On a class of similarity solutions of the equation ut(|u|m-1ux)x with m -1

On a Class of Similarity Solutions of the Equation

u, =

with m > - 1

C . J. VAN DULTNt, S. M . GOMESt1, AND ZHANG HONGFElf2

t Department of Mathematics and Informatics, Delft University of Technology, PO Box 356, 2600 AJ Delft, The Netherlands

% Instituto de Pesquisas Espaciais (INPE), CP 515, 12.201 Sao Jose dos Campos (S.P.), Brazil

[Received 28 September 1987 and in revised form 10 March 1988]

Self similar solutions u(x, t) =f{xljt) of the one-dimensional porous-medium equation are studied in this paper. These solutions emerge from initial values that consist of two constant states: one non-positive for x < 0 and one non-negative for x > 0 . With a diffusivity of the form IKI"1"1 we consider, for m>0 sign-change solutions, and for w e (—1,0] non-negative solutions. The method we use is based on a transformation which maps the ordinary differential equation for/into a singular elliptic boundary-value problem with Dirichlet conditions. Special attention is given to the behaviour of / near zero. We also present a number of numerical results.

1. Introduction

IN THIS paper we study the solution of the initial-value problem

where m e ( - l , ° ° ) and where a e [ 0 , l ] if m > 0 and a = l if me(—1,0].§ Equation (1.1) arises in many different physical models for various values of m. For instance when m > 1 and u 5* 0, it models the flow of a gas through a porous medium. In that case, u denotes the density of the gas, see (Aronson, 1987) or (Muscat, 1946). When m = 2 and there is no restriction of the sign of u, it describes the mixing of fresh and salt groundwater due to mechanical dispersion (de Josselin de Jong & van Duijn, 1986). Then u denotes the rescaled velocity of the fluids. For m = \ and u 5= 0 it arises in plasma physics (Berryman & Holland,

On leave from: ' Departamento de Matematica, Facultad de Ingenieria, Universidad de Buenos Aires, Argentine; 2 Department of Applied Mathematics, Tsinghua University, Beijing, China.

§ The fact that the initial value in problem P(m, or) jumps from a -1 to a implies no loss of generality. This can be seen from an easy rescaling: if the initial value has the form u(x, 0) = A < 0 for JC<0 and u(x, 0) = B>0 for x > 0 , one can rescale the solution according to C~xu(x, C1""1*) with

C = \A\ + B. This leads to a = B/(\A\ + B). 147

1 4 8 C. J. VAN DUIJN, S. M. GOMES AND ZHANG HONGFEI

1980) and for m = 0, u s= 0 it occurs in Carleman's model of the Boltzmann equation (Berryman & Holland, 1982).

The existence and uniqueness of solutions of P(m, a) follow from the general existence and uniqueness results for equation (1.1). If m > 1 and a e [0,1] or if m e (0,1) and a e [0,1], the existence of distributional solutions follows from (B6nilan et al, 1984) and (Herrero & Pierre, 1985), respectively. For these cases the uniqueness follows from (Brezis & Crandall, 1979). If m e ( - 1 , 0 ] and a = 1, the existence of a smooth solution (for t > 0) is given by Esteban et al. (1987). They also show uniqueness within the class of maximal solutions: that is, solutions which satisfy the growth condition

(i) for m e ( - 1 , 0 )

um(x,t) = o(\x\) as |*|-*co, (1.3)

and

(ii) for m = 0

-\ogu{x,t)^o(\x\) as |*|-°o, (1.4) uniformly in t e (r, °°) for any r > 0.

It was observed by Vazquez (1984), that if u = u(x, t) solves P(m, a), then also v(x, t):=u{kx, k2t) is a solution of P(m, a) for any Jfc>0. Since solutions are

unique, we must have

u(kx, k2t) = u(x, i) for ( i , ( ) e U x K+,

and for any k > 0. This implies that u is a self-similar solution of the form u(x, t) =f(x/Jt) for (x, t)€ U xU+. Further, if u is a solution of P(/n, a), then/

should satisfy the boundary-value problem

s

<—> t r t V ^ A - T - ;

(15)

where r\ =x//t is the similarity variable and where primes denote differentiation with respect to r\. Thus existence and uniqueness for P(m, or) implies existence and uniqueness for solutions of S(m, a) within an appropriately chosen function class. In this paper, however, we propose to study S(m, a) directly with ordinary-differential-equation methods.

