Analyzing Methods for Solving Integral Equations

Introduction

An integral equation is an equation in which the unknown function φ(x) to be determined appear under the integral sign. The subject of integral equations is most useful mathematical tools in all branches of mathematics. Integral equations have enormous applications in many physical problems. Many initial and boundary value problems associated with ordinary differential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations (Refs. [2], [3] and [6]). The development of science and technology has led to the formation of many physical laws, which, when restated in mathematical form, often appear as differential equations.

Engineering problems can be model mathematically by differential equations, and thus differential equations play very important roles in the solution of practical problems. For example, Newton’s law, states that the rate of change of the momentum of a particle is equal to the force acting on it, can be model into mathematical language as a differential equation. Similarly, problems arising in electric circuits, chemical kinetics, and transfer of heat in a medium can all be represented mathematically as differential equations. A typical form of an integral equation in φ(x) is of the form

φ(x) = f (x) + λα∫_α( x)^(β (x))▒〖K(x,t)φ(t)dt〗 (1.1)

where K(x,t) is called the kernel of the integral equation (1.1), and α(x) and β(x) are limits of integration. It can be easily observed that the φ(x) appears under the integral sign. It is to be noted here that both the kernel K(x,t) and the function f (x) in equation (1.

1) are given functions; and λ is a constant parameter. The essential objective of this text is to determine the unknown function φ(x) that will satisfy equation (1.1) using a number of solution techniques. We shall devote considerable efforts in exploring these methods to find solutions of the unknown function.

Classification of the Integral Equations

An integral equation can be classified as a linear or nonlinear integral equation as we have seen in the ordinary and partial differential equations. It is to be noted here that, the differential equation can be equivalently represented by the integral equation. Therefore, there is a good connection between these two equations. The most frequently used integral equations fall under two major classes, namely

Volterra integral equations and FIEs. Of course, we have to classify them as homogeneous or nonhomogeneous; and also linear or nonlinear. In some practical problems, we face with singular equations also. Generally, we have two major types of integral equations the two main classes and two related types of integral equations. In particular, the two main classes are given below:

• Volterra integral equations
• FIEs

and the related types are Singular Integral Equations Integro-differential Equations

We shall outline these equations using basic definitions and properties of each type.

Volterra integral equations

The most standard form of Volterra linear integral equations is of the form

u(x)φ(x) = f (x) + λ∫_a^x▒〖K(x,t)φ(t)dt〗 (1.8)

where the limits of integration are function of x and the unknown function φ(x) appears linearly under the integral sign. If the function u(x) = 1, then equation (1.8) simply becomes

φ(x) = f (x) + λ∫_a^x▒〖K(x,t)φ(t)dt〗 (1.9)

and this equation is known as the Volterra integral equation of the second kind; whereas if u(x) = 0, then equation (1.8) becomes

f (x) + λ∫_a^x▒〖K(x,t)φ(t)dt〗 = 0 (1.10)

which is known as the Volterra equation of the first kind. MM-165 CH001.tex 26/4/2007 13: 43 Page 66 Integral Equations and their Applications

FIEs

The most standard form of Fredholm linear integral equations is given by the form

u(x)φ(x) = f (x) + λ∫_a^b▒〖K(x,t)φ(t)dt〗 (1.11)

where the limits of integration a and b are constants and the unknown function φ(x) appears linearly under the integral sign. If the function u(x) = 1, then (1.11) becomes simply

φ(x)= f (x)+ λ∫_a^b▒K(x,t)φ(t)dt (1.12)

and this equation is called FIE of second kind; whereas if u(x) = 0, then (1.11) yields

f (x)+ λ∫_a^b▒K(x,t)φ(t)dt=0 (1.12)

which is called FIE of the first kind.

