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A new algorithm for matching boundary conditions in decomposition solutions
A New Algorithm for Matching Boundary Conditions in Decomposition G. Adomian*
Solutions
and R. Rach
General Analytics
Corporation
ABSTRACT A computationally to boundary
convenient
conditions
and partial differential
and simple procedure
in two-point equations.
boundary
is presented
value problems
The nonlinear
case is discussed
to match solution
for linear differential elsewhere.
For decomposition method solution of problems [I-3] modeled by ordinary or partial differential equations, this article proposes a computationally more efficient technique to evaluate coefficients to match boundary conditions. L
ORDINARY
DIFFERENTIAL
EQUATION
We are concerned with methodologv here. The obviously simple examples are used to illustrate procedures more clearly. Ll,
c12u/ctx2 + pu = 0
In the customary operator equation is written
4[,)
= h
fL( 6,)
= b,.
notation
of the decomposition
method,
the above
Lu + Ru = 0, that is, L = d2/dx2,
R = p. Solve for Lu and apply L-’
to both
sides,
*Address correspondence to G. Adomian at 155 Clyde Rd., Athens, G,4 30605
APPLZED MATHEMATZCS AND COMPUTATZON 586-68 0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas,
New York, NY 10010
61
G. ADOMIAN
62
AND K. RACH
where
L-l(*)
where
=
co +
clx
+
Il (-) dxdx
=
C”
ClX
+
1,2(e),
+
Z,” means the twofold integration
JT’Lu(*) = L-‘Ru u = co +
clx
z;zpu.
+
We can decompose u into u = C:= Ok, to write C:=O~, Z:pC”,,,u,, or use double decomposition [2,3] to write
EiL n=O
u(
c
) =
@‘+
5 C(l)n)- z,2p 2 2 up.
x
n=O
m=O
n, = 0
m=O
= cO + c1 x -
m=O
NOTE. In other work u’,‘“) has been written u,, m. Note also that
c
u,
=
cup.
5 n=O
n=O
m=O
We have
u,
=
~(n:-‘~) and
?
m=O Therefore,
cc mm
C u, = C C uv-“‘I. n=O m=O n=O
we have u(o) = @’
uo= Ul
=
u, =
@
+
0
@’
= ($’
+
+ xc$v
xcp-
I,‘( puo)
e
u’,“-“’ = cim;“’ + xc(,“)- z,2pu,_1.
m=O
The
approximant
4 ,,,+I = C:=,u,.
to the Thus,
solution +,+1
u is c$,+ ,[u] or simply 4, + I, where = (b, + u,. The exact boundary conditions
Matching
Solutions for Boundary
which we approximate
Hence
&
Conditions
by
= uO, and we write
For the next stage of approximation
Since,
+l(&l>
=
c/+ = 41 + ul:
b,, clearly ~~(6~) = 0. Also 4,(
Since
63
+I( t2) = b,,
s,>
= +I( 52) + 4
it is clear that u1(12) k+
= 0. We can continue
I( CT,> = 4&C !C,> + %L( SJ
4,+,(L) Again since c$,(.$~> = b, Summarizing results
s2> = 62.
= 4L(E2> and 4,(t2)
4 445)
+ %n( s,> = b,,
4
S,> = b,
4
6,)
t,>
= b1 = b2.
we have u,((~)
= b,
= u1( 8,)
= 0
= %X82.)
= 0
to
= u~(,$~)
= 0.
63
G. ADOMIAN
AND R. RACH
Since u,,, = c0(“‘) + XC{“~) - Z,?pu,,,~, and ug = c{:)) + XY!“’ 211= c;;’ + xc(,‘) - pcg)x”/2!
- pc(:‘)x”/3!
111
u
=
c
(
_p)ll{Cbl)l--l)X*i1/(2.)!+
cyr’)X*~1+‘/(2TL
1,‘
n = 0
For m & 1,
implying cgv + &cI”) zz b, = b’,“’ ~‘0” + &cy) In matrix form,
if det g # 0. Then,
=
b, = b’,O).
+ l)!}.
Matching
Solutions for Boundary
Conditions
65
where
if det 2 # 0. Continuing,
we can go to mth order writing
implying
where
by0
= -
E
( -P)“{c6”-“)5;2n/(2n)!+
4”-“‘5;“+1/(2n
+ l)!)
(-P)“{cl;m_n)~~n/(2n)!+
.yq;n+l/(2,
+-l)!}
n=l
.
m
@;“’
=
-
c n=l
66
G. ADOMIAN AND R. RACH
or
and
2(m) =
(
,)jl&W
if det g z 0. To accelerate ations, compute
convergence
and avoid additional boundary-matching
evalu-
so that
PARTIAL We
DIFFERENTIAL
are concerned
used to illustrate
EQUATION
with methodology
procedures
EXAMPLE here.