The case in which m > 1 and or = 1 was considered by van Duijn & Peletier (1977). There existence and uniqueness was demonstrated by matching solutions on U~ and U+ together at r\ = 0, such that both/and (fm)' are continuous. As a

result they found that there exists a unique number u> eU~ such that = 0 for —oo < T) «£ (0, is positive, smooth and strictly increasing for to < T; < oo, and near to the solution satisfies (Craven & Peletier, 1972)

l i m - ^ - ( / " -1) ' ( ^ ) = - ^ (W. (1.7) This behaviour is to be expected from the general theory for non-negative

solutions of the porous-medium equation (1.1). The constant w in (1.6) defines an interface f (f) = <ojt such that

= 0 for-°o<;t=s£(0, t^O .>0 for £(t)<x«x>, {3=0.

«(*,'){.

for/ 6 (or- 1, a),

on (a- - 1, a),

(1.9) (1.10) (1.11) Equation (1.7) gives the interface equation

lim vx(x, t)q = - £(f) for t > 0, (1.8)

x i 6(0

where

1 m_1

is related to the pressure for the porous-medium application (see (Aronson, 1987; Peletier, 1981) for survey papers on this subject). Similar results hold when m > 1 and a = 0: to see this we need only replace / by —/ and i\ by — r\.

In studying S(m, a), we base our analysis on a transformation which maps this problem into a singular elliptic boundary-value problem on the interval (a — 1, a) with Dirichlet boundary conditions. Following Bouillet & Gomes (1985) we are led to consider the transformed problem

T(m, or)

where primes denote differentiation with respect to /.

A solution y of T(m, a) must be understood in the sense that y e C[a — I, a], satisfying (1.10) and (1.11), and y' is locally absolutely continuous on (a - 1 , a) such that equation (1.9) holds almost everywhere.

In Section 2 we prove for T(m, a) the existence and uniqueness of solutions y. When m>\ and a=\ the right derivative y'(0+) exists and similarly, when

m > 1 and a = 0 the left derivative y'(0~) exists. Note that these are precisely the cases covered by the results of van Duijn & Peletier (1977). For all other combinations of m and a we find that

y'(a-l) = +°° and y'(a)=-oo. (1.12)

Restricting to cases where (1.12) holds, we show below how to obtain the solution / o f S(m, a). Since any solution y of T(m, or) is strictly concave on (or — 1 , or), it

follows that y' is a continuous and strictly decreasing function on (a —I, a). Then we define / : I R - » ( a r - l , a), according to

/ ( / ( 1 ) ) = -»J for i| 6 R. (1.13) It is straightforward to verify that (i) / is strictly increasing on U such that /(-oo) = a - 1 and/(+°°) = a, (ii)/and \f\m~lf are absolutely continuous on R

and (iii) equation (1.5) is satisfied almost everywhere. Conversely if/is a solution of S(m, a) in the sense indicated above, then y : [a — 1 , ar]-» [0, °o) is defined

150 C. J. VAN DUUN, S. M. GOMES AND ZHANG HONGFEI

according to v(/(r,)) = 2 I/I"-1 /'(»?) forr/eR, (1.14) and one verifies that y satisfies T(m, a).

In Section 3 we discuss the behaviour of / when it tends towards zero. In the case of a sign-change solution (m > 0, a e (0,1)) we set

<o:=-y'(0) and A: = £y(0)>0. Then

/(<o) = 0 and |/r"

/'(<») = ^

We find that \fr^f'()k

1 J Z

r s i g n / ^ ^

m r] — co

Observe that for ar= 1 and WJ > 1, the flux A = 2v(0) is 0, and expression (1.15) reduces to (1.7).

In terms of the solution u of the initial-value problem P(m, a-) equations (1.15) and (1.16) imply the following.

Let £ : [0, °°)-» U be given by

Then

in {(x, t) | -°° <*<£(/;

in{(x,t)\£(t)<x<°o

Thus £ denotes the interface separating the regions where u > 0 and u < 0. Equation (1.15) defines the interface equation

for f>0.

