10.7 Section 2 Background .1 Cosmology .11 Math Section 10.9

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 8 Fredholm Integral Equations

Consider the Fredholm Integral Equations of the first kind

(8.1)

and Fredholm Integral Equations of the second kind

(8.2)

Use finite sums ([2], [13], (17]) in order to replace the definite integral by a finite sum with the aid of one of the quadrature formulae

(8.3)

where the xj lie within the interval [a,b] and Aj, j = 1, 2, . . . , n are quadrature coefficients which do not depend on F(x). Replace the integrals in (8.1) and (8.2) using (8.3):

(8.4)/(8.5)

where yi = y(xi), Kij = K(xi,xj), fi = f(xi).

Solving this system of linear algebraic equations in yi, for example, by Gauss elimination or iteration, a table of approximate values of yi is obtained. An approximate solution of (8.1) in the form of an interpolation polynomial and of (8.2) in the form is

(8.6)

Depending on the choice of the quadrature formula (8.3), the coefficients Aj and abscissae xj are:

1. For the Trapezoidal Rule:

2. For Simpson's Rule:

3. For Gaussian Quadrature:

where the xj(n) are the Gaussian abscissae and Aj(n) the Gaussian coefficients for the interval [0,1}, respectively.

The error in the approximation depends on the choice of the quadrature. For the use of the formulae consult [17], for error estimates ([2], [17]).

Example 1: Use Simpson's Rule with n = 2 to solve the equation

(8.7)

Solution: For Simpson's Rule:

whence (8.7) becomes

Setting x = xi yields the system

which after simplification assumes the form

Hence

,

while the exact solution is

Solving this system, one finds y0 = 1, y1 = 1.002, y2 = 1.0.095. By (8.6), the approximate solution of (8.7) is

Example 2: Apply Gauss' quadrature with n = 2 to the integral equation

(8.8)

Solution: Gauss' formula yields

and (8.8) the system

Substitution for x1 and x2 now yields the system

with the solution

The analytical solution of (8.8) is

The approximate solution can be given the form

Example 3: Using the rectangle formula for n = 12, find the approximate solution of the equation

(8.9)

Solution: For the rectangle formula with n = 12, h = 2p/12 = p/6. Thus (8.9) becomes

(8.10)

The values of Kij are given in Table 115.

The number of unknowns in this system can be reduced significantly by taking into consideration that the solution is symmetric. It can be shown that, if the function y(x) is the solution of (7.9), then the function y(-x) is also a solution of this equation. Hence, by virtue of the uniqueness of the solution of an integral equation, one has y(-x) = y(x), i.e., y(x) is even. Let

and show that

(8.11)

In fact,

Setting -s = t and using the fact that y(x) is even, one finds

Then

whence by setting p + s = -t, one obtains

(8.11) now yields

i.e., the graph of the sought solution is symmetric with respect to the straight lines x = 0, x = ? p/2, as is shown by the figure:

Figure 21.

f

By virtue of the symmetry, one can write

Setting y1 = y(0), y2 = y(p/6), y3 = y(p/3), y4 = y(p/2) and using (8.12), the system (8.10) becomes

where the values of K(xi,xj) = Kij in equations (9) are given in the table:

Table 115

Substituting the values of Kij and computing the coefficients of yi (1, 2, 3, 4), one obtains the system

whence

Using (8.12), one also obtains the values of y at the remaining points and the approximate solution of (8.9) bcomes

which can be simplified to

at the points

 

Exercises

Find approximate solutions of the integral equations by means of the indicated quadrature formulae: