Abstract
Abstract We study a Volterra convolution equation of the second kind, based on a combination of Riemann–Liouville integrals. The problem can be reformulated involving the Caputo fractional derivative, hence the equation becomes of differintegral type. The modeling interpretation is based on a non-Markovian state function, where the Riemann–Liouville multi-orders are memory coefficients that decrease hazard risks of change. We prove the validity of the reformulations with fractional-calculus theory, local existence with fixed-point tools, and global uniqueness with a Gronwall-type argumentation. We show some examples and their associated physics. We also solve the general linear equation by means of the algebraic formalism of Mikusiński operational calculus, which is superior to Laplace transforms or Picard’s iterations. Multivariate Mittag–Leffler functions play a key role. We relate the emerging closed-form solution with the fractional power series that one may expect for these types of models.
Published Version
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