Part of the Fundamental Theories of Physics book series (FTPH, volume 60)
Integral equations of Volterra type arise quite naturally in physical applications modelled by initial-value problems. Consider the linear Volterra equation of the second kind. (Fredholm equations of the second kind which are associated with boundary value problems for a fmite interval [a,b], are similar except that the upper limit is b.)
with a ≤ x, y ≤ b, and let λ = 1. Using decomposition , we identify φ 0 = f(x) assuming f(x) ≠ 0,
then and write as the solution, or with an m-term approximant . In some cases, exact solutions are determinable. Consider an example:
Thus the two-term approximant is
or φ = sin x as is easily verified either by calculating more terms or by substitution.
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