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

**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 *x*_{j} lie within the interval [*a,b*] and *A*_{j}, *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 *y*_{i }= *y*(*x*_{i}), *K*_{ij} = *K*(*x*_{i},*x*_{j}),
*f*_{i} = *f*(*x*_{i}).

Solving this system of linear algebraic equations in *y*_{i}, for example, by **Gauss
elimination** or **iteration**, a table of approximate values of *y*_{i
}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 *A*_{j} and
abscissae *x*_{j} are:

1. **For the ****Trapezoidal Rule:**

2. For** Simpson's Rule:**

3. For **Gaussian Quadrature:**

where the *x*_{j}^{(n)} are the Gaussian abscissae and *A*_{j}^{(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* = *x*_{i} yields the system

which after simplification assumes the form

Hence

,

while the exact solution is

Solving this system, one finds *y*_{0} = 1, *y*_{1} = 1.002, *y*_{2} = 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 *x*_{1 }and *x*_{2} 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*
= 2*p*/12 = *p*/6. Thus (8.9) becomes

(8.10)

The values of K_{ij} 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 *y*_{1} = *y*(0), *y*_{2} = *y*(*p*/6),
*y*_{3} = *y*(*p*/3), *y*_{4}
= *y*(*p*/2) and using (8.12), the system (8.10) becomes

where the values of *K*(*x*_{i},*x*_{j}) = *K*_{ij}
in equations (9) are given in the table:

Table 115

Substituting the values of *K*_{ij}* *and computing the coefficients of *y*_{i}*
*(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: