Research Article
Numerical Solution of Second-Order Fredholm
Integrodifferential Equations with Boundary Conditions by
Quadrature-Difference Method
Chriscella Jalius and Zanariah Abdul Majid
Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to Zanariah Abdul Majid; zana majid99@yahoo.com
Received 15 September 2016; Accepted 8 December 2016; Published 11 January 2017
Academic Editor: Mehmet Sezer
Copyright © 2017 C. Jalius and Z. Abdul Majid. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
In this research, the quadrature-diference method with Gauss Elimination (GE) method is applied for solving the second-order
of linear Fredholm integrodiferential equations (LFIDEs). In order to derive an approximation equation, the combinations of
Composite Simpson’s 1/3 rule and second-order fnite-diference method are used to discretize the second-order of LFIDEs. Tis
approximation equation will be used to generate a system of linear algebraic equations and will be solved by using Gauss Elimination.
In addition, the formulation and the implementation of the quadrature-diference method are explained in detail. Finally, some
numerical experiments were carried out to examine the accuracy of the proposed method.
1. Introduction
Integrodiferential equation is an equation where the
unknown function appears under the sign of integration and
it contains the derivatives of the unknown function. In this
study, we will focus on second-order linear Fredholm inte-
grodifferential equations (LFIDEs) and they can be defned
as follows:
() = ()
() + () () + ()
+∫
(,)()
(1)
with Dirichlet boundary conditions
() =
1
,
() =
1
,
(2)
where the functions (), (), and () and kernel (, )
are known, is a real parameter, and , are constant, while
() is an unknown function to be determined.
In this study, we have discovered four types of problems
in second-order LFIDEs such as the following.
Type 1. When () = 0 and () = 0
() = () + ∫
(,)().
(3)
Type 2. When () ̸ =0 and () = 0
() = () () + () + ∫
(,)().
(4)
Type 3. When () = 0 and () ̸ =0
() = ()
()+()+∫
(,)().
(5)
Type 4. When () ̸ =0 and () ̸ =0
() = ()
() + () () + ()
+∫
(,)().
(6)
Nowadays, there are many researchers who study in the
feld of integrodiferential equations since it has emerged
in many scientifc and physical engineering applications.
Hindawi
Journal of Applied Mathematics
Volume 2017, Article ID 2645097, 5 pages
https://doi.org/10.1155/2017/2645097