Applied Mathematical Methods in Theoretical Physics || Integral Equations of the Volterra Type

✍ Scribed by Masujima, Michio

• Publisher: Wiley-VCH Verlag GmbH & Co. KGaA
• Year: 2009
• Tongue: German
• Weight: 109 KB
• Edition: 2
• Category: Article
• ISBN: 352740936X
• DOI: 10.1002/9783527627745.ch3

✦ Synopsis

Integral Equations of the Volterra Type

3.1 Iterative Solution to Volterra Integral Equation of the Second Kind

Consider the inhomogeneous Volterra integral equation of the second kind,

Also, define

Note that the upper limit of y integration is x. Note also that the Volterra integral equation is a special case of the Fredholm integral equation with

We will prove in the following facts for Eq. (3.1.1):

(1) A solution exists for all values of λ.

(2) The solution is unique for all values of λ.

(3) The iterative solution is convergent for all values of λ.

We start our discussion with the construction of an iterative solution. Consider a series solution of the usual form φ (x) = φ 0 (x) + λφ 1 (x) + λ 2 φ 2 (x) + • • • + λ n φ n (x) + • • • .