DELFT UNIVERSITY OF TECHNOLOGY
REPORT 00-13
On the applicability of the method of separation
of variables for partial difference equations
W.T. van Horssen
ISSN 1389-6520
Reports of the Department of Applied Mathematical Analysis
Copyright 2000 by Department of Applied Mathematical Analysis, Delft, The Netherlands. No part of the Journal may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from Department of Applied Mathematical Analysis, Delft University of Technology, The Netherlands.
ON THE APPLICABILITY OF THE METHOD OF SEPARATION OF VARIABLES FOR PARTIAL DIFFERENCE EQUATIONS
W.T. VAN HORSSEN
Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis, Delft University of Technology, Mekelweg 4, 2628 CD Delft,
The Netherlands,
(W.T.vanHorssen@math.tudelft.nl)
In this paper it will be shown that the method of separation of variables for partial dif-ference equations (as for instance described by R.E. Mickens [1] and R.P. Agarwal [2]) can be applied to a much larger class of problems as is generally assumed. To show how the method should be applied two problems are treated and solved. In [1] and [2] these two problems were considered and claimed to be not solvable by applying the method of separation of variables.
Keywords: Separation of variables, partial difference equation. AMS Subject Classifications: 39-01, 39A12.
1
Introduction
The method of separation of variables is the oldest systematic method to find nontriv-ial solutions for (linear) partnontriv-ial differentnontriv-ial equations. To study waves and vibrations D’Alembert, Daniel Bernoulli, and Euler used this method in the middle of the eighteenth century. The method has been considerably refined and generalized during the last cen-turies, and remains a method of great importance and frequent use today. Like most methods for differential equations there is a more or less similar method for difference equations. Mickens [1] and Agarwal [2] give the following description of the method of sep-aration of variables for partial difference equations. Both authors consider the following equation
ψ(Ek, El, k, l)u(k, l) = 0 , (1.1)
where ψ is a polynomial function of the operators Ek and El. By definition we have
Eku(k, l) = u(k + 1, l), Elu(k, l) = u(k, l + 1), and EkmElnu(k, l) = u(k + m, l + n). Then
two assumptions are made.
Assumption 1.
A nontrivial solution u(k, l) of the partial difference equation(1.1) has the structure u(k, l) = V (k)W (l) . (1.2)
Assumption 2.
After substitution of (1.2) into (1.1) an equation of the type f1(Ek, k)V (k) f2(Ek, k)V (k) = g1(El, l)W (l) g2(El, l)W (l) (1.3) is obtained.
It then follows from (1.3) that V (k) and W (l) have to satisfy f1(Ek, k)V (k) = αf2(Ek, k)V (k) ,
g1(El, l)W (l) = αg2(El, l)W (l) ,
(1.4)
where α is an arbitrary constant. A simple argument (not used in [1] and [2]) to obtain (1.4) from (1.3) is the following. When the difference operator ∆k = Ek− 1 is applied to
(1.3) we obtain ∆k f1(Ekk)V (k) f2(Ek, k)V (k) ! = 0 ⇒ f1(Ek, k)V (k) f2(Ek, k)V (k) = constant = α , (1.5) and so (1.4) is obtained. Of course, also the difference operator ∆l = El− 1 could have
been used to obtain the same result. Using an adapted form of the method of separation of variables we will treat and solve the following two partial difference equations
u(k, l) = p u(k + 1, l − 1) + q u(k − 1, l + 1) with p + q = 1 , (1.6) and
u(k + 3, l) − 3u(k + 2, l + 1) + 3u(k + 1, l + 2) − u(k, l + 3) = 0 , (1.7) in section 2 and in section 3 of this paper respectively. In [1] and [2] these two equations were claimed to be not solvable by the method of separation of variables. Based upon the results obtained in these sections we will reformulate assumption 2 such that the method of separation of variables can be applied to a much larger class of partial difference equations. This reformulation can be found in section 4 together with some remarks on the applicability of the method of separation of variables to partial differential equations.
2
A second order partial difference equation
In this section we consider the following partial difference equation
u(k, l) = p u(k + 1, l − 1) + q u(k − 1, l + 1) , (2.1) where p and q are constants with p + q = 1. Substituting u(k, l) = V (k)W (l) into (2.1), and dividing by V (k − 1)W (l − 1) yields
V (k)W (l) = p V (k + 1)W (l − 1) + q V (k − 1)W (l + 1) ⇒ (2.2) V (k) V (k − 1) W (l) W (l − 1) = p V (k + 1) V (k − 1) + q W (l + 1) W (l − 1) . (2.3) Obviously (2.2) or (2.3) is not an equation of the type (1.3). When we, however, apply the difference operator ∆k to (2.3) we obtain
W (l) W (l − 1)∆k V (k) V (k − 1) ! = p∆k V (k + 1) V (k − 1) ! , (2.4)
which is an equation of the type (1.3) if we divide (2.4) by ∆k(V (k)\V (k − 1)). It should
be observed that by applying the difference operator ∆l to (2.3) also an equation of the
type (1.3) is obtained. From (2.4) it follows that
W (l) = αW (l − 1) , α∆k V (k) V (k − 1) ! = p∆k V (k + 1) V (k − 1) ! , (2.5)
where α is an arbitrary separation constant. Substituting (2.5) into (2.2) and noticing that W (l + 1) = α2 W (l − 1) we obtain W (l) = αW (l − 1) , αV (k) = pV (k + 1) + qα2 V (k − 1) , which can easily be solved, yielding
W (l) = c1αl , V (k) = c2αk+ c3 q pα !k ,
where c1, c2, and c3 are arbitrary constants. So, nontrivial solutions u(k, l) = V (k)W (l) of
the partial difference equation(2.1) are αk+l and q p
k
αk+l. Since the difference equation
is linear we can sum over all solutions to obtain the general solution u(k, l) = f (k + l) + q
p
!k
g(k + l) ,
where f and g are arbitrary functions. To obtain this solution by using the method of separation of variables we only need to apply the difference operator ∆k once.
