D. Ramkrishna
School of Chemical
Engineering,
Purdue University,
West Lafayette, Ind. 47907
N. R. Amundson
Department of Chemical
Engineering,
University of Houston,
Houston, Texas 77004
A Non-Self-Adjoint Problem in
Heat Conduction
A non-self-adjoint heat conduction problem is solved by an expansion in terms of
the root vectors of the conduction operator and its adjoint using their biorthogonality properties. The completeness of the root vectors follows from the fact
that the non-self-adjoint boundary conditions satisfy a regularity condition, which
is sufficient to guarantee it.
Introduction
Linear boundary value problems in heat conduction have adjoint system. Although the boundary conditions lead to a
been investigated extensively for a diverse variety of boundary non-self-adjoint problem, they satisfy a somewhat comconditions, but almost invariably these have been of the type plicated regularity criterion, which guarantees the comthat produce self-adjoint problems. The property of self- pleteness of the root vectors [2]. The solution to the boundary
adjointness makes it possible to obtain readily the solution of value problem is then obtained as a biorthogonal expansion in
steady and unsteady-state heat conduction problems in the terms of the root vectors of the original and the adjoint
form of a series expansion of orthogonal eigenfunctions. problems.
Fortunately, in a good many situations, the boundary conOur organization consists of an initial discussion of the
ditions fit the physics naturally. For example, in one- boundary value problem (which defines a differential
dimensional heat conduction problems the boundary con- operator) followed by a spectral analysis of the differential
ditions at the ends are always "unmixed" in the sense that operator and its adjoint and, subsequently, the solution of the
each boundary condition involves only one or the other end- problem as an expansion in terms of the biorthogonal sets of
point. Boundary value problems of this type are always self- root vectors.
adjoint. A situation, in which, self-adjointness arises from
boundary conditions featuring both endpoints is that which Boundary Value Problem
employs the familiar 'periodicity' criteria characteristic of
Consider a slab of finite thickness along the x'-direction
circular domains. In general, however, "mixed" boundary
conditions produce non-self-adjoint problems, for which but infinite in the other two directions y and z, initially at a
orthogonal eigenfunctions do not exist so that the solution to temperature, which is uniform with respect to y and z. The
the problem cannot be obtained by the technique applicable to slab is to be heated from both ends in such a way that the
the self-adjoint problems. Frequently, the non-self-adjoint heating rate at one end is to be adjusted depending on the
problem defines an adjoint problem with a common set of heating rate at the other end, where the temperature is
eigenvalues and sets of eigenvectors which are biorthogonal. specified. Evidently, such a problem may be inherent to
If each of the above sets of biorthogonal vectors is complete, situations in which some controlled heating (or cooling) is
then a solution to the non-self-adjoint problem is possible in required to accomplish a desired objective. It is not our
the form of a series solution. Completeness can certainly not purpose to dilate on the nature of such objectives here, for
be taken for granted since non-self-adjoint problems can there could be a number of them, the exact nature of which is
display very strange behavior in regard to their eigenvalues of no consequence to this work. It is assumed that conduction
and eigenvectors. It is sometimes possible to test for com- is one-dimensional so that the unsteady-state energy is given
pleteness of the eigenvectors by rather powerful theorems (see by
Chapter V of [1]).
ct^r~7K
= —
0<X'<1
t>0
(1)
In this paper, we consider a non-self-adjoint boundary
dt
ax'2
value problem in heat conduction, which could be of
significance in controlled heating or cooling of a solid object. where we have used x' as the coordinate variable to make a
Naturally such considerations would extend as well to subsequent switch to a dimensionless distance x = x'11. The
corresponding situations of diffusive mass transport. We boundary condition at x ' = 0, where the temperature is
obtain the eigenvalues and the biorthogonal sets of eigen- assumed to be specified is given by
vectors or more generally root vectors1 by analyzing also the
T(.0,t) = Togl(t)
(2)
A root vector is defined in the next section.
Contributed by the Heat Transfer Division for publication in the JOURNAL OF
HEAT TRANSFER. Manuscript received by the Heat Transfer Division June 15,
1981.
