On the numerical solution of the heat conduction equations subject to nonlocal conditions

• Published in: Applied numerical mathematics Vol. 59; no. 10; pp. 2507 - 2514
• Main Authors: Martín-Vaquero, J., Vigo-Aguiar, J.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.10.2009; Elsevier
• Subjects: Convergence; Exact sciences and technology; Implicit techniques; Mathematical analysis; Mathematics; Nonclassic boundary value problems; Numerical analysis; Numerical analysis. Scientific computation; Numerical examples; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Numerical examples; Nonclassic boundary value problems; Implicit techniques; Convergence; Second order equation; Heat equation; Boundary condition; Algorithm; Partial differential equation; Parabolic equation; Numerical analysis; Boundary value problem; Applied mathematics; Numerical solution
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2009.05.007

Many physical phenomena are modelled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. Many different papers studied the second-order parabolic equation, particularly the heat equation subject to the specifications of mass. In this paper, we provide a whole family of new algorithms that improve the CPU time and accuracy of Crandall's formula shown in [J. Martin-Vaquero, J. Vigo-Aguiar, A note on efficient techniques for the second-order parabolic equation subject to non-local conditions, Appl. Numer. Math. 59 (6) (2009) 1258–1264] (and this algorithm improved the results obtained with BTCS, FTCS or Dufort–Frankel three-level techniques previously used in other works, see [M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math. 52 (2005) 39–62]) with this kind of problems. Other methods got second or fourth order only when k = s h 2 , while the new codes got nth order for k = h ; therefore, the new schemes require a smaller storage and CPU time employed than other algorithms. We will study the convergence of the new algorithms and finally we will compare the efficiency of the new methods with some well-known numerical examples.