It is very important to note that integral equations arise in engineering, physics, chemistry, and biological problems. Many IBVPs associated with the ordinary and partial differential equations can be cast into the integral equations of Volterra and Fredholm types, respectively. If the unknown function φ(x) appearing under the integral sign is given in the functional form F(φ(x)) such as the power of φ(x) is no longer unity, e.g. F(φ(x)) = φ^n (x), n = 1, or sin φ(x) etc., then the Volterra and FIEs are classified as nonlinear integral equations. As for examples, the following integral equations are nonlinear integral equations:

φ(x)=f (x) + λ∫_a^x▒〖K(x,t) 〖φ〗^2 (t) dt〗

φ(x)=f (x) + λ∫_a^b▒〖K(x,t) sin (φ(t)) dt〗

φ(x)=f (x) + λ∫_a^b▒〖K(x,t) ln (φ(t)) dt〗

Next, if we set f (x) = 0, in Volterra or FIEs, then the resulting equation is called a homogeneous integral equation, otherwise it is called nonhomogeneous integral equation. MM-165 CH001.tex 26/4/2007 13: 43 Page 7 Introduction 7

Methods for Solution of FIEs

Method of Fredholm Determinants

The solution of the FIE of the second kind

φ(x)= f (x)+ λ∫_a^b▒K(x,t)φ(t)dt (1.12*)

Is given by the formula

φ(x)= f (x)+ λ∫_a^b▒R(x,t;λ)f(t)dt (1.13)

Where the function R(x,t;λ) is called the Fredholmresolvent kernel of (1.12*) and is defined as

R(x,t;λ)=D(x,t;λ)/D(λ) (1.14)

Provided that D(λ)≠0. Here D(x,t;λ) and D(λ) are power series in λ:

D(x,t;λ)=K(x,t)+∑_(n=1)^∞▒〖(-1)^n/n! B_n (x,t) λ^n 〗 (1.15)

D(λ)=1+∑_(n=1)^∞▒〖(-1)^n/n! C_n λ^n 〗 (1.16)

Whose coefficients are given by the formulas

B_n (x,t)=⏟(∫_a^b▒〖…∫_a^b▒〗)┬(n times) |■(K(x,t)&K(x,t_1)&■(…&K(t_ ,t_n))@K(t_1,t)&K(t_1,t_1)&■(…&K(t_1,t_n))@■(K(t_2,t)@⋮@K(t_n,t))&■(K(t_2,t_1)@⋮@K(t_n,t_1))&■(■(…@⋱@…)&■(K(t_2,t_n)@⋮@K(t_n,t_n))))|dt_1…dt_2 (1.17)

And

B_0 (x,t)=K(x,t)

C_n=⏟(∫_a^b▒〖…∫_a^b▒〗)┬(n times) |■(K(x_1,t_1)&K(t_1,t_2)&■(…&K(〖t_1〗_ ,t_n))@K(t_2,t_1)&K(t_2,t_2)&■(…&K(t_2,t_n))@■(K(t_3,t_1)@⋮@K(t_n,t_1))&■(K(t_3,t_2)@⋮@K(t_n,t_2))&■(■(…@⋱@…)&■(K(t_3,t_n)@⋮@K(t_n,t_n))))|dt_1…dt_2 (1.18)

The function D(x,t;λ) is called the Fredholm minor, and D(λ) the Fredholm determinant. When the kernel K(x,t) is bounded or the integral

∫_a^b▒∫_a^b▒〖K^2 (x,t)dxdt〗

Has a finite value, the series (1.15) and (1.16) converge for all values of λ and hence are entire analytic functions of λ.

The resolvent kernel (1.14) is an analytic function of λ, except for those values of λ which are zeros of the function D(λ). The latter are the poles of the resolvent kernel R(x,t;λ).

