On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

Автор: Yuldashev Tursun K.

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 2 т.24, 2022 года.

Бесплатный доступ

In this article the problems of the unique classical solvability and the construction of the solution of a nonlinear boundary value problem for a fifth order partial integro-differential equations with degenerate kernel are studied. Dirichlet boundary conditions are specified with respect to the spatial variable. So, the Fourier series method, based on the separation of variables is used. A countable system of the second order ordinary integro-differential equations with degenerate kernel is obtained. The method of degenerate kernel is applied to this countable system of ordinary integro-differential equations. A system of countable systems of algebraic equations is derived. Then the countable system of nonlinear Fredholm integral equations is obtained. Iteration process of solving this integral equation is constructed. Sufficient coefficient conditions of the unique solvability of the countable system of nonlinear integral equations are established for the regular values of parameter. In proof of unique solvability of the obtained countable system of nonlinear integral equations the method of successive approximations in combination with the contraction mapping method is used. In the proof of the convergence of Fourier series the Cauchy-Schwarz and Bessel inequalities are applied. The smoothness of solution of the boundary value problem is also proved.

Еще

Nonlinear boundary value problem, integro-differential equation, degenerate kernel, fourier series, classical solvability

Короткий адрес: https://sciup.org/143178757

IDR: 143178757   |   DOI: 10.46698/h5012-2008-4560-g

Список литературы On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

  • Costin, O. and Tanveer, S. Existence and Uniqueness for a Class of Nonlinear Higher-Order Partial Differential Equations in the Complex Plane, Communications on Pure and Applied Mathematics, 2000, vol. 53, no. 9, pp. 1067–1091. DOI: 10.1002/1097-0312(200009)53:9-1092::AID-CPA2>3.0.CO;2-Z.
  • Galaktionov, V. A., Mitidieri, E. and Pohozaev, S. Global Sign-Changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range, Advanced Nonlinear Studies, 2012, vol. 12, no. 3, pp. 569–596. DOI: 10.1515/ans-2012-0308.
  • Hwang, S. Kinetic Decomposition for Singularly Perturbed Higher Order Partial Differential Equations, Journal of Differential Equations, 2004, vol. 200, no. 2, pp. 191–205. DOI: 10.1016/j.jde.2003.12.001.
  • Karimov, Sh. T. The Cauchy Problem for the Degenerated Partial Differential Equation of the High Even Order, Sibir. Electr. Matem. Izvestia, 2018, vol. 15, pp. 853–862. DOI: 10.17377/semi.2018.15.073.
  • Littman, W. Decay at Infinity of Solutions to Higher Order Partial Differential Equations: Removal of the Curvature Assumption, Israel Journal of Math., 1970, vol. 8, pp. 403–407. DOI: 10.1007/BF02798687.
  • Sabitov, K. B. The Dirichlet Problem for Higher-Order Partial Differential Equations, Mathematical Notes, 2015, vol. 97, no. 2, pp. 255–267. DOI: 10.1134/S0001434615010277.
  • Yan, J. and Shu, C.-W. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives, Journal of Scientific Computing, 2002, vol. 17, pp. 27–47. DOI: 10.1023/A:1015132126817.
  • Yuldashev, T. K. and Kadirkulov, B. J. Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration, Fractal Fractions, 2021, vol. 5, no. 2, ID 58, 13 pp. DOI: 10.3390/fractalfract5020058.
  • Yuldashev, T. K. and Shabadikov, K. Kh.Mixed Problem for a Higher-Order Nonlinear Pseudoparabolic Equation, Journal of Mathematical Sciences, 2021, vol. 254, no. 6, pp. 776–787. DOI: 10.1007/s10958-021-05339-w.
  • Yuldashev, T. K. and Shabadikov, K. Kh. Initial-Value Problem for a Higher-Order Quasilinear Partial Differential Equation, Journal of Mathematical Sciences, 2021, vol. 254, no. 6, pp. 811–822. DOI: 10.1007/s10958-021-05343-0.
  • Zhao, H., Zhu, C. and Yu, Z. Existence and Convergence of Solutions to a Singularly Perturbed Higher Order Partial Differential Equation, Nonlinear Anal., 1995, vol. 24, no. 10, pp. 1435–1455.
  • Benney, D. J. and Luke, J. C. Interactions of Permanent Waves of Finite Amplitude, Journal Math. Physics, 1964, vol. 43, pp. 309–313.
