IJPAM: Volume 113, No. 4 (2017)

Title

LOADED PARABOLIC EQUATIONS AND BOUNDARY
VALUE PROBLEMS OF HEAT CONDUCTION IN
NON-CYLINDRICAL DEGENERATING DOMAINS

Authors

Muvasharkhan T. Jenaliyev
Institute 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

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

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
M.T. Jenaliyev (Dzhenaliev), About one Class of Loaded Elliptic Equations, 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
M.T. Jenaliyev (Dzhenaliev), To Generalized Solvability of Loaded Wave Equation, 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
M.T. Jenaliyev (Dzhenaliev), On Quadratic Functional in Cauchy Problem for Loaded Differential Operator Equation of First Order. I, Differential Equations, 31, No. 12 (1995), 2029-2037 (in Russian).

14
M.T. Jenaliyev (Dzhenaliev), On Quadratic Functional in Cauchy Problem for Loaded Differential Operator Equation of First Order. II, Differential Equations, 32, No. 4 (1996), 518-522 (in Russian).

15
M.T. Jenaliyev, On the boundary value problems for loaded differential equations, J.Korean Math.Soc. (Seoul), 37, No. 6 (2000), 1031-1042.

16
M.T. Jenaliyev (Dzhenaliev), On Loaded Equations with Periodic Boundary Conditions, 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
M.T. Jenaliyev, M.I. Ramazanov, Spectral Volterra Integral Equation of the Second Kind with a Spectral Parameter, In Book Analytic Methods of Analysis and Differential Equations (AMADE 2015), Cambridge Scientific Publishers, UK, (2016), 61-76.

26
J.-L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, V.2, Springer-Verlag, Berlin (1972).

27
M.T. Jenaliyev, Introduction to the theory of linear loaded differential equations, Gylym, Almaty (1995) (in Russian).

28
M.T. Jenaliyev, M.I. Ramazanov, Loaded equations as perturbation of differential equations, Gylym, Almaty (2010) (in Russian).

29
V.A. Solonnikov, A. Fasano, One-dimensional parabolic problem arising in the study of some free boundary problems, Zapiski nauchnykh seminarov POMI, 269, (2000), 322-338 (in Russian).

30
T. Berroug, H. Ding, R. Labbas, B.-Kh. Sadallah, On a degenerate parabolic problem in H $\ddot{\textrm{o}}$lder spaces, Applied Mathematics and Computation, 162 (2005), 811-833.

31
R. Labbas, A. Medeghri, B.-Kh. Sadallah, An $L_p$-approach for the study of degenerate parabolic equations, Electronic Journal of Differential Equations, 2005, 36, (2005), 1-20.

32
A. Kheloufi, B.-Kh. Sadallah, On the regularity of the heat equation solution in non-cylindrical domains: Two approaches, Applied Mathematics and Computation, 218 (2011), 1623-1633.

33
A. Kheloufi, Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of $\mathbb{R}^3$, Applied Mathematics and Computation, 220 (2013), 756-769.

34
A. Kheloufi, B.-Kh. Sadallah, Resolution of a high-order parabolic equation in conical time-dependent domains of $\mathbb{R}^3$, Arab Journal of Mathematical Sciences, 22 (2016), 165-181.

35
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 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
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.

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