Journal of Heat Transfer
Since the heating rate at x' = / is to be adjusted in accordance with that at x' = 0, we assume that
dT
dT
(3)
* T - T ( / , ' ) = • •k-^(Q,t)+hAt)T0
dx
dx
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FEBRUARY 1982, Vol. 104/185
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where h{(t) is taken to be a known function. When the
specific objective of control is identified, it would then be
possible to obtain an equation for hx (/) from the solution for
the temperature profile, which is developed here. The initial
temperature profile of the slab is expressed as
T(x',0)=T0fl(pc')
(4)
In order to solve this boundary value problem, we consider
the eigenvalue problem
-u"
= \u
(5)
u(0) = 0
(6)
M'(0) + M'(1) = 0
(7)
where primes denote differentiation with respect to x, and the
boundary conditions (6) and (7) represent the homogeneous
versions of (2) and (3) respectively. The differential expression
L = {-(fl/dx2)
with the domain D(L) =
[u={u(x)):
«(0) = 0; «'(0) + w'(l) = 0] defines an operator L. 2 I is a
formally self-adjoint differential expression.
However L is a non-self-adjoint operator. Its adjoint
operator L* can be easily seen to be composed of the differential expression L* = L = ( —cP/dx2) and the domain
D* (L) given by
D*(L) = [v={v(x)}:v'(l)
= 0;v(0) + v(l) = 0]
In what follows we denote D(L) more simply by D and
D* (L) by D*. If it is agreed that L automatically operates on
ue£> and L* operates on veD*, then we may write
<Lu,v> = <u,L*v>
(8)
At this point, we digress to make some observations about
non-self-adjoint operators in general. This digression is essential, since the above operator L exhibits certain features
somewhat unfamiliar to engineering analysis.
It is well known to engineers that real symmetric matrices
have certain desirable properties. These are that the eigenvalues are real and regardless of the number of times any of
the eigenvalues is repeated there is always a complete set of
eigenvectors in terms of which arbitrary vectors can be ex2
The boldface symbol u is to denote the function {u(x)) as a vector. Here
D(L) is a dense subspace of L2 [0,1], which is the set of functions u(x) such
that
u (x)dx<°°
pressed as linear combinations. The familiar Sylvester's
formula is available for forming functions of the matrix.
However, this property does not always extend to unsymmetric matrices, whose eigenvalues could be real or
complex and the eigenvectors may or may not be complete.
When the eigenvectors are not complete (a situation which
may arise in the case of repeating eigenvalues) the ordinary
form of Sylvester's theorem is not applicable. Instead a
"confluent" form of Sylvester's theorem is available, the use
of which leads to a simple solution of linear differential
equations featuring unsymmetric matrices. Self-adjoint
boundary value problems are analogous to symmetric
matrices and as pointed out earlier, unsteady-state, onedimensional heat conduction problems with unmixed
boundary conditions come within the foregoing category. The
solution to such problems is obtained as an expansion in terms
of eigenfunctions. Non-self-adjoint problems, on the other
hand, are analogous to nonsymmetric matrices. Thus it is not
certain that solutions can be obtained as expansions in terms
of eigenfunctions because the latter may not be complete. In
the development here the biorthogonal expansions of
solutions in terms of the root vectors (to be defined presently)
are analogous to using the confluent form of Sylvester's
theorem in dealing with unsymmetric matrices with repeated
eigenvalues.
Let L be a general non-self-adjoint differential operator,
whose adjoint is L*. In this case, it is not possible to make
definite assertions about the spectrum of L in regard to
whether or not it is empty, discrete, continuous, or generates a
complete set of eigenvectors. A further complication that can
arise with non-self-adjoint operators is that associated with
repeating eigenvalues. This eventuality comes up when one or
more eigenvalues are multiple roots of the characteristic
equation. With self-adjont operators, an eigenvalue, which
repeats, say, n times, generates n linearly independent
eigenvectors. Such is not necessarily the case with non-selfadjoint operators. Thus it is possible that only one eigenvector
may be associated with an eigenvalue that is repeated n times.
