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

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

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 xj are the abscissae of points in [a, b], and Aj the coefficients of the quadrature formula ( j = 0, 1, . . ., n). Set in (9.1) x = xi and replace the definite integral by a finite sum to arrive at

(9.3)

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

A linear svstem with a triangular matrix has been obtained. The latter fact considerably facilitates its solution. The values of the coefficients Aj 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 = xi, 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 yn:

Table 116
Values of Kij, fi

and obtain from (9.5)

Exercises

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