Ockendon J., Howison S., Lacey A., Movchan A. Applied Partial Differential Equations

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258 PARABOLIC EQUATIONS

The principal eigenfunction is ¢o = cosh px cos vy, with

A0 =

- 1 A

it = tank-1 Q,

v =

tan-1

where 0 < v < 7r/2. Thus Ao = 0 if

tan-1

a = tanh-1 fi, while if Q > tanh(a/2)

then, no matter how large a is, Ao < 0 and the trivial solution is unstable.

6.4.3.3

Transform methods

When boundaries are absent in heat conduction problems, the spatial operator

has a continuous spectrum, which is, of course, why we used Fourier transforms

rather than Fourier series to obtain (6.30). Moreover, solutions are often sought

over an infinite time interval 0 < t < oo, which also suggests the possibility of

Laplace transforms. However, it can often be puzzling to try to decide the best

variables in which to attempt a transform, and experience is the only guide to

which transform is easiest technically. Hence we include some further examples of

transform solutions.

Example 6.4 (Fourier versus Laplace transforms (i)) Suppose that, instead of

taking a Fourier transform, we had tried to derive (6.32) by a Laplace transform

in which

u(x, p) =

fu(x,t)e_Ptdt.

Then we would have obtained

d2u

axe -

P i _ -9(x),

and, since we want u to decay as Ixi -+ oc,

2JPU(x,p) = f g(t)e-d +

dC,

oc x

where f is defined to have positive real part. In order to retrieve (6.32) we now

have the irksome task of reversing the order of integration in the Laplace inversion

of u, which involves using the fact that the inverse Laplace transform of e-fPx/f

is e-x2/41/

9rt.

Example 6.5 (Fourier versus Laplace transforms (ii)) Suppose we try to solve

the boundary value problem

with

_ 02u

Ot Ox2'

u(O,t) = h(t) fort > 0 and

u(x,0) = 0 for x > 0,

by taking a Laplace transform in t. We find

u(x,p) = h(p)e x11_P,

(6.40)

where, as usual,

t fp > 0. Hence, by the convolution theorem,

THE HEAT EQUATION 259

tu(x, t) = fh(s)w(x, t - s) ds, (6.41)

where w is such that w(x, p) = e-=\. After some manipulations of the inversion

integral, it can be shown that

w(x, t) =

f e

pt sin

x VP- dp,

(6.42)

which can in turn be shown to be equal to xe'=2I4t/2

7rt3 (see Exercise 6.13 for

further discussion).

Things are much easier if, motivated by the nature of the spectrum of d2/dx2

with a zero Dirichlet condition at x = 0, we define the Fourier sine transform as

in §5.7.2 by

u, (k, t) = fu(x7t)sinkxdx.

This gives

u,(k, t) = e-k21 f kh(r)ek2, dr,

and, by inversion,

u(x, t) _

f ke-k2t

sin kx

f h(r)ek2T

d7-

dk,

0 o

which is just (6.41) combined with (6.42) when we reverse the order of integration

and write k = f.

Example 6.6 (Another two-point boundary value problem) Let us again solve the

heat conduction equation, but now with the conditions

u(ir,t)=g(t) fort>0, u(0,t)=0

fort >0,

u(x,0)=0 for0<x<ir.

We could try separating the variables, but imposing the boundary condition at

x = 7r is difficult. We could use the Green's function (6.38), but the Laplace

transform is even easier because

u(x, p) = 9(p)

sinh fx

sinh fpr'

and we have a straightforward convolution representation as long as we can invert

sinh fpx/ sinh fprr. The only singularities of this function are simple poles at

f = fin, n = 1, 2,3,..., and the corresponding residues of e' sinh fpx/ sinh 'Pr

are (- 1)"+1(2n/ir) sin(nx)e-n2t. Hence

2 °O

u(x,t) =

J(_1)'n sin(nz)e' ey(t-s)ds.

