Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions
https://doi.org/10.23947/2587-8999-2024-8-1-43-54
Abstract
Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.
Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.
Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.
Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.
About the Author
Z. V. Beshtokova
Institute of Applied Mathematics and Automation, KBNС RAS
Russian Federation
Russian Federation
Zaryana V. Beshtokova, Junior Researcher, Department of Computational Methods
https://www.scopus.com/authid/detail.uri?authorId=57195928671
89a, Shortanova St., Nalchik, 360000
References
1. Vaynberg M.M. Integro-differential equations. Itogi nauki. ser. mat. anal. teor. ver. regulir. 1962, Moscow: VINITI. 1964;5–37. (in Russ.). URL: https://www.mathnet.ru/eng/intv82 (accessed: 16.11.2023).
2. Lavrent’yev M.M. Inverse problems and special operator equations of the first kind. Mezhdunarodnyy kongress matematikov. Doklady sovetskikh matematikov. Moscow: Nauka.1972:130–136. (in Russ.).
3. Vasil’yev V.V. On the solution of the Cauchy problem for a class of linear integro-differential equations. Izv. Vyssh. Uchebn. Zaved. Mat. 1961;4:8–24. (in Russ.). https://www.mathnet.ru/eng/ivm/y1961/i4/p8 (accessed: 17.11.2023).
4. Trubin V.G. Solution of a degenerate integro-differential equation. Differential and integral equations. Irkutsk: Irkut. un.-t.1978;5:94–101. (in Russ.).
5. Guezane-Lakoud A., Belakroum D. Time-discretization schema for an integrodifferential Sobolev type equation with integral conditions. Applied Mathematics and Computation. 2012:218(9):4695–4702. https://doi.org/10.1016/j.amc.2011.11.077
6. Luoa Z.D., Teng F. A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation. Applied Mathematics and Computation. 2018;329:374–383. https://doi.org/10.1186/s13661-019-1176-2
7. Beshtokov M.Kh. Numerical analysis of initial-boundary value problem for a Sobolev-type equation with a fractional-order time derivative. Computational Mathematics and Mathematical Physics. 2019;59(2):175–192. https://doi.org/10.1134/S0965542519020052
8. Grasselli M., Pata V. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems. Ser. A. 2006;15:1079–1088. https://doi.org/10.3934/dcds.2006.15.1079
9. Olmstead W.E., Davis S.H., Rosenblat S., Kath W.L. Bifurcation with memory. SIAM Journal on Applied Mathematics. 1986;46:171–188. https://doi.org/10.1137/0146013
10. Yong J, Zhang X. Heat equation with memory in anisotropic and non-homogeneous media. Acta Mathematica Sinica. English Ser. 2011;27:219–254. https://doi.org/10.1007/s10114-010-0077-1
11. Beshtokov M.KH., Vodakhova V.A. Nonlocal boundary value problems for a fractional-order convection-diffusion equation. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki. 2019;29(4):459–482. (in Russ.). https://doi.org/10.20537/vm190401
12. Douglas J., Rachford H.H. On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc. 1956;82(2):421–43. https://doi.org/10.1090/S0002-9947-1956-0084194-4
13. Peaceman D.W., Rасhfоrd H.H. The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics. 1955;3(1):28–41. https://doi.org/10.1137/0103003
14. Yanenko N.N. Method of fractional steps. Paris: Libraire Armand Colin. 1968. 165 p.
15. Samarskiy A.A. On an economical difference method for the solution of a multidimensional parabolic equation in an arbitrary region. USSR Computational Mathematics and Mathematical Physics. 1963;2(5):894–926. (in Russ.). https://doi.org/10.1016/0041-5553(63)90504-4
16. Marchuk G.I. Splitting methods. Moscow: Nauka; 1988. 263 p. (in Russ.).
17. D’yakonov E.G. Difference schemes with a ‘disintegrating’ operator for multidimensional problems. USSR Computational Mathematics and Mathematical Physics. 1963;2(4):581–607. (in Russ.). https://doi.org/10.1016/0041-5553(63)90531-7
18. Beshtokova Z.V., Shkhanukov-Lafishev M.Kh. Locally one-dimensional difference scheme for the third boundary value problem for a parabolic equation of the general form with a nonlocal source. Differential Equations. 2018; 54(7):891–880. (in Russ.). https://doi.org/10.1134/S0012266118070042
19. Beshtokova Z.V. Locally one-dimensional difference scheme for a nonlocal boundary value problem for a parabolic equation in a multidimensional domain. Differential Equations. 2020; 56(3):354–368. https://doi.org/10.1134/S0012266120030088
20. Beshtokova Z.V. Numerical method for solving an initial-boundary value problem for a multidi-mensional loaded parabolic equation of a general form with conditions of the third kind. Journal of Samara state technical university. Physical and mathematical science. 2022; 26(1):7–35. (In Russ.). https://doi.org/10.14498/vsgtu1908
21. Samarskii A.A. Theory of difference schemes. Moscow: Nauka; 1983. 616 p. (In Russ.).
22. Alikhanov A.A. On the stability and convergence of nonlocal difference schemes. Differential equations. 2010; 46(7):949–961. https://doi.org/10.1134/S0012266110070037
23. Samarskii A.A., Gulin A.V. Stability of difference schemes. Moscow: Nauka; 1973. 416 p. (In Russ.).
Review
For citations:
Beshtokova Z.V. Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions. Computational Mathematics and Information Technologies. 2024;8(1):43-54. https://doi.org/10.23947/2587-8999-2024-8-1-43-54
Views:
500
Email this article 