Expression (1.16) gives the behaviour near the interface as

° "

(l

"

r sign uUm

' °

= ri

~

im

°(l* - ^

r

)l

1/m

)

as x-» £(f), for every / > 0 . This expression shows that u(-, 0 e C1/m(IR) for each t > 0, giving the optimal regularity in space for solutions of the porous-medium equation with sign-change; see also (Be"nilan, 1987).

In the case of a positive solution (m e (—1,1) and a = 1) we have a precise result about the behaviour of/when r\ —» -°°. We obtain

lim

-, .. T-,-»-» \ 1 — m and thus

lim -xu^-

\x,t) = (lt-\^^) for each f>0,

*-.-» \ 1 — ml

from which the growth conditions (1.3) and (1.4) directly follow. Thus u is indeed maximal in the sense of Esteban et al. (1987).

In Section 4 we consider the limit for m \ — 1. Denoting the solution of S(m, 1) (m e ( - 1 , 0 ) ) by fm, we find that

fm \ 0 as m \ — 1 uniformly on (—°° , L),

for every L > 0 . For the corresponding solutions um this implies that for every

L > 0

um\0 as m \ — 1

uniformly on sets of the form {(x, t) | — °° < x < LJt, t > 0}.

Finally in Section 5 we present a number of numerical results. To find a solution / of S(m, a), we proceed as follows. First we solve T(m, a) for y. From this solution we obtain an approximation to y and y' at some point /o e ( a — 1, a). Then we use (1.13) and (1.14) and a shooting argument to solve

S(m, or): that is, we solve the initial-value problems

(I/I"-

/')' + W =0 for r, > -y'(f

) and r, < -y'(f

),

A good check for the accuracy of the values for y(f0) and y'(f0) is that the

boundary conditions /(—°°) = a — 1 and /(+°°) = a are achieved, although in some cases the convergence is slow (see (1.17)). When dealing with a sign-change solution, we choose f0 = 0.

We conclude this introduction with the remark that non-negative solutions of equation (1.5), for various types of boundary conditions, have also been studied with phase plane methods. A clear example of this approach is given by Atkinson & Jones (1974). The emphasis there, however, is more on the analysis of the trajectories and not so much on the qualitative behaviour of solutions of the original differential equation.

2. Existence and uniqueness for T(m, a)

Consider for arbitrary a e [0,1] the auxiliary problem

-yy" = k on ( o r - 1 , a), (2.1)

BVP \y(a-l)=y(a) = 0,

[

where A: is a given measurable function which is defined and non-negative almost everywhere on (a - 1, or). A solution of BVP must be understood in the sense of T(m, a). The following comparison result holds (see also (Bouillet & Gomes,

1985; van Duijn & Peletier, 1977)).

THEOREM 2.1 Suppose that 0^kl^k2 almost everywhere on (or - 1 , a). If, for

1 5 2 C. J. VAN DUIJN, S. M. GOMES AND ZHANG HONGFEI

Proof. Set w=yt — y2. If the assertion is not true, then there exist a, b with a-l^a<b^a such that w > 0 on (a , b) and w(a) = w(b) = 0. Using (2.1) we

k k

w"-—* + —«£0 a.e.on(a,b). yi yi

This implies that w =s 0 on (a , b), which gives a contradiction.

COROLLARY 2.2 Problem T(m, a) has at most one solution.

Next we consider the existence question. We base the analysis on the fixed-point formulation of BVP:

(

() f

'

^

farall*

[«-l,«r], (2.2)

where g(x, •) denotes the Green's function ((a-x)(s-a +

for every x e [a — 1, a].

Our starting point is an existence result for FP, which is given in (Bouillet & Gomes, 1985) for a similar problem and also in (Gomes, 1986), where a multi-dimensional and more general form of BVP is studied. We make it the content of the following lemma.

LEMMA 2.3 Let k^O almost everywhere in {a — 1 , a) such that meas {k >0} > 0 and

k(s)/(a - s)(s - a + 1) e V(a - I , a).

Then FP has a solution y e C\a - 1 , a]. This solution also satisfies BVP. We are now in a position to prove the following.

