Page 441 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes We now classify some of the Kernels as follows:
1. Degenerate Kernels or Poincere Goursat Kernels
The Kernel K(x, u) of the integral equation is said to be degenerate if it can be represented
in the form
x
K(x, u) = g 1 ( ) ( ) g 2 ( )h 2 ( ) ... ...(3)
x
u
h
u
1
2. Difference Kernel
The Kernel of the integral equation is said to be difference Kernel if it depends upon the
difference of the arguments,
K(x, u) = K(x u).
3. Polar Kernels
They are of the form

u
L ( , )
x
x
u
K(x, u) = M ( , ) 0 1 ...(4)
(x u )
where L(x, u) and M(x, u) are continuous on the square
a x b, a u b and L (x, x) 0
4. Logarithmic Kernels
They are of the form
K(x, u) = L(x, u) log (x u) + M (x u) ...(5)
The following generalized Abel equation is a special case of equation (1) with the Kernel
of the form (4)
x
u
y ( )du
= f(x) 0 1 ...(6)
(x u )
0
Example: In case the Kernel K(x, u) and f(x) are continuous then f(x) must satisfy the
following conditions:
(i) If ( , )K a a 0, then ( ) 0f

a
a
(ii) If ( , )K a a K 1 x ( , ) K x 2 ( , )... K n x 1 ( , ) 0, and
a
a
a
a
0 K n x ( , ) ,
a
a
then f(a) = f 1 ( ) ... f n ( ) 0.
a
a
26.2 Reduction of Volterra Equations of the First Kind to Volterra
Equations of the Second Kind
Consider Volterra integral equation of the first kind
x
y
x
u
u
K ( , ) ( )du = f(x) ...(1)
0
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