Power Series Solution to Non-Linear Partial Differential Equations of Mathematical Physics a E. López-Sandoval *a , A. Mello a , J. J. Godina- Nava b a Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro, RJ, Brazil. *[email protected]b Departamento de Física Centro de Investigación y de Estudios Avanzados del Instituto Politecnico Nacional Ap. Postal 14-740, 07000, México, D. F. México Abstract Power Series Solution method has been traditionally used to solve Linear Differential Equations: in Ordinary and Partial form. However, despite their usefulness the application of this method has been limited to this particular kind of equations. We propose to use the method of power series to solve non-linear partial differential equations. We apply the method in several typical non linear partial differential equations in order to demonstrate the power of the method. Keywords: Power series, Non Linear Partial Differential Equations, Symbolic Computation. 1 Introduction Nowadays, the solution of non-linear partial differential equations is considered as a fundamental tool in the research of multidisciplinary areas, because both their implication in the public health problems and social impact in to solve real life problems. In fact, is mandatory to involve mathematical methods in the traditional research methodology of science areas like Biology, Cell Biology, Physiology, Physics, Chemistry, Chemical Physics, etc. which helped by the technological advance in the computation, to incorporate a new age of knowledge in order to tackle real problems.
Transcript
Power Series Solution to Non-Linear Partial
Differential Equations of Mathematical Physics
aE. López-Sandoval*a, A. Melloa, J. J. Godina- Navab a Centro Brasileiro de Pesquisas Físicas,
Rua Dr. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro, RJ, Brazil.
Centro de Investigación y de Estudios Avanzados del Instituto Politecnico Nacional
Ap. Postal 14-740, 07000, México, D. F. México
Abstract Power Series Solution method has been traditionally used to solve Linear Differential Equations: in Ordinary and Partial form. However, despite their usefulness the application
of this method has been limited to this particular kind of equations. We propose to use the method of power series to solve non-linear partial differential equations. We apply the method in several typical non linear partial differential equations in order to demonstrate the power of the method.
Keywords: Power series, Non Linear Partial Differential Equations, Symbolic Computation.
1 Introduction
Nowadays, the solution of non-linear partial differential equations is considered as a
fundamental tool in the research of multidisciplinary areas, because both their
implication in the public health problems and social impact in to solve real life
problems. In fact, is mandatory to involve mathematical methods in the traditional
research methodology of science areas like Biology, Cell Biology, Physiology, Physics,
Chemistry, Chemical Physics, etc. which helped by the technological advance in the
computation, to incorporate a new age of knowledge in order to tackle real problems.
Power Series Solution (PSS) method (PSSM) has been limited to solve Linear Dif-
ferential equations, both Ordinary (ODE) [1, 2], and Partial (PDE) [3, 4]. Linear PDE
has traditionally been solved using the variable separation method because it permits
to obtain a coupled system of ODE easier to solve with the PSSM. Examples of these
are the Legendre polynomials and the spherical harmonics used in the Laplace´s
Equations in spherical coordinates or the Bessel´s equations in cylindrical coordinates
[3-4]. It is known that in Non Linear PDE (NLPDE) is, as we know, because isn´t
possible to apply the separation of variables method.
The methodology to solve the NLPDE is to obtain a solution by using the
approximated analytical method, i. e., non numerical or semi analytic form, in a
indirect, or direct way. In the direct way, there are methods like Inverse Scattering
Transform [5] or the Lax Operator Formalism [6]. In the direct way it can be used for
example the PSS in an asymptotic approximation the Hirota method involving a
bilinear operator technique [7]; the Adomian Decomposition Method [8], and the
Homotopy Analysis method [9, 10]. This last method involves a series expansion with
with a non small parameter perturbation approximation to adjust the convergence.
This method is different of the classical perturbation theory.
Techniques, even more direct, to approximate to a solution in NLPDE are the
Taylor Polynomial Approximation method (TPAM) [11, 12], and the PSSM. In both
techniques a semi analytic solution is obtained implementing the PSSM. However, the
PSSM has been little used to solve non-linear ODE [13-16], or NLPDE [17-19].
In this letter, we propose to apply in order to find particular solutions of typical
PDE widely used in mathematical physics, namely, the equation of a steady state
laminar boundary layer on a flat plate, the Burger’s equation and the Korteveg-de
Vries equations. In all this examples, we were able to find particular values of the
coefficients of the truncated PSSM. We use the symbolic computation package
Matlab® to obtaing the algebraic operations for the truncated series approximation.
This program helps to do easier the tedious algebraic operations.