In the foregoing situation, the set of all eigenvectors of the
non-self-adjoint operator, L, cannot be expected to be
complete. The difficulty is partially overcome by generalizing
the concept of an eigenvector. This generalization consists of
what has been referred to as a root vector (see for example p.
5 of [1]), which is defined as follows. A vector u is called a
root vector of operator L, corresponding to an eigenvalue A,
if there exists an integer n such that
Nomenclature
D,D*
= domain of differential
operators L and L*,
respectively
D(A) = characteristic
matrix
defined by equation (15)
/ , / [ = initial
dimensionless
temperature distribution
in the slab
g,g, = time dependent dimensionless temperature at
slab end, x = 0
h,h\ = time dependent inhom o g e n i e t i e s in the
boundary conditions (3)
and (42), respectively
k = thermal conductivity of
slab material
k(u(x))
= Wronskian vector of
function u (x) defined
above equation (13)
186/ Vol. 104, FEBRUARY 1982
K(U(x))
L
L
t
T
T,
u ,v
U
Wronskian matrix of
vector V(x)
defined
belows equation (15)
differential expression
differential
operator
defined
by L
and
homogeneous boundary
conditions, L* = adjoint
operator of L
time
temperature of the slab,
T0
s
constant with
d i m e n s i o n of
temperature
coolant temperature
typical vectors, usually u
eAvefl"
vector
defined
by
equation (16) containing
linearly
independent
solutions of equation (11)
u v
x. x = root vectors satisfying
( L - X I ) 2 u x = 0 and
(L*-AI)2vx = 0
x,x' = dimensionless and dimensional distance along
slab axis
Greek symbols
thermal diffusivity of
slab material
a
coefficients of expansion
j£j
in equation (26)
11
Eigenvalue,/
h<£ = dimensionless coolant
temperature
9 = dimensionless
temperature defined above
equation (40)
r = dimensionless time
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(L-XI)"u = 0
(9)
While an eigenvector is also a root vector, the converse is
clearly not true. When L is a differential operator consisting
of the differential expression L and its domain D(L), then
equation (9) implies that
(Z,-X)"u = 0
(10)
and that ueD(L). When root vectors exist for a differential
operator, they can be obtained in an interesting manner.
Thus, for example, let Ui(x,K) and u2(x,\) be the linearly
independent solutions of the differential equation
(Z,-X)u = 0,
0<x<l
(11)
in which the differential expression L involves differentiation
w.r.t.x. We let the boundary conditions defining the domain
of L be given by
/3°k(«(0,X)) + |3 1 k(«(l,X)) = 0
+ c2U2(x,\)
(13)
(L-\)2
^ - (*,X) = 0
Thus the candidate for the root vector from equation (10)
with n = 2 is given by
du,
du2
(21)
w ( x , X ) ^ 7 , —'- (x,X) + 7 2 — 2 - (*,X)
ok
ah
where U\(x,\) and u2(x,h) are the solutions of (11). For w(x,
XQ) to be a root vector it must satisfy equation (14), which
implies that
^K(-(O,X3))+^K(-(I,XO)) =0
(15)
in which \](x, X) and K(U(x, X)) are given by
Wi(x,X)~
U(x,X) =
u2(x,\)
"«,(x,X)
u2(x,h)
_"i'(x,X)
ui(x,\)
K(U(x,X)) =
If an eigenvalue X = X0 is repeated once then this implies
that
d ID(X) I
d\
^=^n
=0
(16)
If D(\0) has rank 1, then only one linearly independent
solution can be obtained for the equation
D(\,)c = 0
where
<]
(17)
(22)
Since the Wronskian matrix K is linear in its argument, we
infer the validity of equation (18) from (22). The coefficients
71 and y2 in equation (21) are obtained from the algebraic
equations
d
[D(X)] x=Xo 7 = 0
d\
(23)
where
<<]
(14)
where ID(X) I is the determinant of the matrix
D(X) = /S°K(U)(0,X)) + /J1 K(U(l,X))
(20)
OK
then the characteristic equation for the eigenvalues is given by
ID(X)I=0
(19)
Again using equation (11) on (19) we obtain
(12)
where /3° and /S1 are (2x2) matrices of constant coefficients
(See Cole [3] for such a representation of boundary conditions); k(u(x,h)) is the Wronskian column vector, with the
first component «(x,X) and the second « ' (x,\), the prime
denoting differentiation with respect to x. If the solution
u (x,\) is expressed as
u(x,\) = clul(x,\)
(L-X)-Jj£-(*.X) = «(*,X)
72
The generalization of the foregoing procedure to determining
the root vector for higher order repetitions of an eigenvalue is
self-suggestive; furthermore, it is not required for the present
paper. Thus considerations are henceforth restricted to the
case for which each eigenvalue is repeated at most once.
