Explicit closed-form solution of coupled systems of Volterra integrodifferential systems (Q1182667)

The author considers a system of \(m\) integro-differential equations of Volterra convolution type in which the kernels are polynomials in \((t- s)\). In matrix notation \[ y'(t)=Ay(t)+\int^ t_ 0\sum^{n- 1}_{j=1}(B_ j(t-s)^{j-1}/(j-1)!)y(s)ds, \] where \(A\), \(B_ j\) for \(1\leq j\leq n-1\) are complex (constant) matrices. By setting \(z(t)=(1/(j- 1)!)\int^ t_ 0(t-s)^{j-1}y'(s)ds\) the author reduces the stated integro-differential equation to a non-homogeneous system of differential equations of order \(n\) having constant coefficients, the non-homogeneous terms being polynomials, and by further reduction to canonical form the author constructs a formula, too complicated to be quoted here, for the solution \(y(t)\).