Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

Theory of automatic control. Linear, continuous systems

The textbook presents the basics of the classical theory of automatic control, based on mathematical models of real systems, given in the form of systems of linear differential equations with constant coefficients. Methods based on Laplace and Fourier transforms, stability, controllability, and observability theory, as well as directed graph theory and linear algebra are used. Meets the requirements of the federal state educational standards of higher education of the latest generation. For students of higher educational institutions studying in the areas of training and specialties 15.00.00 "Mechanical Engineering", 27.00.00 "Management in technical systems".

Doppler effect described by the solutions of the Cattaneo telegraph equation

The Cattaneo telegraph equation for temperature with moving time-harmonic source is studied on the line and the half-line domain. The Laplace and Fourier transforms are used. Expressions which show the wave fronts and elucidate the Doppler effect are obtained. Several particular cases of the considered problem including the heat conduction equation and the wave equation are investigated. The quasi-steady-state solutions are also examined for the case of non-moving time-harmonic source and time-harmonic boundary condition for temperature.

Considerations on Computational Accuracy of the Solution for a Partial Differential Equation

In this paper, we will compare the methods of solving with explicit or implicit finite difference of the partial differential equations that define the mechanical models of hydrodynamics movements, thermodynamics or those that define the vibration movements with the ones that use integral transforms. By applying the Laplace and Fourier transforms, finite in sine or cosine, depending on the boundary conditions of the real physical problem, it leads to the algebraic approach of the problem, which reduces the difficulty of solving partial differential equations. The errors obtained for the solution of partial differential equations using different methods are within the standard norms. However, in terms of calculus precision, the use of integral transforms is more advantageous.

Analytical solutions of some general classes of differential and integral equations by using the Laplace and Fourier transforms

This article deals with a general class of differential equations and two general classes of integral equations. By using the Laplace transform and the Fourier transform, analytical solutions are derived for each of these classes of differential and integral equations. Some illustrative examples and particular cases are also considered. The various analytical solutions presented in this article are potentially useful in solving the corresponding simpler differential and integral equations.