Denoting the eigenvectors of L by u, and the root vectors by
u x ., we have the set uJt wx., which could conceivably be a
basis in L2 [0, 1].
Furthermore, the adjoint operator, L*, has the same
eigenvalues (X,) and a corresponding set of eigenvectors and
root vectors (v ; , v x . j . A number of theorems are proved in
the appendix of this paper, the results of which are stated
here. Firstly, (from Theorem 1) u, and u x . are linearly independent vectors. Secondly, (from Theorem 2) fory ^ k
< \ij ,yk> = < ux. ,yk < = < U; ,uX/t > = < uH vXy, > = 0
(24)
Thirdly, (from Theorem 3) we show that
<Uj,\j>=0
(25)
Fourthly, based on the presumption of completeness of the
sets of root vectors (u J f u x ] and {v,,v x .) and expansion
formula may be derived for an arbitrary vector feL2[0, 1].
This derivation is presented below.
We let
f
=£(«y«y + ^u x )
(26)
y=i
consists of the constants in equation (13). This development
of the characteristic equation (14) and calculation of the Forming the inner product of equation (26) with v*
eigenvector (17) is given in detail by Cole [3] for the general
oo
differential operator of even order. In the foregoing situation,
< f . ^ > = E(oij<Uj,vk>+Pj<u>i.,vk>)
(27)
only one eigenvector is obtained for eigenvalue XQ (with
multiplicity 2) through equations (13) and (17). However, a
The use of equations (24) and (25) on equation (27) yields
root vector for X = XQ may exist if it happens that
<f,v,>
(28)
(18)
[D(X)]X
=0
<*\k>"k>
d\
Further, we take the inner product of equation (26) with v x
which is distinct from equation (16) and is not, in general,
implied by it. The existence of a root vector, given equation
<f
(18) is established as follows. Differentiating equation (11)
(29)
> % > = Yt(aj<uJ'Vxk
> +/3;<u x ,VX > )
w.r.t. X we have
7=1
&
Journal of Heat Transfer
=
•
FEBRUARY 1982, Vol. 104 /187
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We again use the biorthogonality conditions (24) to obtain
from equation (29)
«k <»k,V\k>+Pk<
«Xyt ' % > = < f >% >
Since 0k is known from (28), ak is found to be
<f,v X/t > - <uX;, , v ^ > < f , v i t > / < u x < r , v t >
a*
(30)
h(r)m
*,(/)/
;/(*)=/.(*'); gM^g,(0
Thus the dimensionless form of (1) is
a2e
ae
0<x<l,r>0
&? ~ ~ ~a7
(31)
with the boundary conditions
so that the expansion formula (26) is completely established
e ( 0 , r ) = g(r)
ae
y= i
( <f.vx,- > ~ <Ux,- .%• > <i,Vj > / < » x ; ,vy < )U; + <f,Vj >ux
Of course, whether or not
requires proof and is discussed
Returning to the problem
equations (5)-(7) to obtain
teristics equation is given by
(l+cosVX) = 0
(33)
which has the roots VX= (2« — l)ir, each repeated one because
1
- s i n ( 2 « - l ) i r = 0(34)
dX
/X=(2«-I)TT
2(2n-l)it
There is only one eigenvector u„ = (sin(2«-l)7r.»c). There
exists, however a root vector
r dd
sinV?a:
ae
The initial condition transforms to
(43)
The differential equation (40) may be written as
Z,G(T)=-^-9(T)
where 9 ( T ) = {Q(X,T)},
of the form (39), i.e.,
6(r)= £
dr
and L= -eP/dx.
