Finite-difference methods for solving loaded parabolic equations

V. M. Abdullayev1,2, К. Р. Айда-заде1,3
1Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
2Azerbaijan State University of Oil and Industry, Baku, Azerbaijan
3Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

Tóm tắt

Từ khóa


Tài liệu tham khảo

A. M. Nakhushev, Equations of Mathematical Biology (Vysshaya Shkola, Moscow, 1995) [in Russian].

V. A. Nakhusheva, Differential Equations for Mathematical Models of Nonlocal Processes (Nauka, Moscow, 2006) [in Russian].

A. M. Nakhushev, Loaded Equations and Applications (Nauka, Moscow, 2012) [in Russian].

A. M. Nakhushev and V. N. Borisov, “Boundary value problems for loaded parabolic equations and their applications to the prediction of groundwater level,” Differ. Uravn. 13 (1), 105–110 (1977).

V. M. Abdullaev and K. R. Aida-zade, “On the numerical solution of loaded systems of ordinary differential equations,” Comput. Math. Math. Phys. 44 (9), 1585–1595 (2004).

Yu. A. Anokhin, A. B. Gorstko, L. Yu. Dameshek, et al., Mathematic Models and Methods for Controlling Large- Scale Bodies of Water (Nauka, Novosibirsk, 1987) [in Russian].

A. I. Egorov, Fundamentals of Control Theory (Fizmatlit, Moscow, 2004) [in Russian].

K. R. Aida-zade and V. M. Abdullaev, “On an approach to designing control of the distributed-parameter processes,” Autom. Remote Control 73 (9), 1443–1455 (2012).

A. I. Kozhanov, “Nonlinear loaded equations and inverse problems,” Comput. Math. Math. Phys. 44 (4), 657–678 (2004).

M. Z. Khudalov, “Nonlocal boundary value problems for loaded parabolic equations,” Vladikavkaz. Mat. Zh. 4 (4), 59–64 (2002).

Kh. Zh. Dikinov, A. A. Kerefov, and A. M. Nakhushev, “On a boundary value problem for the loaded heat equation,” Differ. Uravn. 12 (1), 177–179 (1976).

A. A. Alikhanov, A. M. Berezgov, and M. Kh. Shkhanukov-Lafishev, “Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods,” Comput. Math. Math. Phys. 48 (9), 1581–1590 (2008).

A. A. Kerefov, M. Kh. Shkhanukov-Lafishev, and R. S. Kuliev, “Boundary value problems for the loaded heat equation with nonlocal Steklov type conditions,” International Seminar on Nonclassical Equations of Mathematical Physics Dedicated to the 60th Birth Anniversary of Professor Vladimir N. Vragov, Novosibirsk, Russian, October 3–5, 2005 (Novosibirsk, 2005).

A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).

S. K. Godunov and V. S. Ryaben’kii, Difference Schemes: An Introduction to the Underlying Theory (Fizmatgiz, Moscow, 1962; North-Holland, Amsterdam, 1987).

E. A. Bondarev and A. F. Voevodin, “A finite-difference method for solving initial-boundary value problems for loaded differential and integro-differential equations,” Differ. Equations 36 (11), 1711–1714 (2000).

E. A. Bondarev, A. F. Voevodin, V. V. Katyshev, and A. V. Petlina, “Numerical method for solving initial–boundary value problems with a nonlocal boundary condition,” Obratn. Zadachi Inf. Tekhnol. 1 (2), 31–40 (2003).

M. T. Dzhenaliev, “Optimal control of linear loaded parabolic equations,” Differ. Uravn. 25 (4), 641–651 (1989).

V. M. Abdullaev and K. R. Aida-zade, “Numerical solution of optimal control problems for loaded lumped parameter systems,” Comput. Math. Math. Phys. 46 (9), 1487–1502 (2006).

M. Kh. Shkhanukov-Lafishev, “Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions,” Comput. Math. Math. Phys. 49 (7), 1167–1174 (2009).

E. Rothe, “Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,” Math. Ann. 102 (1), 650–670 (1930).

A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

V. M. Abdullaev, “On the application of the method of lines to boundary value problems with nonlocal conditions for loaded parabolic equations,” Izv. Nats. Akad. Nauk Az. Ser. Fiz.-Tekh. Mat. Nauk 28 (3), 76–81 (2008).