3
A third order partial difference equation
In this section we consider the following partial difference equation
u(k + 3, l) − 3u(k + 2, l + 1) + 3u(k + 1, l + 2) − u(k, l + 3) = 0 . (3.1) Substituting u(k, l) = V (k)W (l) into (3.1), and dividing by V (k)W (l) yields
V (k + 3)W (l) − 3V (k + 2)W (l + 1) + 3V (k + 1)W (l + 2) − V (k)W (l + 3) = 0 ⇒ (3.2) V (k + 3) V (k) − 3V (k + 2) V (k) W (l + 1) W (l) + 3 V (k + 1) V (k) W (l + 2) W (l) − W (l + 3) W (l) = 0 . (3.3) Obviously (3.2) or (3.3) is not an equation of the type (1.3). And when we for instance apply the difference operator ∆k to (3.3) we obtain
∆k V (k + 3) V (k) ! − 3W (l + 1) W (l) ∆k V (k + 2) V (k) ! + 3W (l + 2) W (l) ∆k V (k + 1) V (k) ! = 0 , (3.4)
which is still not an equation of the type (1.3). If we, however, compare (3.3) with (3.4) then it is obvious that the number of positions where we have functions in l decreases. We can decrease this number even further by dividing (3.4) by ∆k(V (k + 1)/V (k)), and by
applying the difference operator ∆k to the so-obtained equation, yielding
∆k ∆k V (k + 3) V (k) ! ∆k V (k + 1) V (k) ! − 3W (l + 1) W (l) ∆k ∆k V (k + 2) V (k) ! ∆k V (k + 1) V (k) ! = 0 . (3.5)
W (l + 1) = αW (l) , ∆k ∆k V (k + 3) V (k) ! ∆k V (k + 1) V (k) ! − 3α∆k ∆k V (k + 2) V (k) ! ∆k V (k + 1) V (k) ! = 0 , (3.6)
where α is an arbitrary separation constant. To simplify the calculations we substitute (3.6) into (3.2) (notice that: W (l + 3) = α3
W (l) and W (l + 2) = α2 W (l)), yielding W (l + 1) = αW (l) , V (k + 3) − 3αV (k + 2) + 3α2 V (k + 1) − α3 V (k) = 0 , which can easily be solved, giving
W (l) = c1αl , V (k) = c2αk+ c3kαk+ c4k2αk ,
where c1, c2, c3, and c4 are arbitrary constants. So, nontrivial solutions u(k, l) = V (k)W (l)
of the partial difference equation(3.1) are αk+l, kαk+l, and k2
αk+l. Since the difference
equation is linear we can sum over all solutions to obtain the general solution u(k, l) = f (k + l) + k g(k + l) + k2
h(k + l) ,
where f, g, and h are arbitrary functions. To obtain this solution by using the method of separation of variables we only need to apply the difference operator ∆k two times.
4
Conclusions and remarks
In this paper it has been shown that the method of separation of variables can be applied to a much larger class of partial difference equations as is generally assumed. Based upon the results obtained in section 2 and in section 3 of this paper we propose to replace As-sumption 2 (as given in section 1) by
Assumption 2:
After substitution of (1.2) into (1.1), and after applying the difference operator(s) ∆k,
or ∆l, or both sufficiently many times to the equation, an equation of the type (1.3) is
eventually obtained.
For partial differential equations a similar procedure can be developed. After substitution of a separated solution into the partial differential equations, and after differentiating the
equation sufficiently many times with respect to some of the independent variables, we can finally reduce the problem to ordinary differential equations. This procedure also seems to be not (well-)known in the literature for partial differential equations. Preliminary calculations indicate that the method can be applied to a lot of interesting problems. For instance for problems like
utt− uxx = 0 , 0 < x < l , t > 0 , ux(0, t) = α ut(0, t) , t ≥ 0 , u(l, t) = 0 , t ≥ 0 , (where α is a positive constant)
or wtt+ 2γwxt+ (γ2− 1)wxx = 0 , 0 < x < l , t > 0 , w(0, t) = w(l, t) = 0 , t ≥ 0 , (where γ is a constant with 0 < γ2
< 1)
the method of separation of variables turns out to be successful.
References
[1] R.E. Mickens, Difference Equations, Van Nostrand Reinhold Company, New York, 1987.
[2] R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Appli-cations, Marcel Dekker Inc., New York, 1992.
