Pergamon
Inr. J. Non-Linear
Mechanics, All rinhts
Vol. 33, No. 2, pp. 315-326, 1998 \?J 1997 Elsevier Science Ltd reserved. Printed in Great Bntain 002&7462/98 $17 00 + 0.00
PII: SOO20-7442(97)00013-9
SEPARATION OF VARIABLES FOR THE l-DIMENSIONAL NON-LINEAR DIFFUSION EQUATION Philip W. Doyle*? and Peter J. Vassilioul TDepartment of Mathematics, University of Hawaii, Honolulu, HI, U.S.A.; fSchoo1 of Mathematics and Statistics, University of Canberra, Belconnen, ACT 2616, Australia (Received 4 December 1996)
Abstract-The class of separable solutions of a l-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under a subgroup of the generic group. An equation which admits separation of variables in some field coordinate has separable solutions not invariant under any subgroup, as in the linear case. The class of separable equations significantly extends the class of equations having non-generic symmetry, i.e. those with exponential or power law diffusivities, for which separation of variables is a trivial process. We derive a complete list of canonical forms for diffusion equations which admit separation of variables in some coordinate, and we describe the separation mechanism for these equations. It involves the integration of a fixed third order ordinary differential equation, generally non-linear, and the subsequent integration of a first order ordinary differential equation which depends on the particular solution of the third order equation. The procedure yields a 3-parameter family of separable solutions of the given diffusion equation. Several non-symmetric examples are analyzed in detail, leading to explicit non-invariant solutions. 0 1997 Elsevier Science Ltd. Keywords: separation of variables, non-linear diffusion
1. INTRODUCTION
This work is a study of separation equation
of variables for the second order partial differential
(1) for a single function u of two variables x and t. The scalar evolution equation (1) is the differential form of a conservation law. It is a mathematical model for sourceless diffusion in a homogeneous i-dimensional domain, where the field variable u represents concentration, and the independent variables x and t represent space and time. The difluusiuity k is a positive smooth (Cm) function which represents the factor of proportionality between the concentration flux and negative concentration gradient. See Crank [l] for discussion of the mathematics of diffusion. We use the abbreviation u, = (44u,),, or ut = k(u)u,, + k’(u)u:.
(2)
In the case of constant diffusivity, we have the linear equation uz = ku,,, but (2) is non-linear in the more realistic case where k is not constant. See Samarskii et al. [Z] for a detailed study of non-linear diffusion equations, including an extensive bibliography of recent results. Here we find all diffusivities for which (2) admits separation of variables in some field coordinate, describe the separation mechanism for these equations,
Contributed by W. F. Ames. *Author to whom correspondence should be addressed:
[email protected] 315
316
P. W. Doyle and P. J. Vassiliou
and construct explicit solutions. This project is analogous to the work of Miller and Rubel 131, where all functions fare found for which the non-linear wave equation a,, - u,* =f (4 admits separation in some field coordinate. The separability problem for the wave equation was also recently studied by Grundland and Infeld [4], and by Zhdanov [S]. Our approach is more geometrical than the methods used by these authors. See Ibragimov [6] for a survey of results on symmetry and equivalence for non-linear diffusion equations.
2. EXAMPLE
As an introduction,
we examine the non-linear equation u, = ((1 - u2)-1uX)X,
(3)
or u, = (tanh- lu),,. The constant functions u(x, t) = c, ICI< 1, and the non-constant time-independent functions u(x, t) = tanh IU(X+ a),
o # 0,
are solutions. A non-zero function with the separable form u(x, t) = u(x)w(t) is a solution of (3) if and only if 2u’(x)2w(t)3 w’(t) = u(x)(l - V(X)2W(#) + (1 - u(x)2w(t)2)2’ u”(x)w(t)
It can be shown that there is no pair V(X),w(t) with w’ # 0 which satisfies (4), so there are no time-dependent solutions of (3) which are separable in the given coordinate u. In the coordinate U(u) = 24/$-=Y?) equation (3) has the form u, = (1 + C2)fi,, - LX,“,
(5)
which is neither a sourceless diffusion equation nor linear. A non-zero function U(x, t) = r(x)w(t)
(6)
w’(t) = (u”(x)/u(x))w(t) + (u(x)u”(X) - u’(x)2)w(t)?
(7)
is a solution of (5) if and only if
The quantities p = vu! - of2
il = u’ljv,
(8)
are invariants (first integrals) of the third order ordinary differential equation U
11,
=
V’lYJU.
