Page 461 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

1. Solve the following integral equation

1
y ( ) m y ( )dt 1.
x
t
0
Also find the Neumann series for y(x)
1
t
Hint : y ( )dt constant.
0
29.3 Summary

 The iterative method gives the solution of the function in terms of the powers of the
parameter of the equation.
 We can either get an iterative power series in the wave function or the iterated Kernel.

 After iterating it nth times we get the solution as limiting as n tends to .
 In this way we get the Resolvent Kernel in the nth iteration when n is very large.

29.4 Keyword

The successive method helps in getting the solution of the problem as a power series in terms of
powers of the parameter known as Neumann series. The estimate of the radius of convergence of
the Neumann series gives an estimate of the accuracy of the solution.

29.5 Review Question

The Fredholm integral equation is

2
x
t
x
y ( ) K ( , ) ( ) dt f ( )
x
y
t
0
where ( , )K x t v 2 sin ( ) sin [(v 1) ]
vs
t
v 1
find K (x, t), the third iterative Kernel.
3
Answer: Self Assessment
1
2
3
x
1. y ( ) , the Neumann series is y(x) = 1 + + + ... .
1 u
29.6 Further Readings

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

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