Journal of Mathematical Analysis and Applications 251, 65᎐83 Ž2000. doi:10.1006rjmaa.2000.7020, available online at http:rrwww.idealibrary.com on
Exact Integration of Reduced Fisher’s Equation, Reduced Blasius Equation, and the Lorenz Model Lance Roman-Miller and Philip Broadbridge School of Mathematics and Applied Statistics, Uni¨ ersity of Wollongong, Northfields A¨ enue, Wollongong NSW 2522, Australia E-mail:
[email protected], Phil
[email protected] Submitted by William F. Ames Received May 2, 2000
This paper further develops the construction of exact solutions of nonlinear ordinary differential equations by explicitly constructing integrating factors using convergent power series expansions and the Picard iteration method. The method is used to find the general solution of the reduced Fisher’s equation and the reduced Blasius equation. A further generalization of the above methods using Jacobi’s last multiplier theorem is used to solve the Lorenz model. 䊚 2000 Academic Press
1. INTRODUCTION Although the Frobenius power series method is a standard tool for solving ordinary linear differential equations ŽODEs., a direct substitution of a trial power series solution does not give rise to the general solution of a nonlinear equation. However, the indirect approach of solving for an integrating factor has, in some notable cases, been amenable to power series methods. The use of series expansions to construct integrating factors has a long history. The original work of Abel w1x contains various solutions of nonlinear equations, since referred to as equations of the Abel type, by way of series expansions and the use of recursion relations to define the series coefficients. This tradition has been continued in recent years to find the exact solution of the Van der Pohl equation using the method derived by Roman-Miller ŽRoman-Miller and Smith w2x. and also to produce solutions for the Abel equation ŽBriskin et al. w3x.. 65 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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ROMAN-MILLER AND BROADBRIDGE
In the general case, if the individual coefficients of the power series expansion are sought, as in the solution of the Van der Pohl equation, then the system of equations obtained by equating coefficients is often indeterminate. However, the recurrence relations can be rendered determinate by applying Picard’s iteration theorem to specify the unwanted free coefficients ŽRoman-Miller and Smith w2x.. Many ODEs which are derived from symmetry reductions of practical partial differential equations, such as the Volterra equation, the one dimensional Poisson Boltzmann equation, the Langmuir equation, the Blasius equation, Fisher’s equation, Hille’s equation and Frommer’s equation, are expressed in terms of rational functions. In these cases it is often necessary to reduce the equations to a canonical form of the reduced type as defined by Briot and Bouquet ŽInce w4x.. Under those circumstances a simple change of variable of the form x s expŽ t . subsequently enables a reduction to an analytic equation which is then amenable to exact solution by the method of Roman-Miller and Smith w2x or, in the case of Abel equations with polynomial coefficients, by the method of Briskin et al. w3x. As we shall demonstrate, by this device a much larger class of nonlinear ODEs may be exactly integrated. For the first time we are able to construct exact integrating factors for practically interesting equations that have long defied exact analysis. We conclude this paper by generalizing the above results using Jacobi’s last multiplier theorem and Picard’s iteration to obtain the implicit solution of the Lorenz model.
2. THE POWER SERIES METHOD OF SOLUTION The power series method of solution for the construction of integrating factors can be defined as follows: Consider the nonlinear first order ordinary differential equation dy y f Ž x, y . dx s 0.
Ž 2.1.
We seek an integrating factor Ž x, y . such that dy y f Ž x, y . dx when multiplied by Ž x, y . becomes an exact differential: dF Ž x, y . s Ž x, y . dy y Ž x, y . f Ž x, y . dx.
Ž 2.2.
The general solution is given by C s F Ž x, y ., with C an arbitrary constant. The above differential equation is exact if y
ž
⭸ ⭸y
Ž x, y . f Ž x, y . y Ž x, y .
/
⭸ ⭸y
f Ž x, y . s
⭸ ⭸x
Ž x, y . . Ž 2.3.
EXACT INTEGRATION OF NONLINEAR ODEs AND SYSTEMS
67
Note that in order to solve Ž2.1. in general, we need only find a single non-zero solution Ž x, y . of Ž2.3., by any available method. At this stage one may attempt to solve this equation for a power series solution for Ž x, y . in one of the following forms:
Ž x, y . s
⬁
⬁
Ý Ý am , n x m y n ms0 ns0
Ž x, y . s
⬁
Ý
␣m Ž y . x m
Ž 2.4.
ms0
Ž x, y . s
⬁
Ý
␣m Ž x . y m .
ms0
By the standard method which has been classically used in the solution of second order linear equations, recursion relations are generated for the defining coefficients of either am , n , ␣ m Ž y .
or
␣m Ž x .
after substitution of Ž x, y . into Ž 2.3. .