PROPOSITION 2.4 Let k^O almost everywhere in (a — 1, a) such that

meas {k > 0} > 0 and k e L}(a — 1 , a). Then there exists a solution y of FP and of BVP, which satisfies

y'(ar - 1) < 0° if and only ifk(s)/(s - a + 1) e V(a -I, a), and

y'(a)>—<x> if and only if k{s)l (a - s) eL,\a — 1, a). Proof. For sufficiently large n e M we define

f 1 11 k(x) i f j t e a - l + - , a r — ,

J L n n\

0 ifxe (a-l,a-l + -)u(a--,aj.

yn e C1[cr — 1 , a] such that

for

Since yn is concave on (or - 1 , a), we have for each x e (a - 1 , a)

, , s-a + 1 ..

x - a

a-s

oc — x if x =£ s =s a. Using this in (2.3) gives

, ( o r - * ) ( * - a + 1)

_{(x - a + 1) f}

0 <yn(x) ^ {(or - *)(ac - or

(2.3)

(2.4)

(2.5)

follows for all x € (a - 1, a) and for all sufficiently large neN. Thus {yn} is a

non-decreasing sequence (Jcn \ k and apply Theorem 2.1) of uniformly bounded

functions. Consequently, there exists a function y defined on [a — 1, a], such that

yn S* y as n —» », pointwise on (or — 1, a).

Next fix n0 e N, large enough. Then we have, for all n»n0 and x e [a - 1, a],

•, a — Ks <x,

In (Bouillet & Gomes, 1985; Gomes, 1986) it was shown that, on (a - 1, a), yno(s)/(a— s)(s — a + 1) is bounded below by a positive constant. Hence there

exists C > 0 such that

g{x,-)-<*CkeV(a-l,a)

yn

for every x e [a - 1, a] and all n 5* n0. Hence we may pass to the limit on both

sides of (2.3). This establishes existence for FP. A direct verification shows that y also satisfies BVP. The assertions about the derivatives at the end points follow from an argument of Atkinson & Peletier (1974); see also (Bouillet & Gomes, 1985; Gomes, 1986).

154 C. J. VAN DUUN, S. M. GOMES AND ZHANG HONGFEI

THEOREM 2.5 Let m>0 and a e [0 ,1]. Then T{m, a) has a solution y. If m > 1 and a = l, then y e C'[0 ,1) and if m > 1 and a = 0, then y e C ' ( - l , 0]. In all other cases y'(a - 1) = +°° and y'(a) = — °°.

Next we consider the case in which m e (—1,0] and a = 1. Then the right-hand side of (1.9) is not in L'(0 ,1). We approximate it by an increasing sequence {kn}

of L'(0, l)-functions:

kn(x) = 2 min {n, xm~1} for* e [0,1] and n 3= 1.

By the previous results there exists a non-decreasing sequence {yn} of solutions

of FP and BVP, with k replaced by kn.

First we establish a uniform bound on yn. LEMMA 2.6

0<y

(x)< ( Y T ^ )

f°

(°' *)

Proof. A trivial computation shows that

( ^ y

, *^0, (2.6)

satisfies — yy" = 2xm~l, v > 0 for x > 0 and _y(0) = 0. A comparison argument as in

the proof of Theorem 2.1 then gives the second inequality. The first inequality follows from the concavity of the solutions.

From this lemma and the monotonicity of {yn},

ynSy asn-*•<*>, pointwise on (0,1). (2.7) Next we prove the following.

LEMMA 2.7 For each x e ( 0 , l ) there exists N eN such that for every n>N

where C is a postive constant independent of x and n.

Proof. First we construct a lower bound for yn on [0,1]. Fix n e M, and set

Then kn(x) = 2xm~1 for xe({xo,l). From equation (2.2) and the definition of

g(x, s) it follows that

- ^ f d r

x

,1). (2.8)

•'o yn\s)

For x 3= x0, we have

where we have used Lemma 2.6. Thus

y

(x) >

{ ^ Y ^ } *

• (1 - (i/2)*

"

V

For x < x0, we have

- x).

3 — m -x).

Next we estimate t h e integrand in (2.2). Fix x e ( 0 , l ) a n d choose neM sufficiently large so that x0 < x. T h e n we use the lower bounds t o obtain

g(x,s)^f{ =

i2(l-x)-1

\ 0<s<x

,

Xo < S < X,

i2(l-s)-where Q and C2 are positive constants only depending on m.

