On a boundary value problem for the nonhomogeneous heat equation in an angular domain

Authors

  • M. A. Sadybekov Institute of Mathematics and Mathematical Modeling
  • M. G. Yergaliyev Institute of Mathematics and Mathematical Modeling
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Keywords:

Heat equation, Green’s function, classical solution

Abstract

Due to the fact that the results find theoretical and practical applications, great attention is paid
to the study of boundary value problems for parabolic equations. Also the relevance of studying
such problems is justified by their physical application in the modeling of such processes as the
propagation of heat in homogeneous and nonhomogeneous media, the interaction of filtration
and channel flows, and other. Therefore, at the present stage of its development, the theory of
partial differential equations is one of the important branches of mathematics and is actively
developed by various mathematical schools. However, a number of significant problems in the
theory of partial differential equations remain, as before, unresolved. In the paper we study a
boundary value problem for the nonhomogeneous heat equation in an angular domain. Note that
the problem does not have the initial condition. It is caused by the form of the domain. We obtain
a boundary condition for the nonhomogeneous heat equation considered in the angular domain.
It is proven that the heat potential is a unique classical solution to this problem.

Author Biography

M. A. Sadybekov, Institute of Mathematics and Mathematical Modeling



References

[1] M.M. Amangalieva, M.T. Dzhenaliev, M.T. Kosmakova, M.I. Ramazanov, ”On one
homogeneous problem for the heat equation in an infinite angular domain”, Sib. math.
jour. 56 (6) (2015): 982–995.
[2] A. Friedman, Partial differential equations of parabolic type (Englewood Cliffs, N.J.:
Prentice-Hall, 1964), 262.
[3] Kal’menov T.Sh., Arepova G.D., ”On a heat and mass transfer model for the locally
inhomogeneous initial data”, Vestnik YuUrGU. Ser. Mat. Model. Progr. 9 (2) (2016):
124–129.
[4] Kal’menov T.Sh., Suragan D., ”Boundary conditions for the volume potential for the
polyharmonic equation”, Differential Equations. 48 (4) (2012): 604–608.
[5] Kal’menov T.Sh., Suragan D., ”Transfer of Sommerfeld radiation conditions to the
boundary of a bounded domain”, Zh. Vychisl. Mat. i Mat. Fiz. 52 (6) (2012): 1063–
1068.
[6] Kal’menov T.Sh., Suragan D., ”Initial boundary value problems for the wave equation”,
Electronic journal of differential equations. 2014 (48) (2014): 1–6.
[7] Kal’menov T.Sh., Suragan D., ”On permeable potential boundary conditions for the
Laplace-Beltrami operator”, Siberian Mathematical Journal. 56 (6) (2015): 1060–1064.
[8] Kal’menov T.Sh., Suragan D., ”To spectral problems for the volume potential”, Dokl.
Math. 80 (2) (2009): 646-649.
[9] Kal’menov T.Sh., Suragan D., ”A boundary condition and spectral problems for the
Newton potentials”, Oper. Theory, Adv. Appl. 216 (2011): 187–210.
[10] Kal’menov T.Sh., Tokmagambetov N., ”On a nonlocal boundary value problem for the
multidimensional heat equation in a noncylindrical domain ” , Siberian Mathematic
Journal. 54 (6) (2013): 1023–1028.
[11] Suragan D., Tokmagambetov N., ”On transparent boundary conditions for the high-order
heat equation” , Siberian Electronic Math. Reports. 10 (2013): 141–149.

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How to Cite

Sadybekov, M. A., & Yergaliyev, M. G. (2018). On a boundary value problem for the nonhomogeneous heat equation in an angular domain. Journal of Mathematics, Mechanics and Computer Science, 96(4), 31–36. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/567