2 The Power Series Solution method
We know that almost the totality of the NLPDE have not a solution with an analytic
expression, i.e. a solution in terms of know functions in a closed form. Our proposal
is to construct one solution using a power series, taking advantage of the capacity of
power series to represent any function with an algebraic series and develop the idea to
construct an approximate solution. It also has the possibility to approximate a
solution, inclusive if it not exists in analytic form. In a similar way in which we apply
normally the Taylor series (TS) to some function:
0 0
11...1
1 2
1
1)...()(...),...,(
n n n
n
dd
n
nnd
d
d
daxaxaxxf (1)
where
!!...!
),...,(...
21
1
1
...
...
1
1
1
d
dn
d
n
nn
nnnnn
aaxx
f
ad
d
d (2)
are their corresponding coefficient´s expansions. It is necessary to know the values of
all the derivatives of the function at (a1,…,ad) that represent its center in a open disc.
The Taylor´s series not always guarantee per se that the function represented has an
exact approximation to distant points about its central value. However, considering
that the PSS need to satisfy the NLPDE with an initial values condition (IVC) in time
or space values, or with the boundary values conditions (BVC) in the space, we can to
construct a well posed problem for to obtain an accurate solution. Also the Taylor´s
series approximation is a smoothed function and therefore, with this we can to
guarantee the existence of a solution.
When the differential equation (DE) under study is an IVC problem, it is possible
to obtain a solution finding these IVC (derivatives of the first orders as is showed in
the eq. (2)), using the DE to obtain the higher derivatives orders, and substitute them
in eq. (1); but this procedure is useful only to solve trivial problems [1]. That
disadvantage can be overcome using the PSSM, because this method make possible to
obtain the others values with the recurrence relation technique.
The PSSM represent a general solution with unknown coefficients, and when the
equation (1) is substituted in the PDE we obtain a recurrence relation for the
expansion coefficients. These coefficients in their form should be expressed in
function of the coefficients result of the IVC application (because as we say before,
the coefficients of the TS are the derivative of the function solution, Eq. (1)). But
these coefficients also can be obtained with the evaluation of the BVC, or of any
other point in the space whose value are known, and it will depend on the kind of the
problem under study. In this way, we obtain a system of equations in function of
these initial value based coefficients. The number of equations should be correlated
with the data, i. e., with the IVC or BVC. In order to obtain and to solve a coherent
algebraic system of equations, we also need the same number of both: coefficients and
equation. All these conditions in first instance help us to guarantee that the PDE is a
well posed problem, i. e., the existence, uniqueness and smoothness of its solution is
well defined [20].
The PSSM is a proposal to find a semi analytic solution as an asymptotic
approximation (in the space and time) of a finite series with a minimal error in the
expansion of terms of the series.
3 The PSSM applied to NLPDE
In order to illustrate the use of the PSSM, we start exemplifying with the solution of the equation of a steady state laminar boundary layer on a flat plate [21, 22]:
3
3
2
22
y
U
y
U
x
U
yx
U
y
U
(3)
The PDE has two variables x and y, therefore the proposal for PSS is:
n m
mn
nm yxayxU ),(
(4)
Substituting eq. (4) in (3):
n m
mn
nm
n n
mn
nm
m
mn
nm
mn n
mn
nm
m
mn
nm
m
yxammm
yxammyxanyxamyxam
3
21111
)2)(1(
)1(
(5)
We use the symbolic algebraic solver of Matlab® to solve the series multiplication un-til reach the l=4 (l=n+m=4) power degree, following the method. Solving for each of the terms and matching each coefficient of the same power degree, we obtain:
