Explicit solutions of Volterra integro-differential convolution equations

We find an explicit form of solution of a convolution type integro-differential equation of the first order partial derivative Q(t, z)/partial derivative z = alpha(t, z) - integral(t)(0) beta(t - u)Q(u, z)du, (t,z) is an element of (0, T) x (0, T), (1) where T > 0, and alpha, beta are two given locally integrable functions and Q satisfies a boundary condition Q(t , 0) = gamma (t) for a locally integrable function gamma. To achieve this goal we introduce the notion of a biconvolution algebra of locally integrable functions on R-+(2). We investigate the properties of the biconvolution algebra and study the Volterra integral equations of the second kind associated with the biconvolution operation. Finally, we present an explicit form of solution of integro-differential equation associated with a linear operator Lambda(n) = partial derivative(n+1)/partial derivative z partial derivative t(n) + Sigma(n-1)(i=0)lambda(i)partial derivative(i)/partial derivative t(i), lambda(i) is an element of R, i <= n - 1, n is an element of N. (2) (C) 2021 Elsevier Inc. All rights reserved.