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<p>The article discusses a&#xa0;mathematical model and a&#xa0;finite-difference scheme for the heating process of a&#xa0;plate limited in three spatial variables. The disadvantages of using the classical parabolic heat equation for this case and the rationale for using a&#xa0;mixed equation are given. The implicit difference scheme uses an integro-interpolation method to reduce errors. The quasilinear scheme is used to solve an equation with the nonlinear thermal conductivity coefficient. The first boundary conditions are on the left boundary and on the right boundary of the plate on the space variable x. The third boundary conditions are on the upper and lower boundaries on the space variable y. The variable z is used as parameter. Initial conditions are specified. The heat source in the parabolic part of the equation is 0, and in the hyperbolic part of the equation, sharp heating begins. The problem is solved numerically in the Mathcad-15 package using the locally one- dimensional scheme. The paper presents the results of the calculation program for the finite plate in the form of graphs and tables of the temperature field, and a&#xa0;certificate of state registration of the program is received.</p>

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Mixed initial-boundary value problem for a three-dimensional nonlinear hyperbolic-parabolic equation

  • 期刊论文

  • 发表日期：

    2026年2月2日

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  • Vladislav N. Khankhasaev,
  • Safron A. Bairov

摘要

The article discusses a mathematical model and a finite-difference scheme for the heating process of a plate limited in three spatial variables. The disadvantages of using the classical parabolic heat equation for this case and the rationale for using a mixed equation are given. The implicit difference scheme uses an integro-interpolation method to reduce errors. The quasilinear scheme is used to solve an equation with the nonlinear thermal conductivity coefficient. The first boundary conditions are on the left boundary and on the right boundary of the plate on the space variable x. The third boundary conditions are on the upper and lower boundaries on the space variable y. The variable z is used as parameter. Initial conditions are specified. The heat source in the parabolic part of the equation is 0, and in the hyperbolic part of the equation, sharp heating begins. The problem is solved numerically in the Mathcad-15 package using the locally one- dimensional scheme. The paper presents the results of the calculation program for the finite plate in the form of graphs and tables of the temperature field, and a certificate of state registration of the program is received.