Extension of the adaptive Huber method for Volterra integral equations arising in electroanalytical chemistry, to convolution kernels exp [−α (t-τ)] erex {[β(t-τ)]1/2} and exp [−α(t-τ)] daw {[β(t-τ)]1/2}

Abstract

Integral transformation kernels having convolution forms exp[-α(t -τ)]erex{[β(t - τ)]1/2} and exp[-α(t- τ)]daw{⌊β(t - τ)⌋1/2}, where erex(z) = exp(z2)erfc(z), daw(z) is the Dawson integral, and α ≥ 0 and β ≥ 0, occur in integral equations of Volterra type, pertinent to electroanalytical chemistry. An ability to solve this sort of integral equations has been added to the recently developed adaptive Huber method. Relevant formulae for the method coefficients are reported, and computational tests of the convergence and numerical stability of the method for these kernels are presented. Practical accuracy orders are close to 2. In cases when integral equations contain contributions from the kernel exp⌊- α(t - τ)⌋daw{[β(t - τ)]1/2}, such that the total kernels are increasing functions of t -τ, the method may become unstable. In cases when α > 0, β > 0, β/α ≫ 1 and integration steps are very small, β/α cannot exceed a certain threshold due to machine precision. Similarly, in the case of kernel exp[-α(t - τ)]erex{⌊β(t -τ)⌋l/2}, when α > 0, β > 0, α ≠ β, but α is very close to β, |β/α -1| cannot be smaller than a certain threshold.