n=1

260

PARABOLIC EQUATIONS

*6.4.4

Convection-diffusion problems

The inclusion of a convective term (v V)u alongside the time derivative in the

heat equation makes the spatial partial derivatives non-self-adjoint. This makes

it awkward to use eigenfunction expansions because non-orthogonality means we

have no obvious recipe for the coefficients. Of course, if there are no boundaries

and v is constant, then it is possible to remove the convective term by changing

position to x - vt. To illustrate the intricacies that can arise when v is not con-

stant, we conclude this section by taking the Laplace transform of the simplified

convection-diffusion equation (6.4) for pollutant flux in a river. For simplicity, the

term Oc/8x2 is neglected and the river is assumed to be two-dimensional, so that

c = c(z, y, t). We assume that the flow is in the x direction with speed v(y) > 0,

that the river occupies 0 < y < 1, x > 0, that there is no pollutant flux at the

banks y = 0, 1 and that the water is initially pure. Thus the Laplace transform

c(x, y, p) = for' c(z, y, t)e-PI dt satisfies

D- 8

for0<y<1,x>0,

with

=o aty=o,1 and

e =iq(y,p)

at z = 0,

where co(y, t) is the input of pollutant at x = 0.

This complicated problem for c can be best solved by an eigenfunction expan-

sion resulting from separating the variables in x and y. The result is

c(x, y, p) _

(y, p),

(6.43)

M=0

where 0,n are normalised eigenfunctions satisfying

D (P - \.v(y))*m = 0,

with

dim

=0 at y = 0,1,

and

. =

Cotkm dy

It is only at all easy to find An when p is small; this can be shown to correspond

to t being large, which is often a limit that is of great practical interest. As shown

in Exercise 6.19, for small p

ao =

Dope

.}.

vo v:

THE HEAT EQUATION 261

where vo = fo v(y) dy, and D° is as given in Exercise 6.19. This means that the

inverse Laplace transform of the first term in (6.43) satisfies the famous Taylor

diffusion model

at +v0ax = D°ax2;

we can see this simply by taking the Laplace transform of the Taylor diffusion

model and examining its behaviour for small p. The effect of the y dependence

of it, which represents the `shear' in the flow, is thus equivalent to longitudinal

diffusion in the x direction.

Convection-diffusion problems become even more difficult to solve when the

velocity changes sign in the region of interest. I" Suppose, for example, we consider

the problem above in the steady state but with v(y) = 2y - 1. This means we have

to solve a parabolic equation which is forwards in x in y > 1 and backwards in

x in y <

Hence we do not expect to be able to prescribe an input everywhere

at x = 0 but only the values of c(0, y) for a < y < 1, and similarly c(L, y) for

0 < y < 1 for some L > 0.114 In practice, if we try to separate the variables as in

the earlier examples, we find

C(x, Y) _ E Ame-"-z0m(y),

where the eigenfunctions w. satisfy

Dd2oln

+ am(2y - 1)Vj,,, = 0,

dOm

(0) =

dV;m

(1) = 0.

dye dy

In fact 1,bm is an Airy function (see Exercise 4.10) and the eigenvalues \,n are both

positive and negative, whereas they were all positive in the simpler cases discussed

earlier. Although the V,,,, are complete in 0 < y < 1, they are not complete in

either 0 < y < 1 or .1 < y < 1, and hence the boundary conditions

CA Y) = EAmtI'm(y),

c(L,y)

.4me_.1'"Llp.(y)

do not yield the coefficients Am in a straightforward way.

It is an interesting attribute of parabolic equations that several of the examples

above are susceptible to a totally different method of solution which can even be

applied to nonlinear problems, and this is what we will describe in the next section.

113This happens frequently in industrial processes such as distillation, where 'counter-current'

mass transfer takes place.

114There are many other more general 'forward-backward' problems, often called Cevrey

parabolic problems, such as steady problems in which v changes sign across a general bound-

ary.