From this lemma it follows that for each x e (0,1) we may use the dominated convergence theorem and pass to the limit for n —> °° in the approximate version of equation (2.2). Consequently, y is a solution of T(m, 1). Repeating the first part of the proof of Lemma 2.7 gives

1 + m for* e [0,1]- (2.9)

Then clearly y'(0+) = °°. Using again an argument of Atkinson & Peletier (1974)

one also finds that y'(l~) = — °°. Thus we have shown the following.

THEOREM 2.8 Let m e ( - 1 , 0 ] . Then T(m, 1) has a solution y which satisfies _y'(0+) = oo and y'(1~) = - » .

3. Behaviour o f / near zero

We consider two cases.

(i) a e (0,1) and m>0

Let / denote the solution of S(m, a) with / ( - » ) = a-l<0 and / ( + » ) = a>0. Further let a>, keU be given by

1 5 6 C. J. VAN DUIIN, S. M. GOMES AND ZHANG HONGFEI

where A denotes the flux at r\ = co. The behaviour of/near co is given by the next two results.

THEOREM 3.1

Proof. Integrating equation (1.5) from co to r\ =£ co gives

l/r-Vto) - A + hrifin) ~ I f/(*) * = 0, (3.1)

or

Since / is strictly increasing, the right-hand side of this equation can be simply estimated. This gives

from which the result immediately follows.

COROLLARY 3.2

r )

„

Proof. Integration of equation (3.1) yields

- | / r sign/(»j) - A(iy - co) + \ f\n - 2s)f(s) ds = 0. (3.2)

Since

l/l"'~7'(7?) = 2.v(/(j?)) ^ 2ymax On R » where _ymax denotes the maximum of y on [a—I, a], we have

I / M l ^ (2'">'inax)l/m \r) — co\Vm for f/ 6 U.

Using this in equation (3.2) gives

l | / r signer?) A ^2 f j (i ) 1 / m | i )_f f l r

REMARK 3.3 A formal expansion of / near r/ = co of the form

fm(r}) = k(r) — co){\ +a(r) — co)a + b(r] - co)p + . . . } , i) > co,

requires

and /3 = - , b = -(oa^Vm)-1mUm/2(m + 2) ifme(l,°o), m or j8 = l + - , b = -mkiUm)-1mVm/2(m + i)(2m + l) i f m € ( 0 , l ) , m or j8 = 2, b = -(a)a + $)/6 ifm = l. (ii) a = 1 and m e ( - l , 1)

Here we show that the similarity solution satisfies the asymptotic behaviour given by (1.17). This is a direct consequence of the next result.

LEMMA 3.4

x

\"o"

\~ \-m)

Proof. Since y satisfies equation (1.9), we obtain after integration

— - d s for* e (0,1), (3.3) v(s)

where b e (0,1) denotes the point where y attains its maximum. In this expression we use the upper bound (2.6). For x e ( 0 , b) this gives

and

lim inf xK

—V (x) > (2 • f

x j o \ 1 — m / Next we construct an upper bound. Lemma 2.6 implies that

on (0,1), or

(y —y*)' is strictly decreasing on (0 ,1).

Now suppose that there exists x e ( 0 , l ) for which (y-y*)'(x)^0. Then by the previous result, {y -y*)'(x)>0 for every xe(0,x). Integration gives (y ~y*)(x)> 0 f°r every x e (0, x), contradicting Lemma 2.6. Therefore

y'(x) <y*'(x) = (2 • l±^) x*

-»

158 C. J. VAN DUUN, S. M. GOMES AND ZHANG HONGFEI

for every x e (0,1) and lim!

I-ml ' this proves the lemma.

4. Behaviour for m \ - 1

Let/m and ym denote the solutions of S(m, 1) and T(m, 1), respectively, and let

bm =fm(0) denote the point where ym attains its maximum. We first prove an

estimate for bm. LEMMA 4.1

1

0 < 6m< (1 + m) for all me ( - 1 , 0 ) . m

Proof. We only show the upper bound. In (van Duijn & Peletier, 1977: Theorem 8) the following inequality was proved for solutions y of BVP:

rb fl

sk(s)ds«(1 - b ) \ k(s)ds.

Jo h For T(m, 1) this results in

bm<(-

^)(1

m)(l

from this the estimate follows.

Next we can prove the following.