— c„ o- vs_( .2.« - .1)«:(
,....,
In order to obtain < 6 ( T ) , V„ > K^/X <e(7),v x > , we first
recognize that for veD*(L)
<LQ
(T),v>-<e(r),Lv
=1
i*
i*
ae
ae
r
[ 9 ( X , T ) V ' ( X ) - — ( x , r M x ) Jx
| =0
v„ = { c o s ( 2 / i - l ) i r ( l - x ) )
(36)
(1-*)
(1 X
' s i n ( 2 w - l M l -x)\
(37)
"
C 2(2«
2 ( 2 « --11) T) 7T
J
The vectors v„ and v x satisfy the adjoint boundary
conditions constituting D* \L). It is now possible to make a
direct verification of the orthogonality conditions (24), which
is a straightforward procedure. An additional feature of
interest, however, is the relationship
<ux„.v x „> =
4(2«-1) TT
(46)
<e(T),£v„>+v„(0)/Kr)
v„'(0)g(r)=--e(T),v„>
d\
(47)
Since Lv„ = X„ v„ and 9(0) = f from (43) the solution of (47)
is written as
<9(T),v„> = < f , v „ > e ~ ^ r
[ t
,
n
'(0)g(r')-v„(,0)h(T')]dT'
(48)
2
Similarly, the inner product of (44) with v x yields
x(l - X ) C O S ( 2 / J - l)7rxsin(2«- l)7r(l
-x)dx-Q
(38)
which appears to be a result specific to the problem. The
expansion formula simplifies in this case to
<f,v Xn >u„ + < f , v „ > u x
<«x„.v„>
(39)
It is this expansion, which leads to the solution of the
boundary value problem of interest. The validity of the expansion is contingent on the completeness of the root vectors
(u„,u x ). Mathematical research on non-self-adjoint differential operators [2] has established that the boundary
conditions (6) and (7) ensure the required completeness. The
boundary value problem is non dimensionalized, using the
customary dimensionless variables
*'
= v(0)h(T)-v'(0)g(T)
which results from v '(1) = 0, u(0) = - y ( l ) and the boundary
conditions (41) and (42). Equation (46) may be used to find
the quantities < 6 ( r ) , v„ > and 9 ( r ) , vx > in the expansion
(45). Thus we form the inner product of (44) with v„ to obtain
in the light of (46)
+ j V ^ - 0
1
2
ox
(35)
'
which can be readily shown to satisfy the boundary conditions
(6) and (7). ThusuX/? e£>(L).
Similarly for the adjoint problem, we have
f
(45)
/X=(2n-1)
2(2«-l)7
=
(44)
We seek a solution
<e(T),v Xn >u„ + < 9 ( T ) , V „ > U X
ax
x
(42)
0(x,O)=/(x)
(32)
the root vectors are complete
later.
of specific interest, we solve
u(x,X) = sinVXx:. The charac-
(1+cosVX)
(41)
^ - ( 1 , T ) + — ( 0 , T ) = A(T)
ax
dx
<Ux,,v>>
(40)
at
o,
188/Vol. 104, FEBRUARY 1982
.