262

PARABOLIC EQUATIONS

6.5 Similarity solutions and groups

It is an immediate observation that, if u(x, t) satisfies

8u _ 02u

at Oxa'

then so does the function u(px, p2t), where it is an arbitrary constant. This sug-

gests the possibility of finding solutions for which

u(x, t) = u(px, µ2t) (6.44)

for all x and t. Now, for any particular value of t, we can always set It = 1/ f, so

this identity means that u = F(x/ f) for some function F. In fact, when we write

rl=x/f we soon find

F+2dF=0'

i.e. F=Aerf(2)+B,

where erf y is the error function (2/Va) fo e-°' ds, and A and B are constants.

This observation puts us in a position to solve the beat equation subject to any

initial/boundary conditions for which the data is invariant under the transforma-

tion

x= 1x', t=

t',

u=u'.

For example, with u(0, t) = I and u(x, 0) = 0,

rx/2f

u(z,t)=1-=J e-8

2 ds=erfc(2

where erfc y is the complementary error function.

(6.45)

This solution procedure can be somewhat mystifying at first sight, so we now

give a more systematic account of what are commonly called similarity solutions.

The basic idea is very simple. We just ask the following question.

`What changes of the dependent and/or independent variables make, or leave,

the equation autonomous in one or more independent variable(s), i.e. cause

those variable(s) to appear in the equation only through differentiation?'

The reason that this is interesting is that if the equation is autonomous in x then

it has solutions independent of x. Of course, these are special solutions, but firstly

they are much easier to find than the general solution, and secondly we may be

lucky enough to be able to use them to fit the initial and boundary conditions.

Indeed, we have seen this process in action in (4.15), where we reduced the number

of independent variables from two to one and found the R.iemann function of the

telegraph equation by simply solving an ordinary differential equation. In another

vein, even if our solution fails to satisfy all the requisite conditions, it might still

give us an approximate answer in some limits, such as for small or large time.

SIMILARITY SOLUTIONS AND GROUPS 263

Indeed, in (6.34) we have already seen the solution of a general initial value problem

tending to a function that has a special form as time tends to infinity.

Although no systematic answer is available to the question just posed, one

very helpful observation can be made. This comes from studying what happens

to a function of just one variable, say f (x), when we make an arbitrary change of

variable. If f (x) appears as part of the solution of a differential equation. and hence

needs to be differentiated and subject to other manipulation, any change of variable

x' = g(x) in general generates all the complexities associated with 'functions of a

function'. However, these mostly melt away if we consider a particular family of

transformations,

x' = 9(x,,\),

(6.46)

where A is a continuously varying parameter' 11 chosen so that g(x, 0) = x, provided

that g satisfies one, admittedly stringent, condition. This is that

199

(x; A) =

Fi (.\) F2 (9(x; A))

for some F, and F2. If this is the case we can reparametrise so that

= F(9),

a1\

(6.47)

and, using the fact that g(x, 0) = x, we can see what happens to f . After a little

manipulation of the relevant series, we find

f (x') = f (9(x; A))

2 29

x+a a

+ a, a2

+...

= f

& x=o 2. as

1X=0

f(x)+A 0gl

f'(x)

as=o

+ ! (aa9 I

a=o

f ,(X)

aa(3)2I

af"(x))

+ .. .

Now, if we define the infinitesimal generator U by the operation

of (x) = (L0 ) f (x) =

then equation (6.48) collapses into

(6.48)

(6.49)

f(x') = f(x)+Auf(x)+

21U2f(x)+

(6.50)

This is because

15The parameter µ in (6.44) is equal to e', as we shall see below; this ensures that g(x, 0) = x.

