Page 474 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 31: Fredholm Equations with Poincere Goursat Kernels

 It is seen that the resolvent Kernel can be expressed in terms of quotient of two polynomials Notes
of the nth degree in and denominator is independent of the independent variables.

 Also conditions are discussed when is an eigenvalue and the corresponding eigenfunctions
are discussed with respect to P.G. Kernel only.

31.5 Keywords

In this unit the resolvent Kernel of the Fredholm integral equation of the second kind as well as
corresponding conjugate equation is discussed.
In the next unit we shall be studying Fredholm theorem for the existence and uniqueness of the
eigenvalue solution of the problem with only general Kernel.

31.6 Review Question

The Kernel of Fredholm integral equation
2
t
x
x
y ( ) f ( ) K ( , ) ( )dt
x
t
y
0
is given by
1
t
K ( , ) sin(vx ) sin [(v 1) ]
x
t
v 2
v 1
Find the iterated Kernel
K (x, t)
3
sin u
Hint : Use the relation lim u .
0
Answer: Self Assessment
sin(vx ) sin[(v 2) ]
t
1. K ( , )
t
x
2 v 2 (v 1) 2
v 1
31.7 Further Readings

Books Tricomi, F.G., Integral Equations
Yosida, K., Lectures in Differential and Integral Equations

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