(9)
If v is a solution of (9) then I and p are constant, and (7) is a single first order ordinary differential equation w’ = Aw + /Jw3 (10) for w(t), the variable x is no longer present. This accounts for all non-zero solutions of (5) with the form (6). Using the inverse u(ii) =
ii/J?-zF,
Separation of variables for the l-dimensional non-linear diffusion equation
317
we obtain the solutions of the original equation (3) which are separable in the coordinate U. They are the functions
ww
u(x, t) =
Jl
+ u(x)2w(t)2’
where u satisfies (9) and w satisfies (lo), with Aand ,Ugiven by (8). The non-constant solutions thus obtained are the functions u(x,t) = U(o(x + a), W2(t + b)),
(11)
where a, b, and o # 0 are arbitrary constants, and U is one of the functions U (x, t) = tanh x,
(12.1) (12.2)
(12.3) sinh x
U(x, t) =
(12.4)
cosh’x + eC2” sinh x
U(x, t) =
t > 0,
(12.5)
t 0 and g are arbitrary smooth functions. Equation (2) is the special case g =f’. The form (18) is preserved by affine transformation of x, oriented affine transformation of t, and arbitrary smooth transformation of u. In a coordinate 9, the equation is ii, =f(i+i,,
+ g(C))u:,
(19)
where 7(C) =f(@)) and cj(ii) = u’(ii)g(u(ii))
+ u”(ii)f(u(q)/u’(ii).
Proposition
There is a coordinate zi(u), defined on the entire common interval of definition off and g and unique up to affine transformation, in which the equation & =S(n)u,,
+ g(M,
f > 0,
has the form ii* =
(rqti)u,),, I?> 0.
Proof
Note that S =f
if and only if U(U)is a regular solution of the linear equation ,_” = ((g -f’)/f)(U)U’.
(20)
The non-zero solutions are the regular functions U(u) = +
exp
J-i
J
((g -f’)/f)(u)du
I
du. Cl
The sourceless diffusion equation is thus a canonical form for the class of equations (18). The form (2) is preserved by the group of time-oriented scale transformations (x, r, u) I-P (61x + El 9&t +
E2,
&u
+
d>
where &, & # 0, and d2 > 0. The generators (14) span the symmetry algebra of the generic equation (18). A function has the additively separable form zi(x, t) = u(x) + w(t) if and only if it satisfies the condition
uxt= 0.
(21)
The additive form is identified with the multiplicative form U(x, t) = u(x)w(t) by the change of variable U H e”. In an arbitrary coordinate U,equation (2) is given by (19), and its solutions additively separable in U are the joint solutions of (19) and (21). For example, stationary solutions are separable in any field coordinate. In general, we do not expect time-dependent solutions of the overdetermined system of equations (19) and (21). We will find equations (19) which are compatible with the separability condition (21) in the sense that the two equations have a maximal family of time-dependent joint solutions. Transforming the result will give us the diffusion equations having maximal families of solutions separable in some fixed field coordinate.
320
P. W. Doyle and P. J. Vassiliou 5. COMPATIBILITY
A joint solution of equations (18) and 41 = 0
(22)
also satisfies the equation 4, = (f’(u)%
+ s’(u)G%
Hence the solutions of (18) additively separable in u are the solutions of the system 4 =_f(44x
+ g(u)4
u,t = 0
(23)
%r = (f(4u.U
+ s(u)ul)(f’(u)uxx
+ 044).
i
These independent equations define a 5-dimensional submanifold 9 of the space J of 2-jets of smooth mappings R2 + R, where the variables x, t, u, ux, u,, uxx, uxt, u,, are coordinates on J. See Olver [8] for the basic facts about jet bundles, contact structure and prolongation. The relations (23) express the t-derivatives in terms of the variables x, t, u, uX,uXX,so we use the latter as coordinates on B. The pullback of the contact structure on J to L&Jis the differential system C spanned by the l-forms O=du-u,dx-u,dt, 8, = du, - u,, dx, 8, = du, - u,, dt, where u, and u,, are given by (23). We have 8, =f(u)8,,mod
13,19,,
where
e,, = du,, + so C is also spanned by 8, 8,, and Q,,. Note that C is 3-dimensional, regardless off and g, hence is a 2-codimensional differential system on a 5-dimensional manifold. The 2-dimensional integrals of C are (locally) the second order prolongations of the solutions of (23). There is thus at most a 3-parameter family of solutions of (18) additively separable in u. The maximal case occurs when C is integrable, in which case the equations (18) and (22) are compatible. See Olver [11] for discussion of differential constraints and compatibility conditions. Theorem 1
The equations u, =f(u)u,,
+ g(4uL
f > 0,
(24)
and U Xf =
(25)
0
are compatible if and only if ((f’ + 2g)lf)’ = 0,
(g’lf)’ = 0.
W-4
Proof
Note that de, de, E 0 mod C, so C is integrable if and only if df?,, 3 0 mod C.
0
The compatibility conditions (26) are a special case of the regular separability conditions derived by Kalnins and Miller [12] for general partial differential equations. They also
Separation of variables for the l-dimensional non-linear diffusion equation
321
predict a 3-parameter family of separable solutions. The geometric approach used here has some advantages. See Doyle [13] for its application to the separability problem for general l-dimensional evolution equations. Equation (24) has a 2-parameter family of stationary solutions. It is straightforward, though tedious, to verify that (24) has at most a 2-parameter family of additively separable (non-stationary) travelling wave solutions, and at most two additively separable similarity solutions. Each similarity solution translates to a 2-parameter family of additively separable solutions. This accounts for all solutions both additively separable in u and invariant under a subgroup of symmetries (13). The conditions (26) ensure the existence of a 3-parameter family of additively separable solutions. Hence the generic additively separable solution of equation (24) with coefficients satisfying (26) is not invariant. Suppose that (26) holds, so that (f’ + 2g)fland g’lfh ave constant values c1and p. If the function u(x, t) = v(x) + w(t)
(27)
is a solution of (24), then differentiation with respect to x implies that v satisfies the ordinary differential equation 21”’+ CIV’V” + /Iv’3 = 0. (28) Fix a solution v(x) of (28). Then u(x) + w is a l-parameter does not depend on the variable v. Note that f(W
+ w)(W
+ w),, + O(x)
family of solutions, because (28)
+ w)(W
+ w)f
is a function of the parameter w only, because f(v)v” + g(v)v’2 is an invariant of (28). Hence (27) is a solution of (24) if and only if w is a solution of the ordinary differential equation w’ = W(w), where W(w) =f(v(x)
+ w)v”(X) + g(v(x) +
w)v’(x)2.
