COMPUTER STUDY OF THE 2D WAVE EQUATION: WHAT IS NEEDED FOR THE ALGORITHM CONSTRUCTION?


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Abstract

The author considers the methodological issues of teaching the topic "Hyperbolic equations" as a part of the mathematical physics equations course for the IT students of the training programs “Applied mathematics” and “Applied mathematics and informatics”. This section of the theory of differential equations is essential as well in such courses as “Continuous mathematical models”, “Mathematical modeling” and some other similar courses forming the curriculum for the bachelor’s and master’s degree candidates. The author uses the example of a homogeneous wave equation with one space variable and arbitrary boundary conditions of the third type to demonstrate the unity of analytical and numerical research methods. Thus, the author considers the problem of finding particular solutions for such equation satisfying the stated boundary conditions with the arbitrary coefficients. Such a problem cannot be obviously solved; therefore we need software for the numerical calculation of spectral numbers – the eigenvalues of the corresponding Sturm – Liouville problem on an interval. It is demonstrated that to write the appropriate software code, it is necessary to know various areas of mathematics. The author discusses situations which may cause the inadequate work of a program what is a motivating factor to learn the relevant mathematical topics. It is concluded about the necessity to use the results of the analytical study of an equation to write the computer program algorithms. The author emphasizes that without such an analysis, the program may, firstly, lead to a wrong solution; secondly, the solution may be incomplete (not all possible values are found) and, thirdly, the program may work in the non-optimal and resource-wasting mode.

About the authors

Sergey Vladimirovich Talalov

Togliatti State University, Togliatti

Author for correspondence.
Email: svtalalov@tltsu.ru

Doctor of Science (Physics and Mathematics), Professor of Chair “Applied Mathematics and Informatics”

Russian Federation

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