Solving initial-boundary mathematical physics’ problems based on Kotelnikov formula (the Nyquist–Shannon formula)

Abstract

Background. Numerical methods for differential equations solving is a topical problem in applied mathematics. The article is devoted to the numerical-analytical methods of the second and third order of accuracy, based on the approximation of nonlinear differential equations by piecewise-linear ones. Materials and methods. Integral transform methods for solving equations of mathematical physics is supplemented by the method of expanding the signal in a series according to the Kotelnikov formula (the Nyquist–Shannon formula). Analytical continuation method and Hilbert integral transform one served as the basis for the description of analytical signals. Results. A new analytical method for solving problems of mathematical physics is proposed, which is a synthesis of Integral Fourier transform method and the expansion method into the Kotelnikov series. An algorithm is proposed: first, find the Fourier image of the initial-boundary data; second, expand the found image in a Fourier series; thirdly, we go back to the original. The proposed algorithm is implemented under the assumption that the Fourier image support is bounded. Thus, we obtain discrete analogs of Poisson integral formulas for solving the Cauchy problem and the Dirichlet problem. A discrete analogue of the Cauchy and Schwarz formulas for an analytic function in a half-plane is obtained in the article. Conclusions. The proposed methods can be useful in creating new numerical methods for Cauchy and Dirichlet problems solving.

Keywords

• kotelnikov formula (the nyquist–shannon formula)
• fourier transform
• cauchy problem
• dirichlet problem
• hilbert transform
• analytical signal