  • Cavalcanti, M. M., Domingos Cavalcanti, V. N. and Ferreira, J. Existence and Uniform Decay for a Nonlinear Viscoelastic Equation with Strong Damping, Math. Methods in the Applied Sciences, 2001, vol. 24, pp. 1043–1053. DOI: 10.1002/mma.250.
  • Whitham, G. B. Linear and Nonlinear Waves, New-York–London–Sydney–Toronto, A Willey-Interscience Publication, 1974.
  • Gordeziani, D. G. and Avilishbili, G. A. Solving the Nonlocal Problems for One-Dimensional Medium Oscillation, Matematicheskoe modelirovanie, 2000, vol. 12, no. 1, pp. 94–103 (in Russian).
  • Abdullaev, O. K. Gellerstedt Type Problem with Integral Gluing Condition for a Mixed Type Equation with Nonlinear Loaded Term, Lobachevskii Journal of Math., 2021, vol. 42, no. 3, pp. 479–489. DOI: 10.1134/S1995080221030021.
  • Assanova, A. T. A Two-Point Boundary Value Problem for a Fourth Order Partial Integro- Differential Equation. Lobachevskii Journal of Math., 2021, vol. 42, no. 3, pp. 526–535. DOI: 10.1134/S1995080221030082.
  • Assanova, A. T., Imanchiyev, A. E. and Kadirbayeva, Zh. M. A Nonlocal Problem for Loaded Partial Differential Equations of Fourth Order, Bulletin of the Karaganda University. Mathematics, 2020, vol. 97, no. 1, pp. 6–16.
  • Assanova, A. T., Sabalakhova, A. P. and Toleukhanova, Z. M. On the Unique Solvability of a Family of Boundary Value Problems for Integro-Differential Equations of Mixed Type, Lobachevskii Journal of Math., 2021, vol. 42, no. 6, pp. 1228–1238. DOI: 10.1134/S1995080221060044.
  • Beshtokov, M. Kh., Beshtokova, Z. V. and Khudalov, M. Z. A Finite Difference Method for Solving a Nonlocal Boundary Value Problem for a Loaded Fractional Order Heat Equation, Vladikavkaz. Math. J., 2020, vol. 22, no. 4, pp. 45–57 (in Russian). DOI: 10.46698/p2286-5792-9411-x.
  • Dzhamalov, S. Z., Ashurov, R. R. and Ruziev, U. S. On a Seminonlocal Boundary Value Problem for a Multidimensional Loaded Mixed Type Equation of the Second Kind, Lobachevskii Journal of Math., 2021, vol. 42, no. 3, pp. 536–543. DOI: 10.1134/S1995080221030094.
  • Heydarzade, N. A. On one Nonlocal Inverse Boundary Problem for the Second-Order Elliptic Equation, Transections National Acad. Sciences of Azerbaijan. Mathematics, 2020, vol. 40, no. 4, p. 97–109.
  • Kabanikhin, S. I. and Shishlenin, M. A. Recovery of the Time-Dependent Diffusion Coefficient by Known Non-Local Data, Numerical Anal. Applications, 2018, vol. 11, no. 1, pp. 38–44. DOI: 10.1134/S1995423918010056.
  • Khalilov, Q. S. A Nonlocal Problem for a Third Order Parabolic-Hyperbolic Equation with a Spectral Parameter, Lobachevskii Journal of Math., 2021, vol. 42, no. 6, pp. 1274–1285. DOI: 10.1134/S1995080221060123.
  • Kostin, A. B. The Inverse Problem of Recovering the Source in a Parabolic Equation Under a Condition of Nonlocal Observation, Sbornik. Mathematics, 2013, vol. 204, no. 10, pp. 1391–1434. DOI: 10.1070/SM2013v204n10ABEH004344.
  • Zaitseva, N. V. Nonlocal Boundary Value Problem with an Integral Condition for a Mixed Type Equation with a Singular Coefficient, Differential Equations, 2021, vol. 57, no. 2, pp. 210–220. DOI: 10.1134/S0012266121020105.
  • Yuldashev, T. K. Solvability of a Boundary Value Problem for a Differential Equation of the Boussinesq Type, Differential Equations, 2018, vol. 54, no. 10, pp. 1384–1393. DOI: 10.1134/S0012266118120108.
  • Yuldashev, T. K. Determination of the Coefficient in a Nonlocal Problem for an Integro-Differential Equation of Boussinesq Type with a Degenerate Kernel, Vladikavkaz. Math. J., 2019, vol. 21, no. 2, pp. 67–84 (in Russian). DOI: 10.23671/VNC.2019.2.32118.
  • Yuldashev, T. K. Nonlocal Inverse Problem for a Pseudohyperbolic-Pseudoelliptic Type Integro-Differential Equations, Axioms, 2020, vol. 9, no. 2, ID 45, 21 pp. DOI: 10.3390/axioms9020045.
  • Yuldashev, T. K. and Kadirkulov, B. J. Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation with Hilfer Fractional Operator, Ural Math. Journal, 2020, vol. 6, no. 1, pp. 153-167. DOI: 10.15826/umj.2020.1.013.
  • Yuldashev, T. K. Nonlocal Mixed-Value Problem for a Boussinesq-Type Integro-Differential Equation with Degenerate Kernel, Ukrainian Math. Journal, 2016, vol. 68, no. 8, pp. 1278–1296. DOI: 10.1007/s11253-017-1293-y.
  • Yuldashev, T. K. Mixed Problem for Pseudoparabolic Integro-Differential Equation with Degenerate Kernel, Differential Equations, 2017, vol. 53, no. 1, pp. 99–108. DOI: 10.1134/S0012266117010098.
Еще
Статья научная
QR-код для установки приложения SciUp Наведите камеру и скачайте бесплатное приложение SciUp