Internat. J. Math. & Math. Scl.
VOL. 12 NO. I (1989) 137-144
137
APPLICATION OF DECOMPOSITION TO HYPERBOLIC, PARABOLIC,
AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
G. ADOMIAN
Center for Applied Mathematics
University of Georgia
Athens, Georgia 30602
(Received September 22, 1987)
ABSTRACT:
The decomposition method is applied to examples of hyperbolic, parabolic,
and elliptic partlal differential equations without use of linearlzatlon techniques.
We consider first a nonlinear dissipative wave equation; second, a nonllnear equation
modeling convectlon-diffusion processes; and flnally, an elliptic partial differential
equation.
KEY
AND
WORDS
PHRASES.
method,
Decomposition
linear
and
partlal
nonlinear
differential equations, parabolic equations, elliptic equations, hyperbollc equations.
1980 AMS SUBJECT CLASSIFICATION CODES. 35L35, 35L70.
I.
INTRODUCTION:
The decomposition method [I] has developed rapldly and is now providing solutions
in the form of converging analytlc series for broad categories of ordinary or partlal
differential
conditions.
of equations with given inltlal/boundary
or
systems
equations
An important advantage is that llnearlzation is not required. The rather
global character is shown by application to some typical examples
equation,
and
equation
modeling
convection-dlffusion
processes,
a dissipative wave
and
elliptic
an
equation Without the use of llnearlzation techniques.
2.
HYPERBOLIC CASE
Consider
f(u(x,t))
and
L
the
dissipative
wave
equation
a continuous bounded function and
3 2/x2
utt
(t,x)
Uxx
[O,T]
+
(/@t)f(u)
R.
Let
0
L
t
with
-2/t2
and vrite
L u- L u
t
x
(8/3t)f(u(x,t))
(2.1)
G. ADOMIAN
138
Using the decomposition method [I] we solve for each linear term thus
L u
L u- (/t)f(u)
L u
L u + (81t)f(u)
(2.2)
x
t
x
t
Operating with the inverses, we have
LtlLt
u
t
u
and
Lx ILxu
u
x’
where
the homogeneous solutions are evaluated from the given Initial/boundary conditions.
Thus (2.2) becomes
t + LILxu
L l(/3t)f(u)
@x + LILtu + L l(lt)f(u)
u
u
Adding and dividing by two
u- (1/2)(
t
x + (1/2)(LlL
+
x
+
)u
L-1L
x t
Ll)(:/t)f(u)
I/2)(L
or if
(I/2)(L ILX + L-ILx t
K
L1)
(1/2)(L-1
C
u
t
(I/2)( t +
0
(2.3)
x
we have
u
u
0
+ Ku + G(lt)f(u)
(2.4)
a result also obtained by operating on (2.1) with
Let
u
u
nffi 0
n
(L
-I
t
with u 0 as defined and let f(u)
.
generated for the specific function f(u) as discussed in
u + K
0
u
nffi 0
u + oCelOt)
n
n=0
A
n
.
L-l).
x
n-O
An,
Ill.
where the An are
Now
(2.5)
We define
u
n+l
Ku
n
+ G(/Bt)A
n
(2.6)
APPLICATIONS OF DECOMPOSITION TO PARTIAL DIFFERENTIAL EQUATIONS
for
n
0
[1,2].
as discussed in
3.
An n-term approximation
to complete the solution.
n-
139
n-!
i-O
ui
suffices
PARABOLIC CASE:
Consider a nonlinear parabolic partial differential equation modeling convection-
diffusion processes given in the form
u
au
t
xx
+ f(u,t,x)u
x
(3.1)
0
on a defined finite region on R with t
function of
O.
Assume a is a constant, f is a smooth
t,x,u, and the Inltlal/boundary conditions are given.
obtain a quantitative solution, the specific form of f is required.
Of course,
to
When it is, any
separable terms in x and t will be designated by -g(x,t) and any remaining term
dependent on u, and multiplying ux can be written
}2/}x2 and write (2.1) as
x
N(U,Ux).
Let L
t
/t
and
L
Ltu- aLxU
g- N(u,u
(3.2)
x)
The decomposition method solves for each linear operator term in turn; thus
L
g +
tu
aLxU- S(U,Ux
LxU -a-lg + a-ILtu + a-IN(u,ux)
Since
u-c
L-1Lt
A
u
u
u.A+L -I g
B
u
/
Cx
u(x,O)
and
u
--LIL
x x
u- B
Cx,
we obtain
/
-1
a-lL-1
L + a-lL-1 N(u uX)
g + a-1 L
tu
x
x
x
A is the initial condition, B and C are evaluated from the remaining two conditions.