<6(r),Lv x > + v x (O)A(r)
-vx/(0)g(r)=--<e(7),vXn>
(49)
The equation Lv„ = X„ v„ implies that Lv x + X„ vx , which
when inserted into (49) obtains
<e(T),v„>+X„<e(7),vXn >
+ v x „(0)/ I (r)-v' X n (0)g(r)
=
_^<e(r),vX)|>
(50)
Equation (50) is solved subject to (43) producing
<e(7),vXn> = < f , v X n > e - x « r
+ £ [Vx-„(0)g(7')-Vx„(0)A(r')
T
-<e(T),v„>;u?- x «( T - T 'W
(5i)
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"1
T
0.1
0.2
~l
r
-0.4
-0.8
<£(r)
©
-1.6
-2.0
0.3
0.4
05
0.6
0.7
T
0
0.2
0.4
0.6
0.8
I.O
Fig. 2
Dimensionless coolant temperature program
x
Fig. 1
Evolution of dimensionless temperature profiles
Equation (48) may be substituted into (51), which leads to
<Q(T),VK„ > =[<f.vx„ > -r<f,v„ >]e" x « T
x = 0 is constrained to be at zero temperature. Furthermore,
we assume that the initial temperature distribution f(x) =x
(see Fig. 1). The solution (53) becomes
9(x,r) = 8 2^«' X " T T- ~cosv^„x
+ £«/T'[ivB(Ote(T')-i;X(i((>)A(T')
-f
e-\{r-T")[vn'{Q)g{.T")-vn{Q)h(.T")}dr")
(52)
Of particular interest is the first term on the right side of
(52), which displays the coefficient Te~x"T characteristic of
repeating eigenvalues. The solution is now essentially complete except for the details of evaluating the known terms in
equations (48) and (52). Thus, we have
where f, (f) is the temperature program to be calculated from
the solution. By defining dimensionless variables
K„(0) = cos(2«-l)7r=-l,y'„(0) = 0)MXn(0) = 0
«VB (0) = + y cos(2« - 1)TT= - —
: u X n , u „ > = [0
« -
cos(2rt-l)7r
+
e^n-r'h{T')dr'
Jo
-shr/X,,*!^/(*')(! - ^ ' ) ^ - s i n V X „ ( l
-x')dx'
+ T\ /(*')cosVX„(l-x')e?x'
Jo
-i[h(T')+ieKT"hiT")dT"]dT'}]
(53)
where X„ =(2«-l) 2 7r 2 . Thus the complete solution to the
non-self-adjoint boundary value problem is obtained.
As a demonstration of the solution, we present calculations
for the case g(r) = h(T) = 0, which implies that the slab is
equally cooled (or heated) from both ends and that the end at
Journal of Heat Transfer
A, W
T {t)
>
T0
1 d_
9(1,7) + 9(1 ,T)
~B dx
The solution (54) yields
0(r) =
1
©(*>?•)= £ - 8 V V ~ X " T [ ^ C O S V A „ X
Jo
'
(56)
we obtain from (55)
8(2«-l)ir
from which the solution to the boundary value problem for
9 (X,T) may be written as
X [ f(x')cosSkJl-x')dx'
h 1
k
2(2/2 — 1)7T
xcos(2n-l)7r(l -x)dx =
(54)
sinvVc(r+l)]
V\
If the heat transfer coefficient at x=l is known, it is
possible to calculate a "temperature program" for the
coolant at this end as function of time to maintain equal
cooling rate at both ends. Thus letting
dT
••h,[T(l,t)-Tm
(55)
(57)
Figure 1 shows the evolution of the temperature profile.
Figure 2 shows the required temperature program of the
control end of the rod.
It is of interest to observe that the above boundary value
problem could also have been solved by the method of
Laplace transform, the solution emerging from an inversion
of its transform, which requires contour integration. The
procedure yields the same solution (53). Indeed, the method
pf non-self-adjoint operator representation, which we have
presented here, requires some discussion in regard to its merit
relative to that of the method of Laplace transform. The two
methods are in essence equivalent, except that the spectral
theory of operators decomposes the procedure into units, each
with a powerfully organized structure. Thus the spectral
resolution of the operator L is a much deeper quest that the
chore of determining the singularities of a complex integrand
for the evaluation of its complex integral through the method
of residues. The climax of the method of eigenvalues and
eigenvectors is the expansion formula (32), a virtually instant
FEBRUARY 1982, Vol. 104/189
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passport to construction of the solution to the boundary value
problem. However, the equivalence of the two methods is a
reflection of the profound generalization that the spectral
resolution of operators (or analytic functions of them
represents of the well-known Cauchy integral formula for an
analytic function. This aspect of spectral theory is brought
out in a elementary way by a superb exposition of the subject
by Lorch [4].
3 Cole, R. H., Theory of Ordinary Differential Equations, AppletonCentury-Crafts, New York, 1968.
4 Lorch, E., "The Spectral Theorem" in Studies in Mathematics, Vol. I,
Math. Assn. Amer., 1962, pp. 89-137.
