Well-posedness results for the wave equation generated by the Bessel operator

In this paper, we consider the non-homogeneous wave equation generated by the Bessel operator. We prove the existence and uniqueness of the solution of the non-homogeneous wave equation generated by the Bessel operator. The representation of the solution is given. We obtained a priori estimates in Sobolev type space. This problem was firstly considered in the work of M. Assal [1] in the setting of Bessel-Kingman hypergroups. The technique used in [1] is based on the convolution theorem and related estimates. Here, we use a different approach. We study the problem from the point of the Sobolev spaces. Namely, the conventional Hankel transform and Parseval formula are widely applied by taking into account that between the Hankel transformation and the Bessel differential operator there is a commutation formula [2].

A minimization approach to the wave equation on time-dependent domains

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for ψ(x) ∈ W1p (1 < p ≤ 2), and it is also shown that if ψ(x) ∈ Lp[0, 1] (1 ≤ p ≤ 2), the formal solution is a generalized solution of the mixed problem.

Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).