264 PARABOLIC EQUATIONS

) 21'(x)U2f (x) _ ( 89 I

8xd

(

8 A

)

0A I

(Ia=o)

f'(x),

a-o

a=o

OJX

and we know that

and, from (6.47),

d Og

dx 0aI

= P(X)

A=O

029 I

= F,(9)' I

= F'(x)F(x).

a=o

Thus (6.47) is the key ingredient that allows (6.48) to be written as (6.50); this

procedure can be applied to all orders in A and the series can be formally summed

f (x') = exp(AU)f (x)

To apply this idea to differential equations, we have to deal with at least two

variables, whether dependent or independent, in which case (6.46) and (6.49) gen-

eralise to

with

x' = 9(x,y;A), y' = h(x,y;A),

(6.51)

and

G(9, h),

Lh = H(9, h),

(6.52)

9(x,y;0) = x,

h(x,y;0) = y,

(6.53)

U = GJa-o 0 + Hja=o

(6.54)

6.5.1

Ordinary differential equations

Suppose that x and y are, respectively, the independent and dependent variables in

an ordinary differential equation. The statement that the equation is autonomous

in x is equivalent to saying that it is invariant under the transformation x' =

x+A, y' = y, which clearly satisfies (6.52). More generally, we might ask `Could the

given ordinary differential equation be made autonomous by a change of variables?'

The answer is yes if it is invariant under a transformation (6.46), because we simply

change to a new independent variable X and a new dependent variable Y such

that UY = 0 and UX = 1. This means that the differential equation for Y(X) is

SIMILARITY SOLUTIONS AND GROUPS 265

invariant when X is translated by an arbitrary constant and Y is left undisturbed.

For example, consider the linear equation

x2 +

4(x)y = 0, (6.55)

which is not autonomous but is invariant with g = x, h = e''y, G = 0, H = h and

U = y 8/8y."6 This tells us that, if we take, for example, Y = x and X = logy in

(6.55), then we obtain an equation autonomous in X; in fact, it is

TX2/\d7_

/\dXl2=4(Y)

In this case we do not use the autonomy directly to seek a solution independent

of X; rather we note that the autonomy ensures that the order of the differential

equation can be lowered, say by considering dY/dX as a function of Y. Equally,

suppose we happen to know that yo(x) satisfies (6.55). Then the equation is in-

variant with g = x and h = y + Ayo(x). This tells us to take, say, Y = x and

X = y/yo(x), which is the well-known rule for lowering the order of a linear ordi-

nary differential equation when one solution is available.

6.5.2

Partial differential equations

To keep things as simple as possible, we begin by just considering transformations

of the independent variables. Thus, suppose that x and y are independent variables

in a partial differential equation for u(x, y) and we apply a transformation (6.51)-

(6.53) which leaves the equation invariant. Now, in order to reduce the number of

independent variables from two to one, all we have to do is solve for a new variable

Y such that lAY = 0, and transform to Y and any other independent variable, X

say. We then have a partial differential equation which is invariant under a one-

parameter change of variables in X; hence it has a solution in which u = F(Y).

For example, in the calculation before (4.15),

8xey+R=0,

the transformation g = µx, It = y/µ leaves the equation invariant. Then, writing

p = ea gives

G=g=eax, H=-h=-e-ay,

U=xB--yya,

and hence Y = xy. More relevant for this chapter, the transformation x' = G =

g = eax, t' = H/2 = h = e2-\t leaves the equation 8u/8t = 82u/8x2 invariant, so

U=xB +2t8

;

thus, as observed at the beginning of this section, a possible choice of the

`autonomous' independent variable, which we called Y above, is x2/t.

116 We write h

= eay rather than It = ay for convenience, because it means that A = 0 is the

identity and no reparametrisation is needed to obtain (6.47).

266

PARABOLIC EQUATIONS

Of course, similarity solutions can always be written down by inspection by

people clever enough to spot them. The point about (6.51)-(6.53) is that it pro-

vides a relatively systematic procedure for finding the similarity solutions once

g and h are available. This method is far from trivial, as will be illustrated on

the following pages, and many computer packages are currently available for au-

tomating the procedure and some of its generalisations. Such generalisations will

not be considered here except to remark that we could consider firstly `extended'

transformations in which the functions g and h depend upon the dependent vari-

able and its derivatives as well as the independent variables, or, secondly, multi-

parameter transformations in which g = 9(x, AI , A2) and (6.47) is replaced by, say,

a9/aA1 = F(11(9) and 09/0A2 = P2)(9).