THEOREM 4.2 For I e (0 , 1 ) , let ri'meU be given by fm(t]'m) = /. Then for every

/ e ( 0 , 1 ) , there exists M e ( - 1 , 0 ) such that for all m e ( - 1 , M)

"™> 2"i(-^)""*"<1-2") (

Proof. Again we use Lemma 2.6. A t x = bm this gives

We choose / e ( 0 , 1 ) . Then we combine equation (3.3), y'm(fm(r))) = - 7 / for TJ e I

and fm{q'm) = I, to obtain

J

^—ds.

This we estimate:

and with (4.1)

From Lemma 4.1 there exists M e ( - 1 , 0 ) such that bm e (0, \l) for all

m e ( - l , A f ) . We use this and the estimate for bm in the above inequality to

obtain the desired result.

COROLLARY 4.3 /m \ 0 aj m \ - 1 , uniformly on intervals of the form (—oo, L) with L > 0.

5. Numerical method and results

In this section we present some results of computations for solutions y of T(m, a) and solutions/of S(/n, or). We consider five cases:

a. m = 2, a = 0-75, b. m = 0-5, a = 0-75, c. m = 0-5, a = \, d. m = -0-25, a=l, e. m = -0-5, a = l.

To solve T(m, a) we use an iterative method combined with the finite-difference approximation for the second derivative.

Let Xi = a-l + iA (with i = Q,l,2, . . . , N and AN = 1) denote the equally-spaced points at which we want to compute the solution. Further, let yjn) denote

the value of the nth iteration at the point *,. Then given an initial estimate _y<0), i = 0, 1, 2 , . . . , N, we solve

-y\l\x) + 2y\n+l) - yt\x) = - 2 |JC,|—'/y^A2 for i = 1, 2, . . . , N - 1, (5.1)

with the boundary condition y$>"+i) = y%+1) = 0.

The number of iterations in this process strongly depends on the required accuracy and on the choice of the initial estimate. For the latter we take a function of the form of the upper bound in (2.5):

yP = {C(m){a - *,)(*, - a + l)}i, i = 0, 1, . . . , N.

Here the C(m) are experimentally determined constants, chosen to reduce the number of iterations. When a e (0,1) we take

ra-l

I*!"1"1 dx. C(m)=\°

This gives C(2) = 9/32 for case a and C(0-5) = 1 + / 3 for case b. In the other cases we take: c. C(0-5) = 0-7, d. C(-0-25) = 5 0 and e. C(-0-5) = 140.

In case b the right-hand side of equation (5.1) becomes unbounded if Oe {.x,}""1- Since by our choice of A this is indeed the case {x50 = Q) we cannot

160 C. J. VAN DUUN, S. M. GOMES AND ZHANG HONGFEI

apply equation (5.1) at this point. We deal with this difficulty as follows: near x = 0 we consider the linear equation

Integrating twice from 0 to x > 0 gives

Further, we replace the derivative by

y(n+1)'(0) = (y(n+1\x) - yln+l\-x))/2x.

This gives, with 0 = xk and x = A = xk+l,

4 Am+1

m[m +1) yk'

The results of the computations are shown in Figs 1 and 2. In each case we take N = 200. For the accuracy in the iterations we choose

max \yW — y("~l)\ < 10~6.

In all cases, the number of iterations required is in the order of 100.

To compute the solutions / of the corresponding problem S(/n, a), one could in principle use the transformation

/ ( / ( ' ? ) ) = -'/• (5.2) Here we use a different approach. At some point xp e {JC,},^!1 we compute y and

i i i i l i i i i i i I I I I I I I I I I

FIG. 2. Solution y of T(m, 1) for (c) m = \, (d) m = - j , (e) m = - |

its derivative. Then we know / (=xp) at the point r\p:= -y'(xp) and the

corresponding flux Ul"1'1 !'(%) = y{xp)l2.

Next we solve two initial-value problems

(I/I"1"1/')' + W = 0 f o r 1 > VP and r/ < y\p,

\f\

-

f'(v

)=y(x

)/2.

When the solution has a sign change, we take as a starting point in the above problem T?P = — y'(Q) a nd /(T/p) = 0- When m > 1, case a, we use a series expansion near r\ = TJP to obtain a non-zero value for fm in a neighbourhood from

which we continue with a Runge-Kutta method. This series expansion is given in Remark 3.3 and also in (de Josselin de Jong & van Duijn, 1986) for m=2. When m e (0,1), case b, we start directly from / = 0 with the Runge-Kutta-solution procedure for/"1. When the solution has no sign change, cases c, d and e, we take for xp the intermediate value xp = 0.24 and proceed as above. The results are shown in Figs 3 and 4. Notice the slow asymptotic behaviour as ?j—• -°° in the cases d and e.