Ne add the two equations for u and divide by two, obtaining a single equation for
u.
If we define
u
LIg a-IL:Ig}
-(1/2){aL: 1Lx + a-lL-1Lt}x
-(I/2){L a-lL-l}x
(I/2){u(x,O)
0
K
G
+ B + Cx +
we have
u
u
0
+ Ku + G
N(u,u
x
(3.3)
G. ADOMIAN
140
The
solution by
.
u
decomposition is
components given by
u
for
)
n
0,
u
with
n-O
Ku
n+l
.
n
+ GA
nffi0
.
n
where the An are defined [I] for N(u,u
U
the given u0 and the remaining
Knu0 + G
x)
x)
with N(u u
(3.4)
A
Thus
n-0
(3.5)
Nu
is the complete solution since the An are easily evaluated for any specific function
f.
A practical solution is given by
n-I
n
)" ui
since
The solution can also be made by operating on (2.2) with
de.slgnate
LtlLt
u
the solution of
u
t
u
u
and
L u
t
L_IL
u
x X
t
x"
0 by
u-
the
solution
converges
(L-l-
-IL-Ix ).
Let us
iffiO
generally quite rapidly.
and of L u
x
0 by
We obtain
x
t
i.e.
#t -eLt L xu -IL x L t u + u #x
N(u ux))
(L 1- c*-lL-1)(g
X
-I
-I
(1/2)(# t +
(l/2)(L-lt a-lLx-1)g
+
#x)
(I/2)(L 1Lx
+
a-IL-1Lxt)u
a-lL-1)N(Ux ’Ux)
--(1/2)(L
or
u
u
0
+ Ku + G
(3.6)
N(U,Ux
Though the result is the same and
t’ Cx are
evaluated from the given conditions,
writing u 0 as
Uo
(1/2)(#t
+
x + (1/2)(L-It -a-lL-1)gx
means the result is not limited to the given parabolic equation
be of any order.
The nonlinear term
N(U,Ux)
(3.7)
the derivatives can
can also be more general.
generated for composite nonlinear functions [I].
The An can be
APPLICATIONS OF DECOMPOSITION TO PARTIAL DIFFERENTIAL EQUATIONS
4.
141
ELLIPTIC CASE:
The
V2u
elliptic equation
+ k(x,y,z)u
+ k(x,y)u
u + u
xx
yy
821ax2,
x
in
some
problems of
0
x2 +
k(x,y)
L
arises
Let’s consider the case:
physics and engineering.
Define
f(x,y,z)
82/a
L
y
[Lx + Ly]U +
and write
ku- 0
Solve for each linear operator term in turn.
LxU
ku
LyU
ku
Then
LyU
(4.2)
L
Apply the inverses
L-ILxxU
u-
@x’
LxU
-I to the
first,
x
L-ILyYU
u-
@y,
L y-I to the second.
where
@x @y
Then since
are defined by the inltlal/boundary
conditions, we have
u
@x
L
u
@y
Ll[x
[x2 + y2]u
2
+ y2]u
Choosing convenient conditions u
sin x at y
0 at x
(4.3b)
0 or y
0 and u
sin y at x
and
1, we have
x
@y
Let
(4.3a)
A+ Bx
x sln y
C + Dy
y sin x
represent the u0 term of the decomposition u
Thus, considering both in parallel
@x and @y
in the two
equations for u in (4.3a,b).
u
u
0
0
x sin y
(4.4a)
y sin x
(4.4b)
142
G. ADOMIAN
so that
o Lx(X
u--
u
u
y)
u.
n=O
y2)
(x 2 +
Ly
0-
+
u
n=O
n
’us
Un+l
Un+l
for
_L
2
-l(x
y
(4.5a)
y2)u n
+
(4.5b)
for (4.5a) uses (4.4a), etc.
0 where u
n
_L l(x 2 + yZ)un
L-Ix (xz
Ul
+
For example,
yZ)u 0
(4.6a)
_L l(x z + yZ)u 0
ul
(4.6b)
The one-term approximation to u is given by
I
(x sin y + y sin x)/2.
using the first terms of the trigonometric series that
I
-x3y3/3!.
term and the second term of the expansion for sin y, we get
u
From the
xSyS/5!,
u2
term
the correct
of
cancellation
[x3/3!)sin y]
and
n
If
solution.
f(u)
third
terms
than
other
calculate several
we
the
series
nonlinearities f(u) are involved, the appropriate
let
the
term
of
y,
sin
we
get
is given by the n-term series for sin xy
To save computation, we can substitute u
plus noise terms.
indeed
-L-IL
x y
The n-term approximant
etc.