Concerning terminology, the requirements (6.47) and (6.51)-(6.53), together

with the existence of inverses, are conditions for g in (6.47) or (g, h) in (6.51)-

(6.53) to form what is called a Lie group or continuous group of transformations.

The trivial manipulations necessary to show that (6.47) is equivalent to the crucial

group-closure condition, namely that, for all A and A, g(g(x; A); µ) = g(x; v) for

some v = v(A, p), are given in Exercise 6.20.

With these thoughts in mind, we are now in a position to describe a more sys-

tematic way of finding similarity solutions of the heat equation than that described

at the beginning of this section. Suppose we consider a general continuous group

of transformations of the independent variables

x' = f (x, t; A),

t' = g(x, t; A),

(6.56)

where the group parameter is such that A = 0 is the identity, so that x =_ f (x, t; 0)

and t = g(x, t; 0), and the closure operation is addition, which can be achieved

without loss of generality as in (6.51)-(6.53). Thus, for small A,

x'=x+AU+ t'=t+AV+

(6.57)

where the components U and V of the infinitesimal generator U = U a/ax+V a/at

are just functions of x and t. This means that, for small A, the chain rules for

changing the variables are, to 0(A),

8t -

+ A 8t)

8t'

+ A 8t ax''

a ' (1

- A

at) at at ax

a_ aV a aU) a

aV 0

(

aU) a

ax =

ax at,

1 + A

ax''

ax' - -A

x at

1 - A

ax l ax'

(6.58)

We can now enforce the crucial invariance property by selecting U and V, and

hence the group, by the condition that the heat conduction equation be left in-

variant under the transformation. Using the chain rule as above, the heat equation

with x' and t' as independent variables is

SIMILARITY SOLUTIONS AND GROUPS 267

av au

aU Ou

a2u

a2U au

&V DU

1 +a

ax,

`ax2ax,+ax28t'/

aU a2u

air

82v

(6.59)

XIM,

to lowest order117 as A -+ 0, and so we require

aV aU

02U

02v

aU (

T = 0'

at = axe ,

8t = axe

+ 2

a'x .

6.60)

At first sight it looks as if we are no better off than we were with the heat conduc-

tion equation, but (6.60) is an overdetermined system of three equations in only

two unknowns and it soon transpires that the only solutions are

U=T +d,

V=ct+e,

(6.61)

where c, d and e are constants. Hence, when d = e = 0, we can use ax'/49a = cz/2

and at'/a\ = ct to find

x' =e Ac/2 X,

t' = ea`tt,

(6.62)

which gives (6.44) with a change in notation. Alternatively, with c = 0, d = v and

e = 1, we have

x'=x+Av,

t'=t+,\,

(6.63)

leading to `travelling wave' solutions of the form u = F(x - vt); such a Galilean

transformation is always possible for autonomous partial differential equations.

Concerning the procedure above, we note the following points.

1. The independent variables such as x/v' and x - vt, corresponding to (6.62)

and (6.63), respectively, could be enumerated quite systematically by solving

for a variable f such that

U =(Ux+v8/f=0. (6.64)

2. In all cases u must satisfy a suitable ordinary differential equation. For (6.63),

we find

d2F

dC2

+vd =0,

so F = A + Be-v(x-91), where A and B are constants.

3. Systems such as (6.60) can easily look formidable, even though they are always

linear and overdetermined. One of the boons of modern symbolic manipulations

is that several packages are available to automate the process of deriving and

solving such systems, and we will return to this in Chapter 9.

117In fact, using the group property (6.51)-(6.53) it can be shown that this implies invariance

to all orders in A.