Page 685 - Handbook Of Integral Equations

◦
2 . When applied to the Volterra equation of the second kind in the Hammerstein form
x

y(x) – Q(x, t)Φ t, y(t) dt = f(x), (31)
a
the main relations of the quadrature method have the form (x 1 = a)
i

y 1 = f 1 , y i – A ij Q ij Φ j (y j )= f i , i =2, ... , n, (32)
j=1
where Q ij = Q(x i , t j ) and Φ j (y j )= Φ(t j , y j ). These relations lead to the sequence of nonlinear
recurrent equations
i–1

y 1 = f 1 , y i – A ii Q ii Φ i (y i )= f i + A ij Q ij Φ j (y j ), i =2, ... , n, (33)
j=1
whose solutions give approximate values of the desired function.
Example 5. In the solution of the equation
x

y(x) – e –(x–t) 2 –x 0 ≤ x ≤ 0.1,
y (t) dt = e ,
0

2
–x

where Q(x, t)= e –(x–t) , Φ t, y(t) = y (t), and f(x)= e , the approximate expression has the form
x i

y (t) dt = e
y(x i ) – e –(x i –t) 2 –x i .
0
On applying the trapezoidal rule to evaluate the integral (with step h = 0.02) and finding the solution at the nodes x i =0,
0.02, 0.04, 0.06, 0.08, 0.1, we obtain, according to (33), the following system of computational relations:
i–1
2
2
y 1 = f 1 , y i – 0.01 Q ii y i = f i + 0.02 Q ij y j , i =2, ... ,6.
j=1
Thus, to find an approximate solution, we must solve a quadratic equation for each value y i , which makes it possible to write
out the answer
i–1 1/2

2
y i =50 ± 50 1 – 0.04 f i + 0.02 Q ij y j , i =2, ... ,6.
j=1
•
References for Section 14.2: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev,
et al. (1975), A. F. Verlan’ and V. S. Sizikov (1986).
14.3. Equations With Constant Integration Limits
14.3-1. Nonlinear Equations With Degenerate Kernels
1 . Consider a Hammerstein equation of the second kind in the canonical form
◦
b

y(x)= Q(x, t)Φ t, y(t) dt, (1)
a
where Q(x, t) and Φ(t, y) are given functions and y(x) is the unknown function.
Let the kernel Q(x, t) be degenerate, i.e.,
m

Q(x, t)= g k (x)h k (t). (2)
k=1

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 668