CHAPTERI
INTRODUCTION
Various kinds of work have been done by researchers in solving integral and integro- differential equation that arises in many mathematical modeling problems. The overview about integral and integro-differential equation are stated as follows:
1.1 Integral equations (lEs)
Definition I. I. (Kanwal, 1971) An integral equation is an equation in which an unknown
.
function appears under one or more integral signs. Naturally, in such an equation there
can occur other terms as well.
Generally, linear integral equation is written as
g(s)it (s)=. r(s)+Af K(s, r)it (I)tit, (1.1)
,1
where it (s) is the unknown function which need to be determined and the upper limit of the integral can be either constant or variable. The functions
. 1', g and kernel K are known functions and A is nonzero parameter. The integral equation can be classified into Fredhohn or Volterra types depending on its upper limit.
1.1.1 Fredholm integral equations
Fredholm integral equation is an integral equation of the form (1.1) with upper limit of integration h, which is a constant. For example
g(s)tr(s)=f(s)+Af K(s, t)it (t)dt, cr<s, t<_h, (1.2)
where kernel K(s, t), and functions f (s) and g(s) are given and A is a parameter. The Fredholm integral equation can be either fist, second or third kind depending on the value
of g(s).
1. The Fredholm integral equation of first kind is the lEs (1.2) with g (s) =0.
f h'(s, t)tt(t)dt=0. n
U
2. The Fredholm integral equation of second kind is the lEs (1.2) with g (s) =1 .
h
it (s)=f(s)+. ', jx(s, r)it (r)dr.
a
3. The Fredholm integral equation of third kind is the lEs (1.2) with g(s) : P, - {O, constaiit }.
-1,
1.1.2 Volterra integral equations
The Volterra integral equation is not much different from the Fredholm integral equation.
It is also an integral equation of the form (1.1) with upper limit of integration s, which is a variable. For example
g(n)u(s)= A. K(s, t) u(t)dt, aSs, t <_b, (1.3)
where kernel K(s, t), and functions f (s) and g(s) are given and A is a parameter. It is also classified as Fredholm integral equation depending on the value of g(s)
1. If g (s) = 0, then (1.3) becomes
f(s)+AJK(s, t)it (t)clt=0,
a
and named as Volterra integral equation of first kind.
2. If g (s) = 1, then (1.3) becomes
tl(S)= f(S)+A f K(s, t)ll(1)dl,
O
and named as Volterra integral equation of second kind.
3. If g(s) # {O, constant{ , then (1.3) is called as Volterra integral equation of third kind.
4
1.2 Differential equations (DEs)
Definition 1.2. (Sahoo & Kannappan, 2011) An equation involving an unknolrn f nrction and one or more of its derivatives is called differential equation.
For example
d'v dv d2y dv
Cývý=+=0,
-+=+sin(ý")=0, +v
dr- dT cLv` dY dv
where v is unknown function of x. The differential equation can be first order, second order, third order and higher order. The order of the differential equation represents the order of highest derivative that appear in the equation.
1.3 Volterra-Fredholm integro-differential equations
Definition 1.3. (Engelberg, 2005) An integro-differential equation is an equation in which integrals and derivatives appear.
Therefore, Volterra-Fredholm IDEs is an integro-differential equation in which the integral parts have both Volterra and Fredholm integrals. It can be linear or nonlinear.
Equations below are the examples of linear and nonlinear Volterra-Fredholm IDEs
ir ýf sb
y 1), (s)
u(s)=f(s)+. ýi f K, (s, t)it (t)dt+/i f h', (s, t)it (t)tit, (linear)
d
=0 s
iaa
ýp 11 (. c) `I
r(s)= Jý(s)+. ý, f iý(s, t, u(t)) dt+. ýf V_, (s, t, u(t))cit, (nonlinear)
, =o cLs 11 ,
5
where it (s) is unknown function and the rest are known while nonlinear term.
1.4 Research background
/E{1.2} indicates
There are many problems in the fields of science and engineering that can be modeled into functional equations such as integral equations (IEs), differential equations (DEs) and integro-differential equations (IDES). Some examples are risk process of an insurance company (Paulsen, 1993), Poisson process (Nilsen & Paulsen, 1996), heat transfer (Majeed, et al., 2015) and radiation and mass transfer (Vedavathi, et al., 2015). The important of the functional equation in modeling problems motivate the interest of many researchers to do research on the numerical fields. Their objective is to find the solution of the equation.
It is well-known that in many cases, finding the closed-form of the solution of
integro-differential equation is difficult. Therefore, numerous method has been developed to approximate the solution such as homotopy perturbation method (Nadjafi &
Ghorbani, 2009), Adomian decomposition method (ADM) (Biazar, 2005), modified decomposition method (Wazwaz, 2005), Lagrange polynomial approximation method (Mustafa & Ghanim, 2014), Taylor approximation method (Darania & Ivaz, 2008) and homotopy analysis method (Liao, 1992).
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There are various kind of equations solved by different kind of methods.
Gachpazan et al. (2014) used finite element method to solve linear Volterra integro- differential equation of the form
-ir"(x)+b(. 1-)rr'(. t-)+c-(x)u(x)=. i(x)+fK(x, t)tr(t)dt,
[l
with boundary conditions
it (a)
=0, il(h)=0,
where ii('ý(. v), j={0,1,2} is the derivative of the unknown function it (x) and functions h(x), r(x), ((x) and kernel K(x, t) are known functions. Yusufoglu (2009) has used homotopy perturbation method to solve first-order Fredholm integro-differential equation of the second kind
n/
W(s)
q(s)u(s)+JK(S,
i)iI(()dl+_ýý(s), a<ssh,
,, subject to the initial condition
u (0)
= A, A is a constant,
where u' (s) is the first derivative of unknown function u(s), while kernel K (s, t) and function f(s) and q (s) are given.