The recursion relations are

B_n (x,t)=C_n K(x,t)-n∫_a^b▒〖K(x,s) B_(n-1) (s,t)ds〗 (1.19)

C_n=∫_a^b▒〖B_(n-1) (s,s)ds〗 (n=1,2,…),

C_0=1,B_0 (x,t)=K(x,t) (1.20)

The Adomian Decomposition Method

The Adomian decomposition method (ADM) was introduced and developed by George Adomian in [2–5]. Here we want briefly introduce the Adomian method. The Adomian decomposition method consists of decomposing the unknown function φ(x) of any equation into a sum of an infinite number of components defined by the following decomposition series

φ(x)= ∑_(n=0)^∞▒〖φ_n (x) 〗,(4.13)

or equivalently

φ(x) = φ_0 (x) + φ_1 (x) + φ_2 (x) + ··· (4.14)

where all the components φ_n (x), n≥0 will be determined recurrently. The Adomian decomposition method concerns itself with finding the components φ_0,φ_1,φ_2,... individually. The determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated. To form the recurrence relation, we substitute (4.13) into the FIE (4.12) to get

∑_(n=0)^∞▒〖φ_n (x)〗 = f(x) + λ∫_a^b▒〖K(x,t) (∑_(n=0)^∞▒〖φ_n (t)〗) 〗dt,(4.15)

or equivalently

φ_0 (x)+φ_1 (x)+φ_2 (x)+ ··· = f(x) + λ∫_a^b▒〖K(x,t)[φ_0 (t)+φ_1 (t)+ ···] dt〗.(4.16)

The zeroth component φ_0 (x) is identified by all terms that are not included under the integral sign. This means that the components φ_j (x),j≥0 of the unknown function φ(x) are completely determined by setting the recurrence relation

φ_0 (x)= f(x),〖φ〗_(n+1) (x)=λ∫_a^b▒〖K(x,t) φ_n (t)dt〗,n≥0,(4.17)

or equivalently

φ_0 (x) = f(x),

φ_1 (x) = λ∫_a^b▒〖K(x,t) φ_0 (t)dt〗,

φ_2 (x) = λ∫_a^b▒〖K(x,t) φ_1 (t)dt〗,

φ_3 (x)= λ∫_a^b▒〖K(x,t) φ_2 (t)dt〗,(4.18)

and so on for other components. In view of (4.18), the components φ_0 (x),φ_1 (x),φ_2 (x),φ_3 (x),... are completely determined. As a result, the solution φ(x) of (4.12) is obtained in a series form by using the series assumption in (4.13). It is clear that the decomposition method converts the integral equation into an elegant determination of computable components. It is important to note that if an exact solution exists for the problem, then the obtained series converges very rapidly to that exact solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to confirm the rapid convergence of the resulting series. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes.

The Modified Decomposition Method

As stated before, the Adomian decomposition method provides the solutions in an infinite series form of components. The components φ_j,j≥0 are easily computed if the inhomogeneous term f(x) in the FIE

φ(x) = f(x) + λ∫_a^b▒〖K(x,t)φ(t)dt〗,(4.62)

consists of a monomial or binomial. However, if the function f(x) consists of a combination of two or more of polynomials, trigonometric functions, hyperbolic functions, and others, the evaluation of the components φ_j,j≥0 requires more work. As presented before, the modified decomposition method depends mainly on splitting the function f(x) into two parts, therefore it cannot be used if the f(x) consists of only one term. The modified decomposition method will be briefly outlined here, but will be used in this section of paper. The standard Adomian decomposition method employs the recurrence relation

φ_0 (x) = f(x),

φ_(k+1) (x)= λ∫_a^b▒〖K(x,t) φ_k (t)dt〗,k≥0,(4.63)

where the solution φ(x) is presented by an infinite sum of components defined by

φ(x) =∑_(n=0)^∞▒〖φ_n (x)〗.(4.64)

In view of (4.63), the components φ_n (x),n≥0 are readily obtained. The modified decomposition method presents a slight variation to the recurrence relation (4.63) to determine the components of φ(x) in an easier and faster manner. For many cases, the function f(x) can be set as the sum of two partial functions f_1 (x) and f_2 (x). In other words, we can substitute

f(x) = f_1 (x) + f_2 (x).(4.65)

In view of (4.65), we introduce a qualitative change in the formation of the recurrence relation (4.63). The modified decomposition method identifies the zeroth component φ_0 (x) by one part of f(x), namely f_1 (x) or f_2 (x). The other part of f(x) can be added to the component φ_1 (x) that exists in the standard recurrence relation. The modified decomposition method admits the use of the modified recurrence relation:

