) nil

Coupled Mixed Diffusion Problems 71

defines a formal solution of problem (Pi). In order to prove that series (51) defines a rigorous solution of problem (Pi), we use the hypothesis (7). By Parseval’s inequality for scalar regular Sturm-Liouville problems [23-251, it follows the existence of a constant A4i such that

IICLII 5 My n 2 1. (52)

By (8), (52), and taking into account that (2n - 1)7r/2 I Xi < (2n + l)n/2, it is easy to show that series

c (-AhA) exp (-XhAt) sin (Xix) ci, ?l>l

c Xk exp (-XkAt) cos (Xkx) ci, 7Ql

g - (Xi)‘exp (-XkAt) sin (X$r) ci

are both uniformly convergent in any bounded domain [0, l] x [c, d] with d 1 c > 0. By the derivation theorem of functional series [26, p. 4031, one gets that series (51) is twice-termwise partially differentiable with respect to the variable z and once with respect to t.

The analysis for obtaining the series solution of problem (Pi) for the case where pi = 1 is analogous with the only difference that Xc = 0 is also an eigenvalue (see Theorem 2.1). In this case, the formal series solution takes the form

ui(z, t) = c&r + C exp (-XiAt) sin (X:X) c;,

9L>l (53)

where

and the additional hypothesis

rank 51 < m and Ker Gi is an invariant subspace of A,

(54)

(55)

in order to ensure that GIAjf(x) = 0, OlxIl, o<j

and that equation

( I - c+q so = c;

is compatible, see (25). Note that after constructing the series solutions ui(x, t) of problem (Pi) for 1 5 i 5 k, one gets the series solution of problems (2)-(5), given by

(56)

Summarizing, the following result has been established.

THEOREM 3.1. Let A be a positive stable matrix in Cmxm, let C be an invertible matrix in Cmxm, and let ~1,112,. . . ,pk be k different real eigenvalues of -CT’B. Let

G,< = C-lB + /.iJ, Q=fiG,,, j=l j#i

72 J. CAMACHO et al.

R = G,,,R and S = KerR. Let G< be the UYxm -matrix defined by (18). Suppose that

rankG < m and KerGPi is an invariant subspace of A, llilk. (57)

Let f(z) be a Cm-valued function satisfying (41), (42), and (39) and let 9i(x) be defined by (38). Then, U(IC, t) defined by (56) is a solution of problems (2)-(S).

EXAMPLE 1. Let A, B, and C be matrices in Cnxn defined by

B=

ri 0 0 ii

-C-lB =

c(A) = (1921, to (-C-‘B) = (1, -1))

In accordance with previous notation, we have

r2 -2 -1 11

-1 -2 -2 1 1 -2 -3 2

1 -3 -2 2 ’

_l -2 -1 1 I

--1 2 1 -1 0 1 1 -1 0 0 2 -1 ’

-001 0 I

Pl = -1, P2 = +1.

R,, = C-‘B + p21 = R,, = C-‘B + /1J = 0 0 2 -2

R=RPIRP,=RP,R,,l= 8 “0 ; 1; . [ I 0 0 2 -2

Condition (48) is satisfied if Rf(z) = 0 and Ker G,i is an invariant subspace of A for i = 1,2. Easy computations show that

Rf(z) = 0 if and only if fs(z) = fd(z), 05xt:l.

As Ker G,i = %(I - Gfi + G,i), in order to prove (57)) we must verify

R,,A (I - R;,R,,) = R,,A (I - Ri,R,,,) = 0.

In this case, we have

(58)

(59)

Ril-;[-; -i _; j], Rig=+ 1; 1; I;]

and computing, one gets that (59) holds true. Condition (42) is satisfied if

~&,f(l) - &f’(l) = -&U(l) + f’(1)) = 0, (60)

~2%,f(l) - %,f’(l) = R&(l) - f’(1)) = 0. (61)

Condition (60) and (61) in this case means that

fl(1) - A(l) = fi(1) - A(l), fi(1) + A(l) = 0, f3(1) + A(l) = 0. (62)

By conditions (58), (62), and (41), we conclude that, for the above matrices, the hypotheses of Theorem 3.1 are satisfied by those functions f which are twice continuously differentiable in [O, l] and

f (0) = 0, f*(l) + f;(l) = 0, 2<i<4,

flu) - f:(l) = f20) - f;(l). (63)

Coupled Mixed Diffusion Problems 73

4. THE NONHOMOGENEOUS PROBLEM

In this section, we consider problems (2)-(5) for the case where f(s) and A,B,C satisfy the hypotheses of Theorem 3.1 and G(z, t) satisfies the conditions

RG(x,t) = 0, O<z<l, t>o, (64) G(0, t) = 0 and piRPiG(l, t) - R,,iG, (1, t) = 0, t > 0, lLi<k, (65) G(z, t), G2(x, t), and G&z, t) axe continuous for 0 I z I 1,

;;; o1 llG&, t>ll; dx = M < +oa J Let us denote by Gi(z, t) the projection of G(x, t) on the subspace Ker (GPi),

G&r, t) = (-1)“~&RPiG(z, t).

By (33) and (64), one gets

G(x, t) = $ G&r, t), t>o, O<x<l. i=l

Now we consider the nonhomogeneous problem

(68)

(69)

dx, t) - A dx, t) = Wx, t>, O<x<l, t>o,

u(0, t) = 0, t > 0,

BU(1, t) + CU,(l, t) = 0, t > 0, (70)

4x7 0) = s(x), 05x51.

