Page 439 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

2. Show that for L Kernel K(x, t) the nth iterated Kernel of Volterra integral equation
2
K (x, t) is also L class.
n 2
25.5 Summary

 Volterra integral equations are obtained by converting a differential equation with initial
conditions.
 For L -Kernels the resolvent Kernel can be found by iterated Kernel in the limit of n .
2
 For degenerate type of Kernels the resolvent Kernel can be obtained in a simpler way.

25.6 Keywords

Kernel that is L class has the same properties as a square integrable integral.
2
The L class nature of the Kernel as well as the function of L class helps finding the solution by
2 2
iteration.

25.7 Review Questions

1. What ae integral equation. Give examples.
2. How will you classify integral equations?

3. Account for volterra integral equations.
4. What are L Kernel and functions? Explain with suitable examples.
2
5. Consider the volterra equation with Kernel function

t
t
K ˆ ( ) K k ( )
where k = 2, = 10 and k indefined by
3
k
1 1
k ( ) exp
t
k 3/2
2t k 4kt
construct a solution function.
25.8 Further Readings

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

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