Interpolatory inequalities for first kind convolution volterra integral equations

We consider the problem of computing a function u(t) which satisfies the equation k * u = ^Z₀^t k(t - s)u(s) ds = f(t), 0 = t < 8, (a) where k(t) is a given kernel and the right-hand side f(t) is only known through some observations which contain observational errors. This problem arises in the study of the rheology of linear viscoelastic materials. It is well known that solving this first kind Volterra integral equation is ill-posed and thus special regularisation techniques are required to solve it in a stable fashion. An important property of many kernels occurring in rheological applications is that they admit solutions h(t) of the interconversion equation ^Z₀^t k(t - s)h(s) ds = t, 0 = t < 8. (b) In such situations, the solution of the Volterra equation takes the following form u(t) = _dt^d2₂ {^Z₀^t h(t - s)f(s) ds }. (c) Problems where the interconversion equation can be solved explicitly includes n k(t) = 1 + X a_j exp(-t/t_j) j=1 with the t_j > 0. A characterisation of the solutions h(t) for such k(t) will be given below. Even when the solution h(t) to the interconversion equation is known, the problem of computing the solution of (a) as (c) is still ill-posed. A consequence is that regularisation techniques are required to compute u(t). Even then for data containing errors, one can at best get error bounds of the form kue - uk = ?(e) e for some unbounded ?(e). The form of such error bounds based, on interpolatory inequalities, will be discussed for the first kind Volterra equations considered here. Finally, a numerical technique to solve such Volterra equations, when the data is sampled and contains errors, will be discussed. The method is obtained from equation (c) by explicit differentiation, it evaluates u(t) = h⁰(0)f(t) + ^Z₀^t h⁰⁰(t - s)f(s) ds + h(0) ^df_dt⁽t⁾ , (d) where the prime in (h⁰) denotes the derivative of the function h etc. Instead of having to perform the numerical differentiation of the kernel k in equation (a) and then solve the resulting second kind Volterra integral equation, or the numerical differentiation of equation (c), the solution can be obtained, when h is known, through the direct evaluation of the right hand side of equation (d).