Wazwaz (2005) considered the nonlinear Fredholm and nonlinear Volterra IEs separately in the form
7
1
u(x, t)=.
f(x, t)+
f K(. a-, t)[R(u(x, t»+N(u(x, t»] dt0
and
u(x, t)=. I'(x, t)+ jK(x,
t)[R(tt(x, t))+N(tt(x, t))] dt 0
respectively. The operators R(i) and N(u) indicate the nonlinear and linear operator of unknown function it (x) respectively. He used modified decomposition method to
solve those equations.
Therefore, for the better understanding of the above problems in more general case, this research will focus on:
1. Linear Volterra- Fredholm lEs of the form
Sh
p(s)u(s)=f(s)+J
K,
(s, t)u(t)dt+ý f K, (s, t)tr(t)dt, n <_s, t <_b, (1.4)aa
where it (s) is unknown function, K, (s, t), K, (s, t) are given kernels, p(s), f (s) are known functions and A1 , 4., are constants.
2. First order linear Volterra-Fredholm IDEs
IS
2:,, ýsýtý, (s)=. / (s>+ý, JK, (s, t)u(t)dt+lL JK, (s, t)u(t)dt, a <_s, t <_b, (1.5)
IU
with initial condition
[] p
tr(a)=y,,, (1.6)
S
where u(s) is the unknown function to be searched with it (s) is the first derivative, p, (s),
.f (s) are known continuous function, K, (s, t), K, (s, t) are the given kernels and A, , 2,2
, y,, are the constants.
3. Second order linear Volterra-Fredholm IDEs in most general form
, '. bq
p, (s) tr" (s) =f(s)+ f
K, (s, 1) ti(t) (11 +4
f Lh',; (s, ti=U
i=o 111-0
U< S, t
< b,
0< P, q < 2,with boundary condition
i(') (t) di,
(1.7)
(1.8)
j [a,,
it(') (a)+, 6,.. il(i)
(h)]
=y, iE {0,11,; =o
where it(') (s), jE {0,1,2} are derivative of the unknown function it (s), while p, (s), jE}f (s) are given functions, K, (s, t), K, (s, I)
and are real constants.
4. High order linear Volterra-Fredholm IDEs in most general form
L, p;
(s)'I"' (s)K1,
(s, 1)11=.
f (s)+' $L
a_<s, t<_b, 0: 5 p, q with boundary condition
are given kernels
sq
(t)dt +2, f ZK-,
(s, t)u(')(t)dt,
, r-o
(1.9)
=y, i=O, l,..., nr-1, (1.10) [a,
1ii(') (u) +13, tiU1(b)]
J=U
9
where u (s) is the unknown function to be determined and u" (s) as its I- ah derivative, 1) , (s), f (s) are given smooth enough functions, K,, (s, t), K,,, (s, t) are
the given kernels and A,, are real constants.
1.5 Problem statement
The research problem is to find approximate solution of equations (1.4), (1.5), (1.7), (1.9) by means of polynomial approximation together with Gauss-Legendre QF as well as deterniining its effectiveness by doing the error analysis and providing numerical examples.
1.6 Scope of the study
This thesis is focused on finding the approximate solution of lEs (1.4), first order IDEs (1.5) with initial condition (1.6), second order IDEs (1.7) with boundary condition (1.8) and high order IDEs (1.9) with boundary condition (1.10). The form of second and high order IDEs is more general which is appear derivative of unknown
function under the integral sign. The method for solving those equations is focused on polynomials approximation method based on Legendre polynomials together with Gauss- Legendre QF and collocation method.
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1.7 Objectives
The objectives of this thesis are:
1. To derive polynomial approximation based on Legendre polynomials together with Gauss-Legendre QF and collocation method for solving linear Volterra- Fredhohn lEs and Volterra-Fredholm IDEs of order 1 and 2, equations (1.4) -
(1.8).
2. To present the operational matrix of derivative for Legendre polynomials for
solving high order, (in >- 3), Volterra-Fredhohn IDEs (1.9) - (1.10).
3. To obtain the convergence of the proposed method for Volterra-Fredhohn lEs (1.4) and IDEs of order one (1.5).
4. To provide numerical examples to validate the accuracy of the methods and compare results with other methods.
1.8 Outline of the thesis
This thesis is organized with seven chapters. The first chapter gives the introduction about the IEs, DEs and IDEs, the research background and the objectives of the research.
The second chapter provides the review about the method that has been used by researchers in solving the functional equation. The preliminaries tools are given in this chapter.
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The third chapter presents the description about the method that used to solve the Volterra-Fredholm IEs. Existence and uniqueness of the solution is proven here. Some
numerical examples related to the Volterra-Fredholm lEs are given in this chapter.
The fourth chapter explains about the details of the method for solving Volterra- Fredhohn IDEs of order one. Numerical examples are provided to show the ability of the method.
The fifth chapter deals with the description of the method on solving Volterra- Fredholm IDEs of order two. The applicability of the method is shown by some numerical examples.
The sixth chapter provides the relation of the derivative of the unknown function with the Legendre polynomial in matrix form. The high order Volten-a-Fredholm IDEs are solving in this chapter with details. This chapter also provides the numerical example related to high order Volterra-Fredholm IDEs to show the accuracy of the method.
The thesis ends by chapter seven with the general conclusion and some suggestion about the future work.
