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)
Approximate solutions to three-point boundary value problems with two-space integral condition for parabolic equations. (English) Zbl 1247.65137
Summary: We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.
MSC:
65M99Numerical methods for IVP of PDE
35K20Second order parabolic equations, initial boundary value problems
35C10Series solutions of PDE
WorldCat.org
Full Text: DOI
References:
[1] Y. S. Choi and K.-Y. Chan, “A parabolic equation with nonlocal boundary conditions arising from electrochemistry,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 4, pp. 317-331, 1992. · Zbl 0757.35031 · doi:10.1016/0362-546X(92)90148-8
[2] R. E. Ewing and T. Lin, “A class of parameter estimation techniques for fluid flow in porous media,” Advances in Water Resources, vol. 14, no. 2, pp. 89-97, 1991. · doi:10.1016/0309-1708(91)90055-S
[3] P. Shi, “Weak solution to an evolution problem with a nonlocal constraint,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 46-58, 1993. · Zbl 0810.35033 · doi:10.1137/0524004
[4] A. V. Kartynnik, “A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations,” Differentsial’nye Uravneniya, vol. 26, no. 9, pp. 1568-1575, 1990. · Zbl 0712.35041
[5] A. A. Samarskiĭ, “Some problems of the theory of differential equations,” Differentsial’nye Uravneniya, vol. 16, no. 11, pp. 1925-1935, 1980.
[6] M. Dehghan, “Numerical solution of a parabolic equation with non-local boundary specifications,” Applied Mathematics and Computation, vol. 145, no. 1, pp. 185-194, 2003. · Zbl 1032.65104 · doi:10.1016/S0096-3003(02)00479-4
[7] M. Dehghan, “Efficient techniques for the second-order parabolic equation subject to nonlocal specifications,” Applied Numerical Mathematics, vol. 52, no. 1, pp. 39-62, 2005. · Zbl 1063.65079 · doi:10.1016/j.apnum.2004.02.002
[8] M. Dehghan, “Numerical techniques for a parabolic equation subject to an overspecified boundary condition,” Applied Mathematics and Computation, vol. 132, no. 2-3, pp. 299-313, 2002. · Zbl 1024.65088 · doi:10.1016/S0096-3003(01)00194-1
[9] M. Denche and A. L. Marhoune, “A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 13, no. 6, pp. 85-89, 2000. · Zbl 0956.35072 · doi:10.1016/S0893-9659(00)00060-4
[10] M. Tatari and M. Dehghan, “On the solution of the non-local parabolic partial differential equations via radial basis functions,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1729-1738, 2009. · Zbl 1168.65403 · doi:10.1016/j.apm.2008.03.006
[11] Y. Zhou, M. Cui, and Y. Lin, “Numerical algorithm for parabolic problems with non-classical conditions,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 770-780, 2009. · Zbl 1190.65136 · doi:10.1016/j.cam.2009.01.012
[12] A. L. Marhoune, “A three-point boundary value problem with an integral two-space-variables condition for parabolic equations,” Computers & Mathematics with Applications, vol. 53, no. 6, pp. 940-947, 2007. · Zbl 1124.35319 · doi:10.1016/j.camwa.2006.04.031
[13] F. Geng and M. Cui, “Solving singular nonlinear two-point boundary value problems in the reproducing kernel space,” Journal of the Korean Mathematical Society, vol. 45, no. 3, pp. 631-644, 2008. · Zbl 1154.34012 · doi:10.4134/JKMS.2008.45.3.631
[14] Y. Lin, J. Niu, and M. Cui, “A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7362-7368, 2012. · Zbl 1246.65122 · doi:10.1016/j.amc.2011.11.009
[15] W. Jiang, M. Cui, and Y. Lin, “Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1754-1758, 2010. · Zbl 1222.65085 · doi:10.1016/j.cnsns.2009.07.022
[16] Y. Lin and J. Lin, “Numerical method for solving the nonlinear four-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3855-3864, 2010. · Zbl 1222.65072 · doi:10.1016/j.cnsns.2010.02.013
[17] M. G. Cui and Y. Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Hilbert Space, Nova, New York, NY, USA, 2009. · Zbl 1165.65300