Geodesic
Three-Dimensional Operational Calculi for Nonlocal Evolution Boundary Value Problems
Mathematics and Education in Mathematics, Tome 40 (2011) no. 1, pp. 169-175.

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Direct algebraic operational calculi for functions u(x, y, t), continuous in a domain of the form D = [0, a] × [0, b] × [0, ∞), are proposed. Along with the classical Duhamel convolution, the construction uses also two non-classical convolutions for the operators ∂2x and ∂2y. These three one-dimensional convolutions are combined into one three-dimensional convolution u ∗ v in C(D). Instead of J. Mikusi´nski’s approach, based on convolution fractions, we develop systematically an alternative approach, based on the multiplier fractions of the convolution algebra (C(D), ∗). *2000 Mathematics Subject Classification: 44A35, 44A45, 35K20, 35K15, 35J25.
Keywords: Duhamel Convolution, Convolution Algebra, Multiplier, Multiplier Fraction, Divisor of Zero, Numerical Operator 
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Dimovski, Ivan; Tsankov, Yulian. Three-Dimensional Operational Calculi for Nonlocal Evolution Boundary Value Problems. Mathematics and Education in Mathematics, Tome 40 (2011) no. 1, pp. 169-175. http://geodesic.mathdoc.fr/item/MEM_2011_40_1_a13/