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UDC 517.9 On algorithm of numerical modelling of the Boussinesq – L’ove waves A.A. Zamyshlyaeva, South Ural State University, Chelyabinsk, Russian Federation, zamyshliaevaaa@susu.ac.ruAbstractThe article is devoted to the description of the software complex «Modeling of the Boussinesq – L’ove waves», which consists of four modules and implements the algorithm of numerical solution of the problem Showalter - Sidorov (Cauchy) with Dirichlet condition on a segment, on a graph, in a rectangle or a circle (user selectable) for the Boussinesq – L’ove equation, depending on the coefficients and the initial data. Specified equation models the longitudinal fluctuations in the elastic rod (in case of a segment), in construction (case of graph), propagation of waves in shallow water or in dispersive environments (case of rectangle or circle). The algorithm implemented the method of phase space and modified Galerkin method. In each of the four modules the eigenvalues and the eigenfunctions for the Laplace operator in the relevant domain are computed, the solution in the form of Galerkin sum by the first several eigenfunctions is found. The program allows drawing a graph for the numerical solution of the specified problems. The results may be useful for specialists in the field of mathematical physics and mathematical modeling. KeywordsShowalter – Sidorov problem, Boussinesq – L’ove equation, Sobolev type equation, phase space method, Galerkin method References1. Wang C. Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Application, 2002, Vol. 274, pp. 846–866. 2. Whitham G. Lineynye i nelineynye volny [Linear and Nonlinear Waves]. Мoscow, Мir, 1977. 624 p. 3. Landau L.D., Lifshits E.M. Teoreticheskaya fisika, VII. Teoriya uprugosti [Theoretical Physics, VII. The Elasticity Theory]. Мoscow, Nauka, 1987. 248 p. 4. Zagrebina S.A. The Initial-Finite Problems for Nonclassical Models of Mathematical Physics [Nachalno-konechnye Zadachy dlya Neklassicheskikh Modeley Matematicheskoy Fiziki]. Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming & Computer Software”, 2013, vol. 6, no. 2, pp. 5–24. (in Russian) 5. Zamyshlyaeva А.А. The Initial-finish Problem for the Nonhomogeneous Boussinesq – L’ove Equation [Nachalno-konechnaya Zadacha dlya Neodnorodnogo Uravneniya Bussineska – Lyava]. Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming & Computer Software”, 2011, vol. 10, no. 37 (254), pp. 22–29. (in Russian) 6. Sviridyuk G.A., Fedorov V.E. [Linear Sobolev Type Equations and Degenerate Semigroups of Operators]. Utrecht, Boston, Köln, Tokyo: VSP, 2003. 268 p. 7. Zamyshlyaeva А.А. Lineynye uravneniya sobolevskogo tipa vysokogo poryadka [Linear Sobolev Type Equations of High Order]. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p. SourceBulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2013, vol. 13, no. 4, pp. 24-29. (in Russ.) (The main) |