Page 456 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 28: Integral Equations

28.5 Summary Notes

 In Fredholm integral equations of first kind and second kind the upper limit of integration
is fixed.

 Fredholm Integral equation can be obtained from linear differential equations by applying
certain boundary conditions.

 Types of Kernels appearing in Fredholm equations are of the type; symmetric Kernels,
difference Kernels, Poincere Goursat Kernels.

28.6 Keywords

n
h
x
t
x
t
Degenerate Kernels or Poincere Goursat Kernels are of the type K ( , ) g i ( ) ( ) where
i
i 1
h
t
g ( ), ( ) are known functions.
x
i i
Symmetric Kernels: The Kernels K(x, t) having the property K(x, t) = K(t, x) are known as symmetric
Kernels.
28.7 Review Question
1. Express the differential equation
x
y ( ) y ( ) 6y x 2 1
x
with (0)y y (1) 0 into Fredholm integral equation.
Answer: Self Assessment

1
x
t
t
G
x
1. G ( ) 4 K ( , ) ( )dt sin3x
0
where ( ) 4 ( ),G x x
t (1 x ) t x
x
K ( , )
t
x (1 t ) t x
28.8 Further Readings

Books Erwin Kreyzig, Introductory Functional Analysis with Application
Tricomi, F.G., Integral Equations
Yosida, K., Lectures in Differential and Integral Equation

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