10.8
Section |
2 Background .1
Cosmology .11 Math |
Section 10.10 |

**Section** 9 **Volterra's Integral
Equation of the second kind**

Consider **Volterra's Integral Equation of the second kind**

(9.1)

It can be shown [[30] [38]) that, if the kernel *K*( (*x, s*) is
continuous in *R*{*a* ? *s* ?
*x* ? *b*}in the interval [*a, b*], then
(9.1) has a unique solution for any value of l. Choose one of the **Newton-Cotes
quadrature formulae**

(9.2)

where *x*_{j} are the abscissae of points in [*a, b*], and *A*_{j} the
coefficients of the quadrature formula ( *j* = 0, 1, . . ., *n*). Set in (9.1) *x* = *x*_{i}
and replace the definite integral by a finite sum to arrive at

(9.3)

where *y*_{i} = *y*(*x*_{i}), *K*_{ij}
= *K*(*x*_{i}, *y*_{j}), *f*_{i}* *=* f*(*x*_{i}).

A linear svstem with a triangular matrix has been obtained. The latter fact considerably facilitates its solution. The values
of the coefficients *A*_{j} depend on the choice of the quadrature formula (Chapter
8).

**Example 1:** Use the trapezoidal rule with *h* = 0.2 on the interval [0, 1]
to approxirnate the solution of

(9.4)

**Solution:** In the trapezoidal rule with *n *= 5

Setting in (9.4) *x* = *x*_{i}, *i* = 1, 2, . . . , 5 yields

Hence

(9.5)

Now using the values of in Table 115 and the system of
Equations (5) we successively find values of *y _{n}*:

Table 116

Values of K_{ij}, f_{i}

and obtain from (9.5)

**Exercises**

Apply the** trapezoidal rule** with *h* = 0.2 in order to find
approximate solutions of the integral equations: