Page 684 - Handbook Of Integral Equations

can always be calculated for the approximate solution ˜y(x), which is known on the interval a ≤ x ≤ x k
and was previously obtained for k – 1 parts. The initial value y(a) of the desired solution can be
found by an auxiliary method or is assumed to be given.
To solve Eq. (23), representation (22) is applied, and the free parameters A i (i =1, ... , m) can
be defined from the condition that the residuals vanish:

x k,j

ε(A i , x k,j )= K x k,j , t, Φ(t, A 1 , ... , A m ) dt – f(x k,j ) – Ψ k (x k,j ), (25)
x k
where the x k,j (j =1, ... , m) are the nodes that correspond to the partition of the interval [x k , x k+1 ]
into m parts (subintervals). System (25) is a system of m equations for A 1 , ... , A m .
For convenience of the calculations, it is reasonable to present the desired solution on any part
as a polynomial
m

˜ y(x)= A i ϕ i (x), (26)
i=1
where the ϕ i (x) are linearly independent coordinate functions. For the functions ϕ i (x), power and
trigonometric polynomials are frequently used; for instance, ϕ i (x)= x i–1 .
In applications, the concrete form of the functions ϕ i (x) in formula (26), as well as the form of
the functions Φ in (20), can sometimes be given on the basis of physical reasoning or defined by the
structure of the solution of a simpler model equation.

14.2-6. The Quadrature Method

To solve a nonlinear Volterra equation, we can apply the method based on the use of quadrature
formulas. The procedure of constructing the approximate system of equations is the same as in the
linear case (see Subsection 9.10-1).

◦
1 . We consider the nonlinear Volterra equation of the second kind in the Urysohn form
x

y(x) – K x, t, y(t) dt = f(x) (27)
a

on an interval a ≤ x ≤ b. Assume that K x, t, y(t) and f(x) are continuous functions.
From Eq. (27) we find that y(a)= f(a). Let us choose a constant integration step h and consider
the discrete set of points x i = a + h(i – 1), where i =1, ... , n.For x = x i , Eq. (27) becomes

x i

y(x i ) – K x i , t, y(t) dt = f(x i ). (28)
a
Applying the quadrature formula (see Subsection 8.7-1) to the integral in (28), choosing x j
(j =1, ... , i) to be the nodes in t, and neglecting the truncation error, we arrive at the following
system of nonlinear algebraic (or transcendental) equations:
i

y 1 = f 1 , y i – A ij K ij (y j )= f i , i =2, ... , n, (29)
j=1
where the A ij are the coefficients of the quadrature formula on the interval [a, x i ], the y i are the
approximate values of the solution y(x) at the nodes x i , f i = f(x i ), and K ij (y j )= K(x i , t j , y j ).
Relations (29) can be rewritten as a sequence of recurrent nonlinear equations,
i–1

y 1 = f 1 , y i – A ii K ii (y i )= f i + A ij K ij (y j ), i =2, ... , n, (30)
j=1
for the approximate values of the desired solution at the nodes.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 667