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Short review of an approach to explicit solving of initial and boundary value problems (BVPs) for some partial differential equations (PDEs) is presented. A combination of two classical methods Å the Fourier method and the Duhamel principle Å is used in the framework of a two-dimensional operational calculus. It gives explicit solutions of some local and(More)
In the paper, on the examples of a string and beam, resonance vibrations in linear elastic systems that result from nonlocal boundary conditions (even if external forces are lacking) are considered. Formulas of exact solutions of the corresponding boundary-value problems are presented. These formulas are used for evaluation of the solutions and for(More)
The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral(More)
The Pommiez operator (∆ f)(z) = (f (z) − f (0))/z is considered in the space Ᏼ(G) of the holomorphic functions in an arbitrary finite Runge domain G. A new proof of a representation formula of Linchuk of the commutant of ∆ in Ᏼ(G) is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant(More)
A review of our operational calculus approach to obtaining periodic and mean-periodic solutions of LODE with constant coefficients is presented. Let P (λ) = a 0 λ n + a 1 λ n−1 + · · · + a n−1 λ + a n be a non-zero polynomial with constant coefficients of degree n and let us consider an ordinary linear differential equation of the form: P d dt y = f (t), −∞(More)
The theory of the nonlocal linear boundary value problems is still on the level of examples. Any attempt to encompass them by a unified scheme sticks upon the lack of general methods. Here we are to outline an algebraic approach to linear nonlocal boundary value problems. It is based on the notion of convolution of linear operator and on operational(More)
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