φ_0 (x) = f_1 (x),

φ_1 (x) = f_2 (x)+λ∫_a^b▒〖K(x,t) φ_0 (t)dt〗,

φ_(k+1) (x)= λ∫_a^b▒〖K(x,t) φ_k (t)dt〗,k≥1.(4.66)

It is obvious that the difference between the standard recurrence relation (4.63) and the modified recurrence relation (4.66) rests only in the formation of the first two components φ_0 (x) and φ_1 (x) only. The other components φ_j,j≥2 remain the same in the two recurrence relations. Although this variation in the formation of φ_0 (x) and φ_1 (x) is slight, however it has been shown that it accelerates the convergence of the solution and minimizes the size of calculations.

Moreover, reducing the number of terms in f_1 (x) affects not only the component φ_1 (x), but also the other components as well. We here emphasize on the two important points. First, by proper selection of the f_1 (x) and f_2 (x), the exact solution φ(x) may be obtained by using few iterations, and sometimes by evaluating only two components. The success of this modification depends only on the proper choice of f_1 (x) and f_2 (x), and this can be made through trials only. A rule that may help for the proper choice of f_1 (x) and f_2 (x) could not be found yet. Second, if f(x) consists of one term only, the modified decomposition method cannot be used in this case.

The Method of Successive Approximations

The successive approximations method, also known as the Picard iteration method. This method provides a scheme that can be used to solve initial value problems or integral equations. This method solves any problem by finding successive approximations to the solution by starting with an initial guess as φ_0 (x), called the zeroth approximation. As will be seen, the zeroth approximation is any selective real valued function that will be used in the recurrence relation to determine the next approximations. The most commonly used values for the zeroth approximations are 0, 1, or x. Of course, other real values can be selected as well. Consider the FIE of the second kind

φ(x) = f(x) + λ∫_a^b▒〖K(x,t)φ(t)dt〗,(4.163)

The successive approximations method introduces the recurrence relation

φ_0 (x) = any selective real valued function,

φ_(n+1) (x) = f(x) + λ∫_a^b▒〖K(x,t) φ_n (t)dt〗,n 0.(4.164)

At the limit, the solution is determined by using the limit

φ(x) = lim┬(n→∞)⁡〖φ_(n+1) (x)〗⁡.(4.165)

It is interesting to point out that the Adomian decomposition method

admits the use of an iteration formula of the form

φ_0 (x) = all terms not included inside the integral sign,

φ_1 (x) = λ∫_a^b▒〖K(x,t) φ_0 (t)dt〗,

φ_2 (x)= λ∫_a^b▒〖K(x,t) φ_1 (t)dt〗

⋮⋮⋮

φ_(n+1) (x) = φ_n (x) + λ∫_a^b▒〖K(x,t) φ_n (t)dt〗.,(4.166)

The difference between the two formulas (4.164) and (4.166) can be summarized as follows:

The successive approximations method gives successive approximations of the solution φ(x), whereas the Adomian method gives successive components of the solution φ(x).

The successive approximations method admits the use of a selective real-valued function for the zeroth approximation φ_0, whereas the Adomian decomposition method assigns all terms that are not inside the integral sign for the zeroth componentφ_0 (x). Recall that this assignment was modified when using the modified decomposition method.

The successive approximations method gives the exact solution, if it exists, by

φ(x) = lim┬(n→∞)⁡〖φ_(n+1) (x)〗. (4.167)

However, the Adomian decomposition method gives the solution as infinite

series of components by

φ(x) =∑_(n=0)^∞▒〖φ_n (x)〗.(4.168)

This series solution converges rapidly to the exact solution if such a solution exists. The successive approximations method, or the iteration method will be illustrating by studying an example at the end of methods.