These
more clearly.
d”U/dX2 f4(,>
+ d2u/dy2 Y) =k(y)
UC&> Y> = b,(y),
= 0
simple
examples
are
Matching
Solutions for Boundary
67
Conditions
where we get “constants of integration” ordinary differential equation example d2u/dx2
+ pu
u(S,>
= b,
u( 6,)
= b,,
c,(y),
c,(y).
Compare
with the
= 0
where the integration constants are ca, cr. By using the concept of partial solutions [l, 2, 41 we can replace p by d2/dy2 and replace c,,, cl, b,, b,, by cO( y), cJ y), b,( y>, b,( y>. Thus, write L,u
=
-(d2/dy2)u
or cO + cr -
so that a partial differential equation equation (0.d.e.). Hence, we write
(d2/ay”)Cu,,
(p.d.e.1 is solved as a one-dimensional
For large matrices (nth order), the inversion can be done by the partitioned decomposition method [5,6] where
68
G. ADOMIAN
AND
R. RACH
Method
and Some
The elements are
For A=
A,, A i 21
A,, A,, )
J))
and
R=(ll
we can write
L=
iA;’
‘i’)
for det L > det R. REFERENCES G. Adomian and R. Rach, A Review of the Decomposition Recent
Results
Math. Applic., G. Adomian, Kluwer G.
21(5):101-127 Solving Frontier
Publications,
Adomian
Problems To appear, -,
for Linear
and
Problems
Equations,
Camp.
of Physics-The
Decomposition
Method,
1993.
Analytic
Dimensions
of partial
Solution
of
by Decomposition,
Nonlinear
Boundary-Value
J. Math. Anal. and Applic.,
Application
solutions
in the decomposition
method,
Cornput. Mnth.
(1990). of the decomposition
Anal. Appl. 108(2):409-421 -,
Differential
1993.
Equality
Appl. 19(12):9-12 -,
Rach,
Partial
(1991).
To appear, R.
in Several
or Nonlinear
A new computational
ModelZing, 7(2-3):113-141
method
to inversion
of matrices,
J, Muth.
(1985). approach for inversion of very large matrices, Math. (1986).
Solutions
and R. Rach
General Analytics
Corporation
ABSTRACT A computationally to boundary
convenient
conditions
and partial differential
and simple procedure
in two-point equations.
boundary
is presented
value problems
The nonlinear
case is discussed
to match solution
for linear differential elsewhere.
For decomposition method solution of problems [I-3] modeled by ordinary or partial differential equations, this article proposes a computationally more efficient technique to evaluate coefficients to match boundary conditions. L
ORDINARY
DIFFERENTIAL
EQUATION
We are concerned with methodologv here. The obviously simple examples are used to illustrate procedures more clearly. Ll,
c12u/ctx2 + pu = 0
In the customary operator equation is written
4[,)
= h
fL( 6,)
= b,.
notation
of the decomposition
method,
the above
Lu + Ru = 0, that is, L = d2/dx2,
R = p. Solve for Lu and apply L-’
to both
sides,
*Address correspondence to G. Adomian at 155 Clyde Rd., Athens, G,4 30605
APPLZED MATHEMATZCS AND COMPUTATZON 586-68 0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas,
New York, NY 10010
61
G. ADOMIAN
62
AND K. RACH
where
L-l(*)
where
=
co +
clx
+
Il (-) dxdx
=
C”
ClX
+
1,2(e),
+
Z,” means the twofold integration
JT’Lu(*) = L-‘Ru u = co +
clx
z;zpu.
+
We can decompose u into u = C:= Ok, to write C:=O~, Z:pC”,,,u,, or use double decomposition [2,3] to write
EiL n=O
u(
c
) =
@‘+
5 C(l)n)- z,2p 2 2 up.
x
n=O
m=O
n, = 0
m=O
= cO + c1 x -
m=O
NOTE. In other work u’,‘“) has been written u,, m. Note also that
c
u,
=
cup.
