In this work, we study a one-dimensional mixed problem for an inhomogeneous parabolic equation with constant coefficients of general form, under non-local and non-self-conjugate boundary conditions. The considered mixed problem consist of two parts. The first problem is a mixed problem with a regular boundary condition, and the uniqueness of the solution is proved through the deduction operator. Then the existence of a solution to the mixed problem is shown, and an exact formula for the solution is found. A second mixed problem is the time delay in the boundary conditions. Since the spectral problem obtained after the integral transformation is not homogeneous, the considered problem is again divided into two problems. Under the minimum conditions at the initial data, by combining the deduction method and the contour integral method, the existence and uniqueness of the solution to the mixed problem is proved, where an explicit analytic representation for it is obtained.
| Published in | American Journal of Applied Mathematics (Volume 11, Issue 3) |
| DOI | 10.11648/j.ajam.20231103.11 |
| Page(s) | 32-39 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2023. Published by Science Publishing Group |
Residual Method, Time Shift, Mixed Problem
| [1] | Mammadov Y. A., Ahmadov H. I., Mixed problem for the heat equation with advanced time in boundary conditions, Russ. Math., Vol. 64, pp. 25-42, 2020. |
| [2] | Mammadov Y. A., Ahmadov H. I., Solution of the mixed problem for a nonhomogeneous heat equation with time advance in the boundary conditions, The reports of national academy of sciences of Azerbaijan, vol. LXXVI, No 1-2, pp. 19-22, 2020. |
| [3] | Mammadov Y. A., Mathematical statement and solution of one heat problem at partially determined boundary regime, News of Baku University, series of Physicso-Mathematical Sciences, No. 3, pp. 5-11, 2005. |
| [4] | Sabitov K. B., Boundary Value problem for a parabolic-hyperbolic equation with a nonlocal integral condition, Differential Equations, Vol. 46, No. 10, pp. 1472–1481, 2010. |
| [5] | Zarubin A. N. Tricomi problem for an advanced- retarded equation of the mixed type with closed degeneration line. Differential Equations, Vol. 51, No. 10, pp. 1306-1318, 2015. |
| [6] | Zarubin A. N. Nonlocal tricomi boundary value problem for a mixed-type differential-difference equation, Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Vol. 25, No. 1. - pp. 35-50, 2021. |
| [7] | Zarubin, A. N. Boundary value problem for a mixed functionally differential advancing-lagging equation with fractional derivative, Russ. Math., Vol. 63, pp. 44–56, 2019. |
| [8] | Zarubin, A. N. Boundary-value problem for functional-differential advanced-retarded tricomi equation, Russ. Math., Vol. 62, pp. 6–20, 2018. |
| [9] | Zarubin, A. N., Kholomeeva, A. A. Tricomi problem for an advance–delay equation of mixed type with variable deviation of the argument, Differential Equations Vol. 52, pp. 1312–1322, 2016. |
| [10] | Koshlyakov N. S., Gliner E. B., Smirnov M. M., Equations in partial derivatives of mathematical physics, Moscow, 1970. |
| [11] | Rasulov M. L., Contour Integral Method and Its Application to Problems for Differential Equations (Nauka, Moscow, 1964) [in Russian]. |
| [12] | Naimark M. A., Linear Differential Operators (Ungar, New York, 1967; Nauka, Moscow, 1969). |
| [13] | Petrovsky I. G., Lectures on Partial Differential Equations (Dover Books on Mathematics) Revised Edition, 1961. |
| [14] | Mikhlin S. G., Course of Mathematical Physics, Nauka, Moscow, 1968. |
| [15] | Rasulov M. L., Applications of the contour integral method, Nauka, Moscow, 1975. |
APA Style
Hikmat I. Ahmadov. (2023). On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions. American Journal of Applied Mathematics, 11(3), 32-39. https://doi.org/10.11648/j.ajam.20231103.11
ACS Style
Hikmat I. Ahmadov. On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions. Am. J. Appl. Math. 2023, 11(3), 32-39. doi: 10.11648/j.ajam.20231103.11
AMA Style
Hikmat I. Ahmadov. On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions. Am J Appl Math. 2023;11(3):32-39. doi: 10.11648/j.ajam.20231103.11
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Download@article{10.11648/j.ajam.20231103.11,
author = {Hikmat I. Ahmadov},
title = {On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions},
journal = {American Journal of Applied Mathematics},
volume = {11},
number = {3},
pages = {32-39},
doi = {10.11648/j.ajam.20231103.11},
url = {https://doi.org/10.11648/j.ajam.20231103.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231103.11},
abstract = {In this work, we study a one-dimensional mixed problem for an inhomogeneous parabolic equation with constant coefficients of general form, under non-local and non-self-conjugate boundary conditions. The considered mixed problem consist of two parts. The first problem is a mixed problem with a regular boundary condition, and the uniqueness of the solution is proved through the deduction operator. Then the existence of a solution to the mixed problem is shown, and an exact formula for the solution is found. A second mixed problem is the time delay in the boundary conditions. Since the spectral problem obtained after the integral transformation is not homogeneous, the considered problem is again divided into two problems. Under the minimum conditions at the initial data, by combining the deduction method and the contour integral method, the existence and uniqueness of the solution to the mixed problem is proved, where an explicit analytic representation for it is obtained.},
year = {2023}
}
TY - JOUR T1 - On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions AU - Hikmat I. Ahmadov Y1 - 2023/06/09 PY - 2023 N1 - https://doi.org/10.11648/j.ajam.20231103.11 DO - 10.11648/j.ajam.20231103.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 32 EP - 39 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20231103.11 AB - In this work, we study a one-dimensional mixed problem for an inhomogeneous parabolic equation with constant coefficients of general form, under non-local and non-self-conjugate boundary conditions. The considered mixed problem consist of two parts. The first problem is a mixed problem with a regular boundary condition, and the uniqueness of the solution is proved through the deduction operator. Then the existence of a solution to the mixed problem is shown, and an exact formula for the solution is found. A second mixed problem is the time delay in the boundary conditions. Since the spectral problem obtained after the integral transformation is not homogeneous, the considered problem is again divided into two problems. Under the minimum conditions at the initial data, by combining the deduction method and the contour integral method, the existence and uniqueness of the solution to the mixed problem is proved, where an explicit analytic representation for it is obtained. VL - 11 IS - 3 ER -
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