Theorem 2 Suppose that the coefficients of the equation
satisfy the conditions
(f’ + 2gv.f= %
d/f = A
(30)
where CIand p are constants. A function u(x,t) = u(x) + w(t)
(31)
is a solution of (29) if and only if v and w satisfy the ordinary differential equations v”’ + av’v” +
/w3 = 0,
w’ = W(w),
(32)
where W(w) =f(v(x)
+ w)v”(X) + g@(x) + w)v’(xy.
q
Note that the symmetries (13) are present in system (32). See Doyle [13] for a geometric interpretation of the compatibility conditions (26), and of the separation mechanism described in Theorem 2. For example, the coefficients of the equation u, = (cash u)u,, - (sinh u)u,2/2
(33)
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P. W. Doyle and P. J. Vassiliou
satisfy (30), with a = 0 and /3 = - l/2, so (31) is a solution of (33) if and only if u11,- v’3/2 = 0
(34)
and w’(t) = {(v”(x) - v’(x)2/2)e”(X)/2)ew”’+ {(v”(x) + v’(x)2/2)e-“‘“‘/2}e-““’
(35)
The quantities A = (v” - v’*/2)ev/2,
p = (v” + v’*/2)e-“/2
are invariants of (34), so (35) is a single equation w’ = IZe”+ pe-“. In the coordinates fi = e”‘* and w = e”‘, we have A = -(l/v3”/(l/Q3,
(36)
p = ($‘/fi3.
(37)
and CC,’ = IW2 + /L
(38)
Solving (37) for each value of p, extracting the corresponding value of il from (36), and then solving the Riccati equation (38) yields the additively separable solutions of (33). The non-constant solutions thus obtained are the functions (1 l), where U is one of the functions U(x, t) = + In (x*/2&
t > 0,
(39.1) (39.2)
u(x, t) = ln (Yr(x)*), U(x, t) = In (Yi(x)* tanh 2t),
t > 0,
(39.3)
U(x, t) = ln(y1(x)*coth2t),
t > 0,
(39.4)
U(x, t) = ln(y2(x)* tan 2t),
0 c t < n/4,
U(x,t) = ln(-y3(x)*tan2t),
- 7c/4 O,
x=
j
x=
:‘j&
Y*>L
:j&,
o 0, (43) where k is one of the following functions: k(u) = uy,
u > 0,
(44.1)
k(u) = e”
(44.2)
k(u) = (1 - u2)-r,
(44.3)
k(u) = (u’ - 1)-r,
(44.4)
u > 1,
k(u) = (u2 + l)- r,
(44.5) Z
k(u) = z(u)e’@), k(u)
=
eaz(“)
u=
u=
cash z(u),
s
s -3/2e-s’2 ds,
0
(44.6) (44.7)
0 # &-1,
z(coshs)-3’2e-“Si2ds, s
z > 0,
1
---co 1,
obtained by our procedure are functions (ll), where U is one of the functions U(x, t) = coth x,
U(x,t) =
j&,
U(x,t)=
J&Y
UC%t) =
Jgp-t?
U(x, t) =
&=-&. sin x
U(x, t) =
- cos2 x + eC2” The non-constant
solutions of U, = ((u2 + l)- 1u,),
obtained by our procedure are functions (1 l), where U is one of the functions U(x, t) = tan x,
U(x, t) =
sinh x -cosh’x
U(& t) =
&yxh:
U(x,t) = $y& U(x,t) =
/J.&.
- eezt’ e_2ty
(48)
326
P. W. Doyle and P. J. Vassiliou
Ibragimov [6] notes the similarity and travelling wave solutions, and also describes the non-local symmetry of (48). The equations with diffusivities (44.6-44.9) are more complicated. For example, in the case (44.7) with (r = 0, we have u, = ((cash z(+,),
,
where (cash z)- 3’2dz.
u(z) =
s The change of variable u HZ(U) transforms (49) to (33), and we obtain the non-constant solutions 24(x,t) = u(U(w(x + a), w2(t + b))), where V(x, t) is one of the functions (39). Acknowledgements-This research was partially supported by the Australian Research Council. The author P. W. D. thanks the faculty of the School of Mathematics and Statistics at the University of Canberra for their friendly hospitality, enjoyed and appreciated during the summers of 1994 and 1995.
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