# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. (English) Zbl 1167.65422
Summary: Many physical phenomena are modelled by non-classical parabolic boundary value problems with non-local boundary conditions. In [M. Dehghan, Appl. Numer. Math. 52, No. 1, 39–62 (2005; Zbl 1063.65079)], several methods were compared to approach the numerical solution of the one-dimensional heat equation subject to specifications of mass. One of them was the (3,3) Crandall formula. The scheme displayed in Eq. (64) in that paper is of order $O\left({h}^{2}\right)$, not of order $O\left({h}^{4}\right)$ as proposed by that author. However, it is possible with several changes to derive a Crandall algorithm of order $O\left({h}^{4}\right)$. Here, we compare the efficiency of the new method with the previous results in the same tests, and we reach errors ${10}^{3}$ to ${10}^{5}$ times smaller with the new scheme.

##### MSC:
 65M06 Finite difference methods (IVP of PDE)
##### References:
 [1] Bouziani, A.: On a class of parabolic equations with a nonlocal boundary condition, Acad. roy. Belg. bull. Cl. sci. 10, 61-77 (1999) · Zbl 1194.35200 [2] Bouziani, A.: Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations, Hiroshima math. J. 27, 373-390 (1997) · Zbl 0893.35061 [3] Bouziani, A.; Merazga, N.; Benamira, S.: Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlinear anal. 69, 1515-1524 (2008) · Zbl 1155.35053 · doi:doi:10.1016/j.na.2007.07.008 [4] Carlson, D. E.: Linear thermoelasticity, Handbuch der physik /2, 297-345 (1972) [5] Copetti, M. I. M.: A one-dimensional thermoelastic problem with unilateral constraint, Math. comput. Simulation 59, 361-376 (2002) · Zbl 1011.74013 · doi:doi:10.1016/S0378-4754(01)00419-0 [6] Choi, Y. S.; Chan, K.: A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear anal. 18, 317-331 (1992) · Zbl 0757.35031 · doi:doi:10.1016/0362-546X(92)90148-8 [7] Cushman, J. H.: On diffusion in fractal porous media, Water resour. Res. 27, 643-644 (1991) [8] Dagan, G.: The significance of heterogeneity of evolving scales to transport in porous formations, Water resour. Res. 30, 3327-3336 (1994) [9] Day, W. A.: Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Quart. appl. Math. 40, 319-330 (1982) [10] Day, W. A.: A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories, Quart. appl. Math. 41, 468-475 (1983) · Zbl 0514.35038 [11] Day, W. A.: Heat conduction within linear thermoelasticity, (1985) [12] Day, W. A.: Parabolic equations and thermodynamics, Quart. appl. Math 50, 523-533 (1992) [13] Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. numer. Math. 52, 39-62 (2005) · Zbl 1063.65079 · doi:doi:10.1016/j.apnum.2004.02.002 [14] Farjas, J.; Rosell, J. I.; Herrero, R.; Pons, R.; Pi, F.; Orriols, G.: Equivalent low-order model for a nonlinear diffusion equation, Physica D: Nonlinear phenomena 95, 107-127 (1996) · Zbl 0888.35005 · doi:doi:10.1016/0167-2789(96)00068-1 [15] Lin, Y.; Xu, S.; Yin, H. M.: Finite difference approximation for a class of nonlocal parabolic equations, Int. J. Math. math. Sci. 20, 147-164 (1997) · Zbl 0880.65070 · doi:doi:10.1155/S0161171297000215 [16] Mesloub, S.: On a singular two dimensional nonlinear evolution equation with nonlocal conditions, Nonlinear anal. 68, 2594-2607 (2008) · Zbl 1139.35365 · doi:doi:10.1016/j.na.2007.02.006 [17] Pao, C. V.: Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. math. Anal. appl. 195, 702-718 (1995) · Zbl 0851.35063 · doi:doi:10.1006/jmaa.1995.1384 [18] Pao, C. V.: Numerical methods for nonlinear integro-parabolic equations of Fredholm type, Comput. math. Appl. 41, 857-877 (2001) · Zbl 0986.65146 · doi:doi:10.1016/S0898-1221(00)00325-4 [19] Rosell, J. I.; Farjas, J.; Herrero, R.; Pi, F.; Orriols, G.: Homoclinic phenomena in opto-thermal bistability with localized absorption, Physica D: Nonlinear phenomena 85, 509-547 (1995) · Zbl 0890.58056 · doi:doi:10.1016/0167-2789(95)00077-H [20] Samarskii, A. A.: Some problems in differential equations theory, Differential equations 16, 1221-1228 (1980) [21] Wang, S.; Lin, Y.: A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation, Inverse problems 5, 631-640 (1989) · Zbl 0683.65106 · doi:doi:10.1088/0266-5611/5/4/013 [22] Wang, S.; Lin, Y.: A numerical method for the diffusion equation with nonlocal boundary specifications, Int. J. Engrg. sci. 28, 543-546 (1990) · Zbl 0718.76096 · doi:doi:10.1016/0020-7225(90)90056-O