0.75

' ' ' i i i i i i FIG. 3. Solution/of S(m, \) for (a) m = {, (b) m = 2

162 C. J. VAN DUUN, S. M. GOMES AND ZHANG HONGFEI

FIG. 4. Solution / o f S(m, 1) for (c) m = \, (d) m = -\, (e) m = -\

The advantage of this method is a very precise picture of the similarity solution /. The asymptotic expansion near zero automatically gives the desired regularity. Also the behaviour as r\ —» ±°° is very precise (a computation of / using the discretized version of expression (5.2) would never give an asymptotic behaviour as shown in Figs 3 and 4).

Our computations also indicate that the values of y are very accurate. A small deviation of y at xp, used as initial value in the above-mentioned shooting

procedure, will never result in the desired end value a as r\—>°° and a— 1 as

r]—»-oo.

REFERENCES

ARONSON, D. G. 1987 The porous medium equation. Some Problems in Nonlinear Diffusion. Lecture Notes in Mathematics 1224 (eds A. Fasano & M. Primicerio).

Berlin: Springer.

ATKINSON, C , & JONES, C. W. 1974 Similarity solutions in some non-linear diffusion problems and in boundary-layer flow of a pseudo-plastic problem. Q. Jl Mech. appl.

Math. 27, 193-211.

ATKINSON, F. V., & PELETIER, L. A. 1974 Similarity solutions of the nonlinear diffusion equation. Arch, ration. Mech. Analysis 54, 373-392.

BENILAN, PH. 1987 A strong regularity If for solutions of the porous media equation.

Contributions to Nonlinear Equations. Research Notes in Mathematics 89 (ed. C.

Bardos). London: Pitman.

BENILAN, P H . , CRANDALL, M. G., & PIERRE, M. 1984 Solutions of the porous medium equation in R" under optimal conditions on initial values. Indiana Univ. Math. J. 33, 51-87.

BERRYMAN, J. G., & HOLLAND, C. J. 1980 Stability of the separable solution for fast diffusion. Arch, ration. Mech. Analysis 74, 379-388.

BERRYMAN, J. G., & HOLLAND, C. J. 1982 Asymptotic behaviour of the nonlinear diffusion equation n, = (n~'/ix)T. J. math. Phys. 23, 983-987.

BOUILLET, J. E., & GOMES, S. M. 1985 An equation with a singular nonlinearity related to diffusion problems in one dimension. Q. appl. Math. 42, 395-402.

BREZIS, H., & CRANDALL, M. G. 1979 Uniqueness of solutions of the initial-value problem for u, - A<j>(u) = 0. /. Math, pures appl. 58, 153-163.

CRAVEN, A. H., & PELETIER, L. A. 1972 Similarity solutions for degenerate quasilinear parabolic equations. J. math. Anal. Applies 38, 73-81.

DE JOSSELIN DE JONG, G., & VAN DUUN, C. J. 1986 Transverse dispersion from an originally sharp fresh-salt interface caused by shear flow. J. Hydrology 84, 55-79.

ESTEBAN, J. R., RODRIGUEZ, A., & VAQUEZ, J. L. 1987 A nonlinear heat equation with singular diffusivity. Preprint, Universidad Autonoma de Madrid.

GOMES, S. M. 1986 On a singular nonlinear elliptic problem. SIAM J. math. Anal. 17, 1359-1369.

HERRERO, M. A., & PIERRE, M. 1985 The Cauchy problem for u, = A(um) when

0 < m < l . Trans. Amer. math. Soc. 291, 145-158.

MUSKAT, M. 1946 The Flow of Homogeneous Fluids Through Porous Media. Ann Arbor: Edwards.

PELETIER, L. A. 1981 The porous media equation. Applications of Nonlinear Analysis in

the Physical Sciences (eds H. Amann et al.) London: Pitman.

VAN DUUN, C. J., & PELETIER, L. A. 1977 A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Anal. Theory Methods Appl. 1, 223-233.

VAZQUEZ, J. L. 1984 The interfaces of one-dimensional flows in porous media. Trans.