In result, we have 13 equations with a set of 13 unknown variables (a03, a13, a04, a14, a23, a05, a33, a06, a15, a24, a07, a34, a25) one physical parameter (ν), and 8 known variables (a00, a01, a10, a11, a02, a20, a12, a21). We could know this set of variables applying the IVC or BVC. Solving this system of equations using newly the symbolic algebraic solver of Matlab® to obtain the recurrence relation in function of the known variables and the physical parameter:
solving the system of equations for the 12 unknown coefficients in function of a00, a01,
a10, a11, and the ν parameter, result in the followings recurrence relation coefficients:
a02 =-(a003a10 + a01a00
2 + 4νa00a10
2 + 2a11νa00 + 4a01νa10)/2ν
a20=(a01 + a00a10)/2ν
a21 =-(a003a10 + a01a00
2 + 4νa00a10
2 + νa11a00 + 3νa01a10)/2ν
2
a12 =(a004a10 + a00
3a01 + νa00
2a10
2 + νa11a00
2 - 4νa00a01a10 - 4νa01
2 - 2ν
2a11a10)/(2ν
2)
a22 =(3a003a10
2 + 4a00
2a01a10 + a00a01
2 + 6νa00a10
3 + 4νa01a10
2 - 2νa11a01)/(2ν
2)
a30 =(a002a10 + a01a00 + νa10
2 +ν a11)/(6ν
2)
a03 =(-a005a10 - a00
4a01 + 6νa00
3a10
2 - νa11a00
3 + 13νa00
2a01a10 + 6νa00a01
2 + 16ν
2a00a10
3+
4ν2a11a00a10+ 12 ν
2a01a10
2- 6 ν
2a11a01)/(6 ν
2) (18)
a31 =-(a002a10
2 + 2a00a01a10 + a01
2)/(2ν
2)
a13 =(- 4a005a10
2 - 6a00
4a01a10 - 2a00
3a01
2 - 16 νa00
3a10
3 + 5 νa00
3a10a11 + 7 νa00
2a01a11 +
29 νa00a012a10 - 24 ν
2a00a10
4+ 24 ν
2a00a10
2a11+ 6 ν
2a00a11
2+ 15 νa01
3 - 16 ν
2a01a10
3+
32 ν2a01a10*a11)/(6a00 ν
2)
a32 =(3a005a10
2 - 3a00
3a01
2 - 6νa00
3a10
3 + 5νa00
3a10a11 - 16νa00
2a01a10
2 - νa00
2a01a11 + 11νa00a01
2a10
- 24ν2a00a10
4+ 20 ν
2a00a10
2a11+ 8 ν
2a00a11
2+ 15a01
3 - 16 ν
2a01a10
3+ 32 ν
2a01a10a11)/(12a00 ν
3)
a23 =(- 5a005a10
2 - 4a00
4a01a10 + a00
3a01
2 - 10 νa00
3a10
3 + νa11a00
3a10 + 16 νa00
2a01a10
2
+7νa11a002a01+ 23 νa00a01
2a10 - 8 ν
2a00a10
4+ 16 ν
2a11a00a10
2+ 3 νa01
3 - 4 ν
2a01a
3+
8 ν2a11a01a10)/12 ν
3
a33 =(- 3a007a10
2 + 3a00
5a01
2 + 12 νa00
5a10
3 + 15 νa00
5a10a11 + 86 νa00
4a01a10
2 + 21 νa00
4a01a11
+ 83 νa003a01
2a10 + 24 ν
2a00
3a10
4+ 72 ν
2a00
3a10
2a11+ 12 ν
2a00
3a11
2+ 15 νa00
2a01
3 +
144ν2a00
2a01a10
3+ 26 ν
2a00
2a01a10a11+ 130 ν
2a00a01
2a10
2- 42 ν
2a00a01
2a11- 48 ν
3a00a10
5+
48ν3a00a10
3a11+ 30 ν
2a01
3a10- 32 ν
3a01a10
4+ 64 ν
3a01a10
2a11)/(36a00 ν
4)
we recover the coefficients of the stationary state as a particular case when we eliminates the coefficients of the t variable.
In the next example, we will try to solve the Korteweg–de Vries equation [25, 26]:
063
3
x
UU
x
U
t
U (19)
The equation admits a traveling wave solution. Then, we can to do the following
transformation:
)(),( ztxU , where tkxz (20)
and kc / is the speed of light. Therefore, substituting Eq. (20) in Eq. (19) results
the following non linear ODE:
063
32
dz
d
dz
dk
dz
dc , (21)
We solve it using the following proposal of PSS:
n
n
n zaz0
)( (22)
Substituting Eq. (22) in Eq. (21):
06)2)(1( 1
10
3
2
21
1
n
n
n
n
n
n
n
n
n
n
n
n znaxazannnkznac (23)
Developing the series until reach n=8, and we obtain the following System of alge-
braic equations for the coefficients:
n=0
13
2
10 66 caakaa
n=1
2204
22
1 212246 caaaaka
n=2
321305
2 3181860 caaaaaak
n=3 (24)
41
2
2406
2 363630 caaaaaaak
n=4
53241507
2 5303030210 caaaaaaaak
n=5
64251608
22
3 636363633618 caaaaaaaaka …
The system of equations is solved considering the known parameters (a0, a1 and a2)
and the physical parameters (k and c):
2
1103 6/)6( kcaaaa
22
12204 12/)36( kacaaaa
4
21
2
1
2
101
2
05 120/)361236( kaakacacaaaa
42
1
2
10
2
2
2
2
2
202
2
06 360/)1590361236( kcaaaakacacaaaa (25)
63
1
2
210
2
1
3
21
22
110
2
1
2
01
3
07
5040/)1801296
2161518108216(
kakaaak
acacakcaaacacaaaa
62
10
2
1
2
0
2
2
2
1
2
2
20
2
2
3
21
22
1
2
20
2
2
2
02
3
08
20160/)75622681188
12962166318108216(
kaaaaaak
aakacacakacaacacaaaa
Therefore, our solution in function of the coefficients (a0, a1 and a2) is:
...)()()()()()(),( 5
5
4
4
3
3
2
210 tkxatkxatkxatkxawtkxaaztxU
Starting from this solution, also we can to obtain a particular solution, just considering
that the parameter a1=0:
2
2204 12/)6( kcaaaa
42
2
22
2202
2
06 360/)361236( kcakaacaaaa
6
2
322
220
222
202
2
02
3
08 20160/)216181296108216( kackcaaackaaacaaaa
