IDE has special importance in engineering and sciences [2,3]. IDE can be ... In this work, we use ADM and Laplace transform method to solve some. PIDEs.
SOLVING PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS USING ADOMIAN DECOMPOSITION METHOD AND LAPLACE TRANSFORM METHOD
Jyoti Thorwe, Dr. Sachin Bhalekar Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India.
National Conference on Mathematical Sciences (NCMS-2012), School of Mathematical Sciences, North Maharashtra University, Jalgaon March 5, 2012.
1. Introduction : Integral equations are one of the most useful mathematical tools in both pure and applied analysis. The theories of ordinary differential equations and partial differential equations are fruitful source of integral equations [1]. Integro-differential equation (IDE) contains both differential and integral operators. IDE has special importance in engineering and sciences [2,3]. IDE can be solved by various methods such as Adomian decomposition method (ADM),Variational iteration method(VIM), Homotopy perturbation method, Laplace transform method, Fourier transform method. The ADM is a non-numerical method which solves a wide variety of functional equations such as linear, nonlinear differential equations, Integral equations, IDE [4,5]. ADM proposed by G. Adomian in 1980 [4,5 ] has many advantages such as it avoids linearization, perturbation and discretization in order to find an explicit solution. Laplace transform gives functional description which often simplifies the process of analyzing the behavior of the system [6]. Laplace transform method is particularly useful when the kernel in equation is of convolution type [6] such as Abel’s equation [7]. Partial integrodifferential equations (PIDEs) are IDEs where dependent variable contains more than one independent variables. In this work, we use ADM and Laplace transform method to solve some PIDEs.
2. Preliminaries: 2.1. Laplace Transform method: Definition: The Laplace transform of a function f(x), is defined by ̅ (p) = ℒ {
}=∫
(If integral on RHS exists) where, x ≥ 0 , p is real and ℒ is called the Laplace transform operator. Convolution Theorem : } = ̅ (p) and ℒ {
If ℒ { ℒ{
*
}=ℒ{
} = ̅ (p) , then }ℒ{
} = ̅ (p) ̅ (p).
2.2. Adomian decomposition method: 2.2.1 ADM is useful in solving functional equations of the form u = f + L(u) + N(u)
(1)
where, L(u) and N(u) are linear and nonlinear function of u respectively, f is known function. It is assumed that the nonlinear operator N(u) can be decomposition as : N(u) = ∑ , where,
∑
are called Adomian polynomials. The ADM admits the use of decomposition series =∑
∴ (1) becomes ∑
= f + L(∑
) + N (∑
=f+∑
+∑
If u0 = f , un = L(un-1) + N(An-1) then ∑
) (2) ,n ≥ 1
satisfies (2).
2.2.2. Modified ADM (MADM): If f = f1 + f2 then we may take u0 = f1 (or u0 = f2) In this case the components of ADM series becomes u1 = f2 + L (u0) + A0 un+1 = L (un) + An , n ≥ 1 which may lead to a rapid convergence [4]. This procedure is called modified Adomian decomposition method. 3. Solving PIDEs: 3.1. Using Laplace transform method: Consider PIDE, ∑
∑
∫
(3.1.1)
with prescribed conditions. where, or the functions of
are known functions. .
’s are constants
Taking Laplace transform on both sides of PIDE (3.1.1) w.r.t. t and using Convolution theorem we get ordinary differential equation in ū(x,p). Now taking inverse Laplace transform of ū(x,p) we get a solution u(x,t) of (3.1.1) 3.2. Using Adomian decomposition method: Integrating equation (3.1.1) twice w.r.t. t from 0 to t we get functional equation of the form (1) which can be solved using ADM and MADM. Examples: 1) Consider the PIDE tt
=
+2∫
–2 ( ,0) = ex ,
with initial conditions
and boundary conditions
(1.1) t(
,0) = 0
(1.2)
(0,t) = cos t
(1.3)
Taking Laplace transform on both the sides of (1.1), ̅ ( , )̅ ( , )=
[∫
∫
=
+2( )ū-2
∫
(
( )
(1.4)
+C]
Therefore the solution of the (1.4) is , ̅ ( , )=(
)
+C
(1.5)
from the boundary condition (1.3) ̅ ( , )=
(1.6)
using (1.5) and (1.6) C=0 (1.5) becomes ̅ ( , )=(
)
(1.7)
Taking inverse Laplace transform of (1.7) =
.
(1.8)
Now, we solve the PIDE (1.1) - (1.3) using ADM System (1.1) - (1.3) is equivalent to the following equation –
=
+∫
+ ∫ (1.9)
In the view of modified Adomian decomposition method = =-
t2 + ∫
+ ∫
= =∫ = Similarly, =
+ ∫
= and so on =∑ = =
(1.10)
The solution of equation (1.1) is plotted in fig.a.
Fig.a. Graph of
=
2) Consider the PIDE ∫ with initial conditions and boundary condition
(2.1) = 0 , ut(x,0) = =t
Taking Laplace transform on both the sides of (2.1)
(2.2) (2.3)
̅ ( , )–
= +
+
+
̅
)̅=
+
(2.4)
Solution of (2.4) is, ̅ ( , )=
+C
(2.5)
From (2.3) ̅ From (2.4)
C=0
̅ ( , )=
(2.6)
Taking inverse Laplace transform on both the sides of (2.6) =
(2.7)
The solution of (2.1) is plotted in the figure (b)
Fig.b. Graph of
=
3) Consider the PIDE +∫
=
–
with initial conditions
(3.1) ,
=0
(3.2)
(3.1) - (3.2) is equivalent to the following equation =
-
-
+∫
+
∫ (3.3)
In the view of modified Adomian decomposition method, = ∫
=
+
∫
= Similarly, = = and so on. =∑ = +
+ --------
= which is a required exact solution. The solution of equation of (3.1) is plotted in following fig.c.
Fig.c. Graph of
=
4) Consider the PIDE =
–
-
+2∫
with initial conditions
(4.1) =1 ,
=0
(4.2)
and boundary conditions
(4.3)
Taking Laplace transform on both the sides of (4.1) ̅( , +
= ̅ =
-
̅ (4.4)
Solution of (4.4) is, ̅( , ) =
+C
(4.5)
From (4.3) ̅ ( , )=
(4.6)
using (4.5) and (4.6) we get C = 0 ̅ ( , )=
(4.7)
Taking inverse Laplace transform of (4.7)
which is the required exact solution. The solution of equation (4.1) is plotted in the following fig.d.
Fig.d.Graph of
Conclusion: Using Laplace transform we get exact solution of PIDE if it contains a convolution kernel, while by modified ADM we get solution of PIDE with any kind of kernel. References: [1] Abdul-Majid Wazwaz, Linear and nonlinear integral equations, Methods and applications, Springer. [2] M.Y.Ongun, The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+T cells, Mathematical and Computer Modelling 53(2011)597-603. [3] Efficient solution of a partial integro-differential equation in finance E.W. Sachs , A.K. Strauss,Applied numerical mathematics,58 [4] G.Adomian, Solving Frontier Problems of Physics: The decomposition method, Kluwer, (1994) [5] G.Adomian and R.Rach, Noise terms in decomposition series solution, Comput.Math.Appl, 24(1992)61-64. [6] L. Debnath and D. Bhatt, Integral Transforms and their applications, CRC Press. [7] M.Rehman, Integral equations and their applications, WIT Press.