The difficulty confronted at this stage is that the defining relations in the general case for the coefficients a m, n , after substitution of Ž x, y . are often underdetermined and the general system of equations is unsolvable. One may not freely choose the undetermined coefficients, as the recurrence relations will in general yield coefficients for a power series that is not convergent in any useful topology. This problem is resolved by generating the first few terms of the series expansion for Ž x, y . using Picard’s iteration method which has long-established convergence properties. Picard’s recurrence relation is used in the form
nq 1 Ž x, y . s 0 Ž 0, y . y q n Ž t , y .
x
H0 ⭸ ⭸y
ž
⭸ ⭸y
n Ž t , y . f Ž t , y .
f Ž t , y . dt ,
/
Ž 2.5.
where Ž0, y . s 1 and 0 Ž x, y . s 1. Note that the line x s 0 is non-characteristic for Eq. Ž2.3., so that by the Cauchy᎐Kovalevskaya theory, there exists a unique solution satisfying Ž0, y . s 1. In the case of solutions wherein the coefficients are grouped together as a coefficient function, ␣ mŽ y . say, then the coefficients of the series expansion of the integrating factor can be constructed from a simple linear recurrence relation. However, as more information is sought in terms of recurrence relations for a m, n or if the number of dimensions of the
68
ROMAN-MILLER AND BROADBRIDGE
domain of a coefficient function increases, then simple recurrence relations without Picard iteration are often insufficient to resolve the increasingly indeterminate system of equations that is generated. In the case of the equations of the Lorenz model, discussed below, a very large system of indeterminate defining equations for the coefficients of expansion is generatedᎏ61 equations in total.
3. CHALLENGES FOR THE POWER SERIES METHOD 3.1. The Requirement of an Analytic Equation There still remain many challenges in the solution of nonlinear equaŽ . tions. To start with, the Picard method assumes that f Ž x, y . in dy dx s f x, y Ž . is analytic. We may generalize this to the case that f x, y is not analytic but rather a rational function or more generally a ratio of analytic gŽ x, y . Ž . functions, which is indeterminate at the origin: dy with g dx s f x, y s h and h analytic in x and y, and taking zero values at the origin. Then we Ž . can transform to an equivalent equation of the form dy dx s q x, y where Ž . q x, y is analytic in x and y. This can be done by the method of Briot and Bouquet to reduce dy Ž . to an equation of the reduced type ŽInce w4x., being an dx s f x, y equation of the type Ž x, y .
x mq 1
d dx
ymq 1 Ž x . s ymq1 Ž x . q ax q higher order terms.
Ž 3.1.
Ž . This equation may then be transformed to the desired form dy dx s q x, y where q is analytic in x and y by a change of variable of the form x s et.
Ž 3.2.
3.2. The Problems of Con¨ ergence A number of difficulties present themselves concerning the problem of convergence. The first question is how do we find the region of convergence? The next question is how large is the region and is it sufficiently large to be of practical use? The region of convergence may be estimated by a number of known techniques which are covered in an excellent exposition by Mikhlin and Smolitskiy w5x. Here it must be noted that the method of solution by way of integrating factors as presented in w2x contains as a special case the classical method of solution that uses straightforward equations for power series coefficients. Particular solu-
EXACT INTEGRATION OF NONLINEAR ODEs AND SYSTEMS
69
tions, which are well presented in the classical works ŽDavis w6x., may be derived from the method of integrating factors by reversion of series. If the region of convergence of a power series solution is relatively small then the solution may be extended by way of the method of analytic continuation ŽDavis w6x.. Alternatively, convergence may be extended to larger regions by way of the selection of different norms used to define convergence ŽWalter w7x.. In the following two sections, we will illustrate these techniques by solving the interesting nonlinear ODEs that arise as symmetry reductions of Fisher’s equation and of the Blasius equation. These reductions, and those of many other topical PDEs, have been summarized by Sachdev w8x. Whilst the equations considered above have different applications, they are all linked by a common theme. That theme is the ability to be reduced and then solved exactly by a combination of symmetry reductions and the construction of integrating factors to produce exact differentials. In fact we have found that many first order ODEs can be treated effectively in this manner provided they can be transformed to one of the Picard analytic forms. This means that full integration is achievable for a much larger class of ODEs than had previously been thought. The next section will consider the solution of Fisher’s equation in moderate detail. The solution of other equations presented in this paper is given in outline form only.
4. FISHER’S EQUATION Fisher’s equation ŽFisher w9x. was designed to model the wave of advance, in 1 q 1 dimensions, of a new advantageous mutation through an established sexually reproducing population. More correctly, it describes the migration of a viral mutant in one dimension Že.g., Broadbridge et al. w10᎐13x and references therein .. Fisher’s equation has also found applications in modeling the neutron population in a nuclear reactor and it also has been found to give rise to solutions which match the steady state of the Kortweg de Vries᎐Burgers equation ŽCanosa w14x.. After rescaling, Fisher’s equation may be written in the form
⭸ ⭸t
us
⭸ 2u ⭸ x2
q uŽ 1 y u. .
Ž 4.1.
Writing u s uŽ s ., where s s x y ct is the d’Alembert wave variable and c is the speed of wave propagation, we obtain an equation of the form d2 ds
2
uqc
d ds
u q u y u 2 s 0,
Ž 4.2.