Example

Solve the FIE by using the successive approximations method

φ(x) = x + ex - ∫_0^1▒〖xtφ(t)dt〗.(4.169)

For the zeroth approximation φ_0 (x), we can select

φ_0 (x)=0. (4.170)

The method of successive approximations admits the use of the iteration formula

φ_(n+1) (x)= x + ex - ∫_0^1▒xtφn(t)dt,n≥ 0. (4.171)

Substituting (4.170) into (4.171) we obtain

φ_1 (x) = x + ex - ∫_0^1▒〖xtφ_0 (t)dt〗 = ex + x,

φ_2 (x)= x + ex - ∫_0^1▒〖xtφ_1 (t)dt〗 = ex -1/3 x,

φ_3 (x)= x + ex - ∫_0^1▒〖xtφ_2 (t)dt〗 = ex +1/9 x,

⋮⋮⋮

φ_(n+1) = x + ex - ∫_0^1▒〖xtφ_n (t)dt〗 = ex +(-1)^n/3^n x.(4.172)

Consequently, the solution φ(x) of (4.169) is given by

φ(x)= lim┬(n→∞)⁡〖φ_(n+1) (x)= ex〗. (4.173)

The Direct Computation Method

In this section of the paper, we will apply direct computation method to solve the FIEs. The method approaches FIEs in a direct manner and gives the solution in an exact form, not in a series form. It is important to note that we can apply this method for the degenerate or separable kernels of the form

K(x,t) = ∑_(k=1)^n▒〖g_k (x) h_k (t)〗.(4.129)

Examples of separable kernels are x - t,xt,x^2 - t^2,xt^2 + x^2 t, etc.

The direct computation method can be applied as follows:

We first substitute (4.129) into the FIE of the form

φ(x) = f(x) +∫_a^b▒〖K(x,t) φ(t)dt〗.(4.130)

This substitution gives

φ(x) = f(x) + g_1 (x)∫_a^b▒〖h_1 (t)φ(t)dt〗 + g_2 (x)∫_a^b▒〖h_2 (t)φ(t)dt〗 + ⋯+g_n (x)∫_a^b▒〖h_n (t)φ(t)dt〗.(4.131)

Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (4.131) becomes

φ(x) = f(x) + λα_1 g_1 (x) + λα_2 g_2 (x) + ··· + λα_n g_n (x),(4.132)

where

α_i =∫_a^b▒〖h_i (t)φ(t)dt〗,1≤i≤n.(4.133)

Substituting (4.132) into (4.133), we obtain a system of n algebraic equations that can be solved to determine the constants α_i,1≤i≤n. Using the obtained numerical values of α_i into (4.132), the solution φ(x) of the FIE (4.130) is readily obtained.

Comparison between Alternative Methods

Now we finished the mathematical analysis of the technics that handle FIEs, we are now ready to carry out a comparison between these methods. When it comes to selecting a preferable method among the five methods for solving linear FIEs, we cannot recommend a specific method. However, we found that if the separable kernel K(x,t) of the integral equation consists of a polynomial of one or two terms only, the direct computation method might be the best choice because it provides the exact solution with the minimum volume of calculations. For other types of kernels, and if in addition the nonhomogeneous part f(x) is a polynomial of more than two terms we found that the

Adomian decomposition method or the Variational Iteration Method, are proved to be effective, reliable and produces a rapid convergent series for the solution. The series obtained by using the decomposition method may give the solution in a closed form or we may obtain an approximation of high accuracy level by using a truncated series for concrete problems. It is worth noting that the decomposition method expands the solution φ(x) about a function, instead of a point as in Taylor theorem.

To compare the decomposition method with the successive approximation method, it is clear that the decomposition method is easier in that we always integrate few terms to obtain the successive components, whereas in the other method we integrate many terms to evaluate the successive approximations after selecting the zeroth approximation. The two methods give the solution in a series form. In addition, we point out that the method of successive substitutions suffers from the huge size of calculations in evaluating the several multiple integrals especially if the function f(x) is a trigonometric, logarithmic or exponential function. However, the method is directly based on substituting the unknown function φ(x) under the integral sign by the given function f(x). It is to be noted that, for a first course in integral equations, we introduced five methods only to handle FIEs, noting that other traditional techniques are left for a further study.