Let (Xk},li be the set of eigenvalues of the regular Sturm-Liouville problem (45) and let {sin (Ai x)},.Q~ be the corresponding eigenfunctions set (adding Aa = 0 and X-J(X) = x if pi=l). Considering the Sturm-Liouville series expansion of each component of Gi(x, t) in series of eigen- functions of problem (45), one gets the representation [23,24]

Gi(x’ t, =

&l Tin(t) sin (% 4 Pi # 19 c,,, Tin(t) sin (A; X) + Ti o(t)x, /.& = 1, (71)

where

Now we use the

s,’ Gi(x, t) sin (Ai x) dx,

Tin(t) =

s,’ sin (XL x) dx

~ , 1

- ’

J; Gi(x, t) xdx

#x2dx ’ n = 0.

eigenfunction method to first construct a formal solution of problem (70) of the

(72)

form &i exp (-A (Xi)’ t) sin (Xk x)&(t), Pi # 1,

w&Y, t> = ‘&l exp (-A (XfJ’t) sin (Xkx)&,(t) + & o(t)x, pi = 1, (73)

where

R&Z= J’ eA(X”)2”T&) ds + 4, n 2 0, (74) 0

where ci is defined by (47) for n 2 1 and by (54) for n = 0. The proposed solution of problems (2)-(5) is then defined by

l&(X, t) = 2 Wi(X, t). (75) i=l

74 J. CAMACHO et al.

Now we discuss the convergence of the formal solution. Note that each function appearing in the general term of wi(z, t) satisfies the boundary value conditions (2) and (3) by the hypotheses (64) and (65). By the results of Section 3, it is sufficient to prove the convergence of the series

+,t) = cexp (-At(xh)2)~,(t)sin(xbs), Ql

(76)

where

~~.,(t)=~exp(A(Xb)2s)~~~(~)ds, ~121. (77)

Note that if we denote

B:(t) = exp (-At (i~i)~)?i~(t) = Jtexp (-A (Xk)” (t - s))Z,(s)ds, 0

then, v~(z, t) = C B;(t) sin (Xi z).

By the Cauchy-Schwarz inequality for integrals, it follows that

(78)

(79)

(80)

(13) and (8), one gets

llexp(_A(X;~2(t_ 4: < e-2a(A)(X32(t-s) F (llAp-)2112J~'2(t-

j=O

(81)

=e -2c~(A)(X3~(t-s) p 2m-2((~;)2(t-4)~

where &,-z(z) is a polynomial of degree 2m - 2 with positive coefficients. Hence,

1’ llexp (_A (xl)’ (t _ s)) 111 & < e-2a(A)(Xi)2 t 1” e2acA)s(x~)‘&,+2 ((xi)’ (t - S)) ds

= J te-2a(A)u(A:)2p 2m-2 ((Xh)'u) da (82) 0

Considering the substitution t (Xa)2 = TJ into the integral appearing in (82) and using formula (9),

one gets

J te-2a(A)u(~:)"p 0

2m_2 ((x;)'~) du = --&l(':')a e-2a(A)uPz,,+z(~) dv n

1

= 2 (Xi)2 a(A) L _ e-zcW(~:) c

1zm--2 Gi-2 (tN12) , (83)

k=O (24~4))~ 1 where

2m-2 g_,(o) L= c

k=O GW4>)k *

Coupled Mixed Diffusion Problems 7s

Since the coefficients of Pz~_z(z) and Pi:_,(z) are nonnegative, for 0 I k 5 m - 2, by (83), it follows that

J ’ e-2a(A)u(X32~ an-2 ((.tz)‘+ du I 2a(af;hh12, 12 2 1, t > 0. (84) 0

By (80) and (84), it follows that

IIWll~ I L J 2 a(A) (X;>2 o t IITin(4ll; ds, nL 1, t > 0. (85)

By the Cauchy-Schwarz inequality for sums, for m 2 no, one gets

By Parseval’s inequality [25], we have

C IIT~&>II; 5 J’ lIG(~, 4ll; dz, tl>l 0

and by (86),(87), it follows

(87)

By the convergence of the series Cn,l (Xf)-2, one gets the convergence of the series (76) and (75). Using the derivation theorem for fun%ional series [26, p. 4031, working in the domain [0, 1) x [0, to],

it is easy to show that under hypotheses (65)-(67), the series (76) and (75) are twice-termwise partially differentiable with respect to Z, once with respect to t at any point (zo,to) with 0 5 ~0 I 1, to > 1. Note that series

C (Xi)‘Bh(t) sin (X~Z) n>l

is also uniformly convergent in [0, l] x [0, to], because by the Parseval’s inequality, one gets

t JC (A;)” IIZn(S)II~ ds 0 n>l

t 62 J /I 2

5 0 z

(G ( ix’s

I( k 2 t

I l a2 JJII

2

t 2,s o o da: (G.( ))/I 2 dxds.

Summarizing, the following result has been established.

THEOREM 4.1. With the hypothesis and notation of Theorem 3.1, let G(x, t) be the function satisfying properties (64)-(67). Then u(x, t) defined by (75) is a solution of problems (2)-(4).

76 J. CAMACHO et al.

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