and the particular solution in function of the coefficients (a0, a2) is:
...)()()()()(),( 8
8
6
6
4
4
2
20 tkxatkxatkxawtkxaaztxU
The solution incorporates an even symmetry according with the parity property of the PDE in the Eq. (19). In this last example, we solve the coupled Korteweg–de Vries equations [26, 27]:
,)(
332
13
3
x
VW
x
UU
x
U
t
U
(26)
,33
3
x
VU
x
V
t
V
.33
3
x
WU
x
W
t
W
This system of equations also admits traveling wave solutions. Similarly, we can
propose the transformation:
)(),( zutxU , )(),( zvtxV , )(),( zwtxW (27)
where ,tkxz and kc / is the speed of the light. Substituting Eq. (27) in Eq. (26)
results in the coupled system of ODE:
,)(
332 3
32
dz
vwd
dz
duu
dz
udk
dz
duc
(28)
,33
32
dz
dvu
dz
vdk
dz
dvc
,
.33
32
dz
dwu
dz
wdk
dz
dwc
Now we implement the use of a system of PSS functions:
n
n
n zazu0
)( , n
n
n zbzv0
)( and n
n
n zczw0
)( (29)
and substituting Eq. (29) in (28), we obtain the following equations:
1
0 11 0
1
1
10
3
2
21
1
3
3)2)(1(2
n
n n
n
n
n
n
n n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
znczbzcznb
znazazannnk
znac
1
10
3
2
21
1
3)2)(1( n
n
n
n
n
n
n
n
n
n
n
n znbzazbnnnkznbc (30)
1
10
3
2
21
1
3)2)(1( n
n
n
n
n
n
n
n
n
n
n
n znczazcnnnkzncc
Matching the coefficients of the same power degree until reach n=8, we obtain the
following system of algebraic equations for the expansion series coefficients:
Therefore, with these coefficients we can to obtain the following solutions:
...)()()()()(),( 5
5
4
4
3
3
2
210 tkxatkxatkxatkxawtkxaatxU
...)()()()()(),( 5
5
4
4
3
3
2
210 tkxbtkxbtkxbtkxbwtkxbbtxV
...)()()()()(),( 5
5
4
4
3
3
2
210 tkxctkxctkxctkxcwtkxcctxW
In a similar way with the previous problems, we can to considerer the
particular case (a1=b1=c1=0), and to obtain an even solution for U, V and
W because the symmetry of the system of equations is odd.
4 Discussion and conclusions
In this letter we have showed that is possible to solve non linear DE with the PSSM.
This is implemented as a general approximate solution for each one, PDE, or ODE,
in a similar way than the solution of Linear ODE. So, we transform one DE problem
into an algebraic system of equations. Therefore, with this method, it is possible to
obtain a well posed problem, because here, when the system of DE is closed, i. e. it
have the same variable function than equations, then the number of algebraic
variables (expansion series coefficients) that we obtain with this method, is the same
with respect to the equations number. Thus, according with the methodology
described, it is possible to obtain a solution.
This PSSM is a semi analytic technique that could permits to obtain, in an easier
and exact way, the solution of difficult differential equations with an approximated
closed form expression. This is especially useful to solve non linear equations,
opening the possibility to describe in an exact and convergent way, the behavior of
chaotic dynamical systems [14, 16].
Although it was not the focus of this article, the solution could be approximated
until the degree necessary into the power series. The convergent of the PSSM depend,
of the number of terms used of the series. Once we know it, we can to determine the
domain of the space and time where the solution is valid.
In summary, we have shown that the PSSM is a general technique which can be
used to solve any kind of non linear differential equations [17]. This opens the
possibility of analyze other characteristics of the NLPDE, and obtaining a better semi
analytic approximations that involves less computational efforts. This procedure will
provide much greater accuracy just adding more terms into the series.
5 Acknowledgements
The authors are extremely grateful to Roman López Sandoval and Rogelio Ospina Ospina for their very careful reading and helpful discussion of the manuscript. Also E. L. S. would like to acknowledge the support of the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) with the Grant of the PCI D-B from 01/01/2012 until now.
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