Решение двумерной граничной задачи теплопроводности в вырождающейся области
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.06Ключевые слова:
нецилиндрическая область, конус, краевая задача теплопроводности, сингулярное интегральное уравнение Вольтерра, регуляризацияАннотация
В работе рассматривается краевая задача теплопроводности вне конуса, то есть в области вырождающейся в точку в начальный момент времени. При этом граничное условие содержит производную по временной переменной. Особенность рассматриваемой задачи состоит именно в наличии подвижной границы и вырождения области решения в начальный момент времени в точку. К этому типу задач в общем случае не применимы известные классические методы. Методом тепловых потенциалов подобные краевые задачи теплопроводности редуцируются к решению сингулярных интегральных уравнений типа Вольтерра второго рода. Под сингулярным уравнением типа Вольтерра подразумевается уравнение, ядро которого обладает следующим свойством: интеграл от ядра уравнения при стремлении верхнего предела к нижнему не стремится к нулю. Доказана теорема о разрешимости рассматриваемой краевой задачи в весовых пространствах существенно ограниченных функций. Исследованы вопросы разрешимости сингулярного интегрального уравнения Вольтерра второго рода, к которому редуцирована исходная задача. Найдено ненулевое решение этого сингулярного интегрального уравнения.
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[3] Kheloufi A. "Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of R3" , Applied mathematics and computation, 220 (2013): 756-769.
[4] Cherfaoui S., Kessab A., Kheloufi A. "Well-posedness and regularity results for a 2m-th order parabolic equation in symmetric conical domains of RN+1" , Mathematical methods in the applied sciences, 40:5 (2017): 6035-6047.
[5] Kheloufi A. "On a Fourth Order Parabolic Equation in a Nonregular Domain of R3" , Mediterranean journal of mathematics, 12 (2014): 803-820.
[6] Kheloufi A., Sadallah B.-K. "Study of the heat equation in a symmetric conical type domain of RN+1" , Mathematical methods in the applied sciences, 37 (2014): 1807-1818.
[7] Chapko R., Johansson B.T., Vavrychuk V. "Numerical solution of parabolic Cauchy problems in planar corner domains", Mathematics and computers in simulation, 101 (2014): 1-12.
[8] Wang Y., Huang J., Wen X. "Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order" , Applied numerical mathematics, 163 (2021): 77-95.
[9] Dehbozorgi R., Nedaiasl K. "Numerical solution of nonlinear weakly singular Volterra integral equations of the first kind: An hp-version collocation approach" , Applied numerical mathematics, 161 (2021): 111-135.
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[12] Kharin S.N., Nouri H., Miedzinski B., Wisniewski G. "Transient phenomena of arc to glow discharge transformation at contact opening" , Proc. of 21st Int. Conf. on Electric contacts, Zurich, Switzerland, (2002): 425-431.
[13] Kharin S. "Mathematical model of electrical contact bouncing" , AIP Conference Proceedings, (2015): 1676.
[14] Sarsengeldin M.M., Kharin S.N., Rayev Z., Khairullin Y. "Mathematical model of heat transfer in opening electrical contacts with tunnel effect" , Filomat, 32 (2018): 1003-1008.
[15] Kavokin A.A., Kulakhmetova A.T., Shpadi Y.R. "Application of Thermal Potentials to the Solution of the Problem of Heat Conduction in a Region Degenerates at the Initial Moment" , Filomat, 132 (2018): 825-836.
[16] Amangalieva M.M., Akhmanova D.M., Dzhenaliev M.T., Ramazanov M.I. "Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity" , Differential Equations, 47:2 (2011): 231-243.
[17] Jenaliyev M.T., Amangaliyeva M.M., Kosmakova M.T., Ramazanov M.I. "On a Volterra equation of the second kind with incompressible kernel" , Advances in Difference Equations, 2015 (2015): article number 71.
[18] Amangalieva M.M., Dzhenaliev M.T., Kosmakova M.T., Ramazanov M.I. "On one homogeneous problem for the heat equation in an infinite angular domain" , Siberian Mathematical Journal, 56:6 (2015): 982-995.
[19] Jenaliyev M.T., Ramazanov M.I. "On a Homogeneous Parabolic Problem in an Infinite Corner Domain" , Filomat, 32:3 (2018): 965-974.
[20] Baderko E.A. "Parabolic problems and boundary integral equations" , Mathematical methods in the applied sciences, 20 (1997): 449-459.
[21] Polianin A.D., Manzhirov A.V. Handbook of integral equations (Boca Raton, CRC Press, 2008).
[22] Gakhov F.D., Chersky Yu.I. Convolution type equations [Uravneniya tipa svyortki] (Moscow: Nauka, 1978).
[23] Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Volume 2. Special Functions (Moscow: Fizmatlit, 2003).
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Опубликован
2021-10-09
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Раздел
Математика
