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Random volterra integral equations
Mathl. Comput. Modelling Vol. 22, No. 8, pp. 101-102, 1995
Pergamon
Copyright@1995 Elaevier Science Ltd Printed in Groat Britain. All rights reserved 08957177/95 $9.50 + 0.00
0895-7177(95)00158-l
Random Volterra Integral Equations G. ADOMIAN General Analytics Corporation 155 Clyde Fbad, Athens, GA 30605, U.S.A. (Received December 1994; accepted January
1995)
Abstract-using the decomposition method, differential and partial differential equations involving stochastic processes have been solved. This paper considers application to random Volterra integral equations. Keywords-Decomposition
A random
Volterra
integral
method, Random integral equation, Adomian polynomials.
equation
can be written
as
t x(&w)
=
h(t,w)
+
I
0
Ic(t,7;~)f(~,2(t,~))dr.
The z(t, w) is decomposed into CzY, z, with 20 identified as h(t; w). The function f is represented by Cp=o A,(f). The A, are the (Adornian) polynomials [l-3]; the sum is a rapidly convergent series which is equivalent to a generalized Taylor series about the function zo [2]. Thus,
x(t;w) = h(t;w) +
tW,T;W)f&LIfW J 0
The components
t s =stt =st s =t
n=O
of x can now be written XI-J =
h(T;w),
21(7;w) =
0
0
0
x2(7-;w) =
0
J
k(t, ‘7 ‘J)f(T, X1(‘-,w)) dT k(t, T; w)& dT
t s 0
xn(T; w) =
k(t,
7;
u)A,_~
0
101
d-r
G. ADOMAN
102
The approximant
to the solution is m-1
(Pm =
C
Xn,
n=O
which converges to z in the limit. Quite generally, a very few terms are sufficient as seen in numerous examples [3]. (The error in a recent calculation by Jinquing and Wei-Guang [4] for the Duffing equation using only four terms, i.e., n = 4, w&s less than O.OOOl%.) Because of the rapid convergence [4,5] so that a few terms can represent an essentially exact solution, the (Pi can be used to obtain the expectation and covariance, and with an assumption of Gaussian behavior, can be used to get higher statistics as well [3].
REFERENCES 1. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., (1994). 2. G. Adomian, Nonlinear Stochastic Operator Equations and Applications to Physics, Kluwer Acad. Publ., (1989). 3. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, (1986). 4. F. Jinquing and Y. Wei-Guang, Adomian’s decomposition method for the solution of generalized Duffing equations, In Proc. 1992 Internat. Workshop on Mathematics Mechanization, Beijing, China, (1992). 5. Y. Cherruault, G. Saccomandi and B. Some, New results for convergence of Adomian’s method, Mathl. Comput. Modelling 16 (2), 83-93 (1992).
Pergamon
Copyright@1995 Elaevier Science Ltd Printed in Groat Britain. All rights reserved 08957177/95 $9.50 + 0.00
0895-7177(95)00158-l
Random Volterra Integral Equations G. ADOMIAN General Analytics Corporation 155 Clyde Fbad, Athens, GA 30605, U.S.A. (Received December 1994; accepted January
1995)
Abstract-using the decomposition method, differential and partial differential equations involving stochastic processes have been solved. This paper considers application to random Volterra integral equations. Keywords-Decomposition
A random
Volterra
integral
method, Random integral equation, Adomian polynomials.
equation
can be written
as
t x(&w)
=
h(t,w)
+
I
0
Ic(t,7;~)f(~,2(t,~))dr.
The z(t, w) is decomposed into CzY, z, with 20 identified as h(t; w). The function f is represented by Cp=o A,(f). The A, are the (Adornian) polynomials [l-3]; the sum is a rapidly convergent series which is equivalent to a generalized Taylor series about the function zo [2]. Thus,
x(t;w) = h(t;w) +
tW,T;W)f&LIfW J 0
The components
t s =stt =st s =t
n=O
of x can now be written XI-J =
h(T;w),
21(7;w) =
0
0
0
x2(7-;w) =
0
J
k(t, ‘7 ‘J)f(T, X1(‘-,w)) dT k(t, T; w)& dT
t s 0
xn(T; w) =
k(t,
7;
u)A,_~
0
101
d-r
G. ADOMAN
102
The approximant
to the solution is m-1
(Pm =
C
Xn,
n=O
which converges to z in the limit. Quite generally, a very few terms are sufficient as seen in numerous examples [3]. (The error in a recent calculation by Jinquing and Wei-Guang [4] for the Duffing equation using only four terms, i.e., n = 4, w&s less than O.OOOl%.) Because of the rapid convergence [4,5] so that a few terms can represent an essentially exact solution, the (Pi can be used to obtain the expectation and covariance, and with an assumption of Gaussian behavior, can be used to get higher statistics as well [3].
REFERENCES 1. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., (1994). 2. G. Adomian, Nonlinear Stochastic Operator Equations and Applications to Physics, Kluwer Acad. Publ., (1989). 3. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, (1986). 4. F. Jinquing and Y. Wei-Guang, Adomian’s decomposition method for the solution of generalized Duffing equations, In Proc. 1992 Internat. Workshop on Mathematics Mechanization, Beijing, China, (1992). 5. Y. Cherruault, G. Saccomandi and B. Some, New results for convergence of Adomian’s method, Mathl. Comput. Modelling 16 (2), 83-93 (1992).