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SOLVING THIRD ORDER LINEAR FREDHOLM INTEGRO-DIFFERENTIAL
SOLVING THIRD ORDER LINEAR FREDHOLM INTEGRO-DIFFERENTIAL
EQUATION USING ADOMIAN DECOMPOSITION METHOD AND
EQUATION USING ADOMIAN DECOMPOSITION METHOD AND
VARIATIONAL ITERATION METHOD
VARIATIONAL ITERATION METHOD
NORLILYANA AYUNI BINTI JASMI SUPERVISOR : W. KHAIRIYAH HULAINI BINTI W.RAMLI
SUPERVISOR : W. KHAIRIYAH HULAINI BINTI W.RAMLI
NORLILYANA AYUNI BINTI JASMI
Methodology & Implementationn
M e t h o d o l o g y & I m p l e m e n t a t i o
A b s t r a c t
Abstract
This study compares the Adomian Decomposition ADOMIAN DEGOMPOSITION METHOD
Method (ADM) and Variational Iteration Method (VIM) VARIATIONAL ITERATION METHOD
in solving third-order linear Fredholm integro- (1) Solves the 3 order LFIDE :
rd
differential equations (LFIDEs), which often appear (1) The difference equation is given as :
in science and engineering. ADM uses Adomian
polynomials to break down the equation, while VIM
applies a correction process using a Lagrange
multiplier. Solutions were calculated using MAPLE, (2) Kernel assumed separable:
and errors were analyzed with Excel. The results (2) Based on correction functional :
show both methods are accurate, but VIM gives
faster and slightly more precise results, making it
more efficient for similar future problems.
(3) Apply inverse operator
rd
(3) Lagrange multiplier for 3 order :
P r o b l e m S t a t e m e n t
Problem Statement
Fredholm integro-differential equations are often
used to model real-world problems in fields like (4) Series form of solution :
physics, engineering, and biology. However, these
equations are challenging to solve because they
combine both integral and differential terms. Finding (4) The initial approximation as :
exact solutions is usually difficult, especially for
higher-order or nonlinear cases. Therefore, this (5) First term :
study explores two reliable methods which is the
Adomian Decomposition Method and the Variational
Iteration Method to solve third-order equations and
compare their accuracy. (5) Compute successive approximations
(6) Next term : u1(x), u2(x), …., un(x)
Objectives
O b j e c t i v e s (6) Using the same steps, few iterations until
To solve the third-order Fredholm Integro- n=3 of solution VIM successively as follows.
Differential equation using Adomian Decomposition
Method. (7) By using the same steps, few iterations until (7) Next iteration until n = 10 is calculated by
To solve the third order Fredholm Integro n = 3 of solution ADM successively as follows. using Maple software.
Differential Equation through Variational Iteration
Method. (8) The next iterations are calculated until n = 10 (8) Final solution is taken from the last
To compare the error between analytical and using Maple and it gives the series solution. approximation which is iteration n=10
numerical value
Result & Discussion
R e s u l t & D i s c u s s i o n
A
A D A
D M D
M M
V V V
I I I
M M M
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m
s
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Conclusionn Recommendationn
o
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n
R
e
In this study, both the Adomian Decomposition Method (ADM) and the Variational Based on the findings, future research is recommended to extend this study to
Iteration Method (VIM) were successfully applied to solve third-order linear higher-order equations, such as fourth-order linear Fredholm integro-differential
Fredholm integro-differential equations. The results showed that both methods equations. In addition to ADM and VIM, researchers can explore the use of other
are effective, as the approximate solutions closely matched the exact solutions semi-analytical methods like the Homotopy Perturbation Method (HPM) for
with minimal errors. However, VIM consistently produced more accurate results comparison. This can help identify even more efficient approaches for solving
with faster convergence and smaller error values. Therefore, VIM is considered complex integro-differential problems in various scientific and engineering
the more efficient and reliable method for solving this type of equation. fields.