5 n=O
n=O
m=O
We have
u,
=
~(n:-‘~) and
?
m=O Therefore,
cc mm
C u, = C C uv-“‘I. n=O m=O n=O
we have u(o) = @’
uo= Ul
=
u, =
@
+
0
@’
= ($’
+
+ xc$v
xcp-
I,‘( puo)
e
u’,“-“’ = cim;“’ + xc(,“)- z,2pu,_1.
m=O
The
approximant
4 ,,,+I = C:=,u,.
to the Thus,
solution +,+1
u is c$,+ ,[u] or simply 4, + I, where = (b, + u,. The exact boundary conditions
Matching
Solutions for Boundary
which we approximate
Hence
&
Conditions
by
= uO, and we write
For the next stage of approximation
Since,
+l(&l>
=
c/+ = 41 + ul:
b,, clearly ~~(6~) = 0. Also 4,(
Since
63
+I( t2) = b,,
s,>
= +I( 52) + 4
it is clear that u1(12) k+
= 0. We can continue
I( CT,> = 4&C !C,> + %L( SJ
4,+,(L) Again since c$,(.$~> = b, Summarizing results
s2> = 62.
= 4L(E2> and 4,(t2)
4 445)
+ %n( s,> = b,,
4
S,> = b,
4
6,)
t,>
= b1 = b2.
we have u,((~)
= b,
= u1( 8,)
= 0
= %X82.)
= 0
to
= u~(,$~)
= 0.
63
G. ADOMIAN
AND R. RACH
Since u,,, = c0(“‘) + XC{“~) - Z,?pu,,,~, and ug = c{:)) + XY!“’ 211= c;;’ + xc(,‘) - pcg)x”/2!
- pc(:‘)x”/3!
111
u
=
c
(
_p)ll{Cbl)l--l)X*i1/(2.)!+
cyr’)X*~1+‘/(2TL
1,‘
n = 0
For m & 1,
implying cgv + &cI”) zz b, = b’,“’ ~‘0” + &cy) In matrix form,
if det g # 0. Then,
=
b, = b’,O).
+ l)!}.
Matching
Solutions for Boundary
Conditions
65
where
if det 2 # 0. Continuing,
we can go to mth order writing
implying
where
by0
= -
E
( -P)“{c6”-“)5;2n/(2n)!+
4”-“‘5;“+1/(2n
+ l)!)
(-P)“{cl;m_n)~~n/(2n)!+
.yq;n+l/(2,
+-l)!}
n=l
.
m
@;“’
=
-
c n=l
66
G. ADOMIAN AND R. RACH
or
and
2(m) =
(
,)jl&W
if det g z 0. To accelerate ations, compute
convergence
and avoid additional boundary-matching
evalu-
so that
PARTIAL We
DIFFERENTIAL
are concerned
used to illustrate
EQUATION
with methodology
procedures
EXAMPLE here.
These
more clearly.
d”U/dX2 f4(,>
+ d2u/dy2 Y) =k(y)
UC&> Y> = b,(y),
= 0
simple
examples
are
Matching
Solutions for Boundary
67
Conditions
where we get “constants of integration” ordinary differential equation example d2u/dx2
+ pu
u(S,>
= b,
u( 6,)
= b,,
c,(y),
c,(y).
Compare
with the
= 0
where the integration constants are ca, cr. By using the concept of partial solutions [l, 2, 41 we can replace p by d2/dy2 and replace c,,, cl, b,, b,, by cO( y), cJ y), b,( y>, b,( y>. Thus, write L,u
=
-(d2/dy2)u
or cO + cr -
so that a partial differential equation equation (0.d.e.). Hence, we write
(d2/ay”)Cu,,
(p.d.e.1 is solved as a one-dimensional
For large matrices (nth order), the inversion can be done by the partitioned decomposition method [5,6] where
68
G. ADOMIAN
AND
R. RACH
Method
and Some
The elements are
For A=
A,, A i 21
A,, A,, )
J))
and
R=(ll
we can write
L=
iA;’
‘i’)
for det L > det R. REFERENCES G. Adomian and R. Rach, A Review of the Decomposition Recent
Results
Math. Applic., G. Adomian, Kluwer G.
21(5):101-127 Solving Frontier
Publications,
Adomian
Problems To appear, -,
for Linear
and
Problems
Equations,
Camp.
of Physics-The
Decomposition
Method,
1993.
Analytic
Dimensions
of partial
Solution
of
by Decomposition,
Nonlinear
Boundary-Value
J. Math. Anal. and Applic.,
Application
solutions
in the decomposition
method,
Cornput. Mnth.
(1990). of the decomposition
Anal. Appl. 108(2):409-421 -,
Differential
1993.
Equality
Appl. 19(12):9-12 -,
Rach,
Partial
(1991).
To appear, R.
in Several
or Nonlinear
A new computational
ModelZing, 7(2-3):113-141
method
to inversion
of matrices,
J, Muth.
(1985). approach for inversion of very large matrices, Math. (1986).