Operational Calculus Operational Method Convolution Duhamel Principle Cauchy Problem Nonlocal Boundary Value Problem Computer Algebra System Symbolic Computation Numerical Computation
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 3, No 4, (2009), 381p-424p
The presented research is related to the operational calculus
approach and its representative applications. Operational methods are considered,
as well as their program implementation using the computer algebra
system Mathematica. The Heaviside algorithm for solving Cauchy’s problems
for linear ordinary differential equations with constant coefficients is
considered in the context of the Heaviside-Mikusinski operational calculus.
The program implementation of the algorithm is described and illustrative
examples are given. An extension of the Heaviside algorithm, developed by I. Dimovski and S. Grozdev, is used for finding periodic solutions of
linear ordinary differential equations with constant coefficients both in the
non-resonance and in the resonance cases. The features of its program implementation
are described and examples are given. An operational method
for solving local and nonlocal boundary value problems for some equations
of the mathematical physics (the heat equation, the wave equation and the
equation of a free supported beam) is developed and the capabilities of the
corresponding program packages for solving those problems are described.
A comparison with other methods for solving the same types of problems is
included and the advantages of the operational methods are marked.
This article presents the principal results of the doctoral thesis “Direct Operational Methods
in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of
mathematics and Informatics, BAS), successfully defended before the Specialised Academic
Council for Informatics and Mathematical Modelling on 23 March, 2009.