70
ROMAN-MILLER AND BROADBRIDGE
which becomes d dx
u 2 y u y cy
y Ž u. s
y
after applying the transformation d
u Ž s . s y. ds Letting y s 1z we obtain an equation of the form dy s y 3 Ž x y x 2 . q cy 2 . Ž 4.3. dx We note that in this and all examples considered below, the variables of the final transformed equation will be written as x and y merely for the purpose of comparing the structure of the transformed equation in question to that of others. After applying the methods contained within ŽRoman-Miller and Smith w2x. we obtain the results for the exact solution of Fisher’s equation to order 12 in x and y, F Ž x, y . s
35 y 9 x 12 648 q x9 y q x8
ž ž ž ž ž ž ž
q x5
qx
q
q
y
q
6
6 y 5c 2 5
23 y 5 c 2
7 y4c 6
12
y
60 3 y5 8
yc y q 3 4
y cy 2 x q y q ⭈⭈⭈ ,
q
y
y3 3
y5 2
960
/ ž 2
560 y
36
y
157y 6 c 120
q c4 y 5
/
5 y7 16
/
/
/
y c y 3 2
/
803 y 8 c
61 y 6 c
q
4 qx
q
8
3 y4c
/
3780
2687y 8 c
5 y7
3
163 y 6 c 3
q
y
1487y 8 c
/
54
5 y6c3
y
y
5 y7
12
28
180
48
5 y7
103 y 7c 2
523 y 7c 2
q
y
ž
35 y 9
q
756
56
42
y5
1375 y 8 c
65 y 7c 2
23 y 6 c
q x4 y 3
48
128
q x 10
108
35 y 9
35 y 9
q x7 y q x6
35 y 9 x 11
y
y3 2
/ Ž 4.4.
EXACT INTEGRATION OF NONLINEAR ODEs AND SYSTEMS
71
where the integrating factor is given by
s y
ž
35 12
q y q q
65c 2
q
/
8
23 y 5 c
y6 x8
q y
103c 2
q
35
y6 q
803 y 7c
ž / ž / / ž / / ž ž / ž / / žž / / ž ž 7
5y4 6
y
q
5
61c 6
q 6c
2
4
q y3cy q y
115c 2
q y 2 q y4c 3 q q 3c 2 y
3
2
8
3661c 2
y 10 c 3 y 5 q y q
2
3
ž ž ž /
4
12
14 c 3
163c 3 10
180
y
q 5c 4 q
157c 20
15
70
y
35
16
x7
y6 x6
y5 x5
y4 x4
8
/ /
y3 x3
y 2 x 2 y 2 cyx q 1 q ⭈⭈⭈
Ž 4.5.
and the recursion relations are defined by d d 0 s ym ␣ m Ž y . m y ␣ my1 Ž y . cy 2 y ␣ Ž y. y3 dy dy my2 q
ž
d dy
ž /
/
ž
/
␣ my 3 Ž y . y 3 y 2 ␣ my1 Ž y . cy y 3 ␣ my2 Ž y . y 2
q 3 y 2␣ my 3 Ž y . .
Ž 4.6.
Reverting F Ž x, y . about y Ž0. s 1 we obtain the following Maclaurin series expansion for y Ž x . accurate to O Ž x 5 . for the case c s 0. The case for other values of c follows similarly. Further accuracy can be obtained by additional iterations of the Picard algorithm for Ž x, y . prior to reversion of F Ž x, y .: 1 1 3 1 yŽ x. s 1 q x2 y x3 q x4 y x5 2 3 8 2 1 65 8 65 9 1105 10 q x6 y x q x y x 6 64 24 192 9815 11 809761 12 37049 13 q x y x q x 864 41472 1152 165677 14 y x q O Ž x 15 . . Ž 4.7. 3456
72
ROMAN-MILLER AND BROADBRIDGE
Figures 1 and 2, for the known exactly solvable cases c s 0 and c s
5
'6 Žsee in particular w15x., contain graphs of the solutions obtained by the method of Roman-Miller plotted against the exact solutions and compared to a numerical approximation using a fourth᎐fifth order Runge᎐Kutta method implemented by Maple. Excellent agreement is seen within the region of convergence. However, these truncated series solutions show significant errors whenᎏ Ž x, y . ᎏis large enough for the neglected terms to be prominent.
5. THE BLASIUS EQUATION The classical Blasius equation ŽBlasius w16x. arose in the study of boundary layer flows. In particular it applies to the study of the flow of an incompressible fluid over a semi infinite plate. The study of such properties is of great significance in considering the problems of skin friction on aerodynamic surfaces. Recently there have been a number of positive advances in the study of this equation. The differential transform method by way of domain splitting has led to accurate methods of improved numerical as well as power series approximations to solutions of the equation ŽY’u and Chen w17x.. Progress has also been made in relation to series approximations to the solution of this equation by improving Bairstow’s technique ŽParlange et al. w18x.. The Blasius equation may be written in the form
q ⬙ s 0.
Ž 5.1.
Sachdev w8x transforms this to an equation of the form d dx
yŽ x. s
yŽ x. Ž2 q x y yŽ x. . xŽ1 q x q yŽ x. .
Ž 5.2.
under a transformation x s ⬘r ⬙, y s ⬘ 2r ⬙, t s ln
