# IJPAM: Volume 113, No. 4 (2017)

# Title

LOADED PARABOLIC EQUATIONS AND BOUNDARYVALUE PROBLEMS OF HEAT CONDUCTION IN

NON-CYLINDRICAL DEGENERATING DOMAINS

# Authors

Muvasharkhan T. JenaliyevInstitute of Mathematics and Mathematical Modeling

Almaty, KAZAKHSTAN

# Abstract

This paper gives an overview of the author's research, his colleagues and pupils for the years 1976-2016. At the beginning the results concerning the loaded differential equations are provided, including the essential-and spectral-loaded equations. Then, the results of research concerning the boundary value problems of heat conduction in non-cylindrical degenerating domains are discussed.# History

**Received: **December 15, 2016
**Revised: **January 30, 2017
**Published: **March 30, 2017

# AMS Classification, Key Words

**AMS Subject Classification: **35K05, 35K20, 47A75
**Key Words and Phrases: **loaded differential equation, boundary value problem, degenerating domain, heat equation, existence and non-unique of solution

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## Bibliography

- 1
- A.M. Nakhushev,
*Loaded equations and their applications*, Nauka, Moscow (2012), (in Russian). - 2
- M.T. Jenaliyev (Dzhenaliev), Optimal Control of Linear Loaded Parabolic Equations,
*Differential Equations*, 25, No. 6 (1989), 641-651 (in Russian). - 3
- M.T. Jenaliyev (Dzhenaliev), An Solvability of Boundary Value Problems for Linear Loaded Equations with non-regular coefficients,
*Differential Equations*, 27, No. 9 (1991), 1585-1595 (in Russian). - 4
- M.T. Jenaliyev (Dzhenaliev), About Boundary Value Problem for Linear Loaded Parabolic Equation with Non-local Boundary Conditions,
*Differential Equations*, 27, No. 10 (1991), 1825-1827 (in Russian). - 5
- M.T. Jenaliyev (Dzhenaliev), Boundary Value Problems and Optimal Control Problems for Linear Loaded Hyperbolic Equations,
*Differential Equations*, 28, No. 2 (1992), 232-241 (in Russian). - 6
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*Differential Equations*, 28, No. 3 (1992), 522-524 (in Russian). - 7
- M.T. Jenaliyev (Dzhenaliev), Boundary Value Problems for Loaded Parabolic and Hyperbolic Equations with Time-Derivatives in Boundary Conditions,
*Differential Equations*, 28, No. 4 (1992), 661-666 (in Russian). - 8
- M.T. Jenaliyev (Dzhenaliev), Nonhomogeneous Boundary Value Problems for Loaded Evolutional Equations of odd Order with Interior Boundary Conditions,
*Differential Equations*, 29, No. 4 (1993), 617-626 (in Russian). - 9
- M.T. Jenaliyev (Dzhenaliev), Boundary Value Problems for Differential Operator Equations of odd Order and Loaded Equations,
*Differential Equations*, 29, No. 8 (1993), 1380-1389 (in Russian). - 10
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*Differential Equations*, 30, No. 4 (1994), 723-724 (in Russian). - 11
- M.T. Jenaliyev (Dzhenaliev), Boundary Value Problems for Loaded Differential Equations of First and second Order and Their Applications,
*Differential Equations*, 31, No. 1 (1995), 160-162 (in Russian). - 12
- M.T. Jenaliyev (Dzhenaliev), About Non-local Boundary Value Problems for Loaded Differential Operator Equations of First Order,
*Differential Equations*, 31, No. 11 (1995), 1902-1906 (in Russian). - 13
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*Differential Equations*, 31, No. 12 (1995), 2029-2037 (in Russian). - 14
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*Differential Equations*, 32, No. 4 (1996), 518-522 (in Russian). - 15
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*Differential Equations*, 37, No. 1 (2001), 48-54 (in Russian). - 17
- M.T. Jenaliyev (Dzhenaliev), M.I. Ramazanov, On Boundary Value Problem for Spectrally Loaded Heat Operator ,
*Siberian Mathematical Journal*, 47, No. 3 (2006), 527-547 (in Russian). - 18
- M.T. Jenaliyev (Dzhenaliev), M.I. Ramazanov, About one Boundary Value Problem for Spectrally Loaded Heat Conduction Operator. I,
*Differential Equations*, 43, No. 4 (2007), 498-508 (in Russian). - 19
- M.T. Jenaliyev (Dzhenaliev), M.I. Ramazanov, About one Boundary Value Problem for Spectrally Loaded Heat Conduction Operator. II,
*Differential Equations*, 43, No. 6 (2007), 788-794 (in Russian). - 20
- D.M. Akhmanova, M.T. Jenaliyev (Dzhenaliev), M.I. Ramazanov, On particular integral equation of second kind with spectral parameter,
*Siberian Mathematical Journal*, 52, No. 1 (2011), 3-14 (in Russian). - 21
- M.M. Amangaliyeva, D.M. Akhmanova, M.T. Jenaliyev (Dzhenaliev), M.I. Ramazanov, The boundary value problems for spectrally loaded heat conduction operator with proximity loaded curve to times axis at zero or at infinity,
*Differential Equations*, 47, No. 2 (2011), 231-243 (in Russian). - 22
- M.M. Amangaliyeva, M.T. Jenaliyev, M.T. Kosmakova, M.I. Ramazanov, About Dirichlet boundary value problem for the heat equation in the infinite angular domain,
*Boundary Value Problems*, 2014, 213 (2014), 1-21. - 23
- M.M. Amangaliyeva, M.T. Jenaliyev, M.T. Kosmakova, M.I. Ramazanov, On a Volterra equation of the second kind with 'incompressible' kernel,
*Advances in Difference Equations*, 2015, 71 (2015), 1-14. - 24
- M.M. Amangaliyeva, M.T. Jenaliyev (Dzhenaliev), M.T. Kosmakova, M.I. Ramazanov, On one homogeneous problem for the heat equation in an infinite angular domain,
*Siberian Mathematical Journal*, 56, No. 6 (2015), 982-995 (in Russian). - 21
- V.I. Korzyuk, M.T. Jenaliyev (Dzhenaliev), I.S. Kozlovskaya, Boundary Value Problems for a Weakly Loaded Operator of a Second-Order Hyperbolic Equation in a Cylindrical Domain,
*Differential Equations*, 51, No. 12 (2015), 1595-1606 (in Russian). - 25
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*Loaded equations as perturbation of differential equations*, Gylym, Almaty (2010) (in Russian). - 29
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lder spaces,
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- M.T. Jenaliyev, Loaded parabolic equations and boundary value problems of heat conduction in non-cylindrical degenerating domains,
*Kazakh Mathematical Journal*, 17, No. 1 (2017), 13-23.

# How to Cite?

**DOI: 10.12732/ijpam.v113i4.2**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2017

**Volume:**113

**Issue:**4

**Pages:**5 - 15

Google Scholar; DOI (International DOI Foundation); WorldCAT.

**This work is licensed under the Creative Commons Attribution International License (CC BY).**