We see
We observe from the
xy.
for
sin xy and verify it is
terms,
it
An polynomials
is
Or,
xy.
sin
easy to
if
see
analytic
are generated and we
A
n-0
To
make
some
have Lx
of
u,
2 + 40xu.
and
the
Operating with
L-I(2)
checks
dZu/dx2
dimensional case
-I
the
with u(-l)
2
function
forcing
yields
u
A + Bx + x 2 and let
A + Bx +
u
u
nffiO
that
the sn is u.
of
accuracy
methodology,
u(1)
0.
we
consider
Here L
the
d2/dx2
one-
and we
This is a relatively stiff case because of the large coefficient
nonzero
L
of
-40xu
We identify
Un+
-I
L
yields
an
additional
L-I(2) + L-l(40xu).
n
Airy-like
Let u 0
function.
A + Bx +
with the components to be determined so
(40xu).
Then all components can be
APPLICATIONS OF DECOMPOSITION TO PARTIAL DIFFERENTIAL EQUATIONS
determined, e.g., u
+
(10/7)x 8, etc.
An n-term approxlmant
given by -0.135649,
-0.083321, for x
(20/3)Ax + (10/3)Bx 4 + 2x 5
for x
143
(80/9)Ax 6 + (200/63)Bx 7
and u 2
n-1
n
u
i
with n- 12 for x-- 0.2 is
0.4 is given by -0.113969,
for x
0.8 is given by -0.050944, and for x
These easily-obtalned results are correct to seven digits.
0.6 is given by
1.0 is, of course, zero.
We see that a better
solution is obtained and much more easily than by variational methods.
The solution
is found just as easily for nonlinear versions without llnearlzatlon.
REFERENCES
I.
ADOMIAN, G., Nonlinear Stochastic Operatpr Equations, Academic Press, (1986).
2.
ADOMIAN,
3.
ADOMIAN, G., Decomposition Solution for Dufflng and Van der Pol Oscillators,
Int. J. of Math. Sclences 9, 4, (1986), 731-2.
4.
Analysis With the
BELLOMO, N. and RIGANTI, R., Semillnear Stochastic Systems:
Method of the Stochastic Green’s Function and Application in Mechanics, J.
Math. Anal. and Appllc. 96, 2, (1983).
5.
BELLOMO,
G.
Applications of the Decomposition Method to the
and 2, (1986).
Equations, J. Math. Anal. and...Appllc., llgp
N. and MONACO R., A Comparison Between Decomposition Methods and
Perturbation Techniques for Nonlinear Random Differential Equations, J. Math.
Anal. adnLAppllc. II0_____,
6.
Navler-Stokes
2, (1985).
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the Transient Behavior,
Analysis of
J. Math. Anal. and Appllc., in press.
7.
RACH, R., A Convenient Computational Form for the An Polynomials, J. Math. Anal.
and Appllc., 102 2, (1984), 415-419.
8.
GABETTA, E., On a Class of Semillnear Stochastic Systems
9.
RACH, R., On Continuous Approximate Solutions of Nonlinear Differential
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Quadratic Type Nonlinearities: Time Evolution of the Probability Density, J.
Math. Anal. and Appllc., in press.
Equations J. Math. Anal. and Appllc., in press.
I0. RACH,
R., On the Adomian (Decomposition) Method and Comparisons with Picard’s
Method, J. Math. Anal. and Appllc., in press.
11. BIGI,
D. and RIGANTI, R., Solutions of Boundary-Value Problems by Decompolstion
Me thod.
12. BONZANI, I., Analysis of Stochastic Van der Pol Osclllators Using the
Decomposition Method, C_o_mplex and Distributed Systems: Analysis, Slmulatlon
and Control, eds. S. C. Tzafestas and P. Borne, Elsevir, (1986).
13. RIGANTI,
R., Analytical Study of the Class of Nonlinear Stochastic Autonomous
one Degree of Freedom, Meccanica, 14, 4, (1979), 180-86.
Oscillators with
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Mechanics, World Scientific Publishing Co., Singapore, (1987).
15..ADOMIAN,
G.,
Physics
and
Applications of Nonlinear Stochastic Systems Theory to.. Phy.s.lcs,
Reidel, Dordrecht, (1988).