APPENDIX
(59)
. In the following theorems, it is understood that we are
dealing with a non-self-adjoint operator, L whose adjoint
operator is L*. Further, L and L* have the same eigenvalues
which repeat, at most, once. (The generalization to higher
order repetitions is self-suggestive). The eigenvalues are
assumed to be real since the generalization to complex values
is unnecessary here, u is an eigenvector of L and u x = du/d\ is
a root vector of L with n = 2. Similarly v is the eigenvector of
L and vx is a root vector with n — 2.
Theorem 1: u and u x are linearly independent; so are v and
>V
Proof: Let a and (3 be numbers such that au + /3ux = 0.
Then 0 = L(cm + /3ux) = aLu + /3Lux.
From equation (19), the above equality yields
0 = aXu + /3u + 0Xux.
along with the boundary condition (6). This too leads to a
complete set of eigenvectors3 with eigenvalues ((2n l) 2 « 2 -7T2 J which occur as single roots, together with another
set obtained from
But 0 = aXu + /3Xux so that j3u = 0.
Thus /3 = 0. Hence, a = 0 so that u and u x are linear independent.
Theorem 2: For X,- # \k, each repeating once we have
Conclusions
The unsteady-state, heat-conduction problem of this paper
has been solved by using the root vectors of the non-selfadjoint operator that appears in the problem with non-selfadjoint boundary conditions. The boundary conditions
satisfy a regularity condition which assures the completeness
of the root vectors.
Other variants of the boundary conditions (2) and (3) may
also be envisaged. For example instead of boundary condition
(7) one may have
u'(0) = a«(0)
IX
^
VXtan — =a
2
Note how the problem whose solution has been obtained is
recovered from here letting a—oo. Since none of the eigenvalues is repeated no root vectors are present and the
eigenvectors are entirely adequate. We have pursued the
problem for a = °° because it accommodates the complication
of repeated eigenvalues and the consequent necessity for
including the root vectors. The prpblem for finite a is solved
by a biorthogonal expansion including only the eigenvectors.
The methodology of this paper may be important for other
non-self-adjoint problems, which may arise in heat conduction or mass diffusion.
<uj,\k > = <u,-,vX/t > = <u Xy ,yk > = <u X ; ,vx^ > =0.
Proof: That <Uj,vk > = 0 is well known since uy and vk are
eigenvectors of L and L* for distinct eigenvalues. Now since
ujeD,vXkeD*
0=<Ln/,Vx/>-<u/,L'vXit>
= h <Uj,\k>-<
so that (X, -r\k)<ujt\Xk>=
and 0 and <u,- ,vx > = 0.
Similary, <u x .,v A : >0.
To prove the last result, we write
0 = < L U x . ,nXk > - <u X ; ,L*vX/t >
or
0=<u
Acknowledgment
We are thankful to Professor M.F. Malone of the
University of Massachusetts who provided us with finite
difference calculation of the problem for comparison.
References
1 Gohberg, I. C. and Krein, M. G., Introduction to Theory of Linear
Nonselfadjoint
Operators, American Mathematical Society, Providence,
Rhode Island, 1969.
2 Dunford, N. and Schwartz, J. T., Linear Operators, Part III, Spectral
Operators, Wiley-lnterscience, New York, 1971, pp. 2344-2345.
3
This assertion requires the establishment of the regularity condition with
which we do not bother in this article.
190 / Vol. 104, FEBRUARY 1982
uy ,yKk >\k< u,, v Xt >
;>vx* > + X / < , V v \ t >
- <»\j Sk > - \ t < l V % >
so that (X, -\k)<ux
,v^ > = 0
from which <u x .,v x< . > = 0
Q.E.D.
Theorem 3: <U/,v, > = 0 when \j repeats
< u x . ,\j > are root vectors.
Proof: Since Lu x . = u, + XyUx.
< L U X ; ,yj > = <Uj,VJ >+\J<UX.
once
and
,vy >
so that
h < % . VJ > = < "j >Vj> + h < %• - v ; >
or < u , , v , > = 0
Q.E.D.
Note that it is also easy to prove that < u x . , v / > = <u,,v x . > .
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