An explicit numerical algorithm to the solution of Volterra integral equation of the second kind
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This paper considers a numeric algorithm to solve the equation y(t)=f(t)+∫^t_0 g(t-τ)y(τ) dτ with a kernel g and input f for y. In some applications we have a smooth integrable kernel but the input f could be a generalised function, which could involve the Dirac distribution. We call the case when f=δ, the Dirac distribution centred at 0, the fundamental solution E, and show that E=δ+h where h is integrable and solve h(t)=g(t)+∫^t_0 g(t-τ)h(τ) dτ The solution of the general case is then y(t)=f(t)+(h*f)(t) which involves the convolution of h and f. We can approximate g to desired accuracy with piecewise constant kernel for which the solution h is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.
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