Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions. (English) Zbl 1182.35143

Zhou, Shiping; Cui, Minggen

Int. J. Comput. Math. 86, No. 12, 2248-2258 (2009).

Summary: This paper develops an iterative algorithm for the solution to a variable-coefficient semilinear heat equation with nonlocal boundary conditions in the reproducing space. It is proved that the approximate sequence \(u_n(x, t)\) converges to the exact solution \(u(x, t)\). Moreover, the partial derivatives of \(u_n(x, t)\) are also convergent to the partial derivatives of \(u\)(x, t). And the approximate sequence \(u_n(x, t)\) is the best approximation under a complete normal orthogonal system.

MSC:

35K58	Semilinear parabolic equations
35A35	Theoretical approximation in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

reproducing kernel space; iterative algorithm; best approximation

References:

[1]	Aronszajn N., Trans. AMS 168 pp 1– (1950)
[2]	Bouziani A., C. R. Acad. Sci. Paris, Ser. I Math 321 pp 1177– (1995)
[3]	DOI: 10.1016/j.na.2007.07.008 · Zbl 1155.35053
[4]	DOI: 10.1137/0724036 · Zbl 0677.65108
[5]	Cui M. G., Nonlinear Numercial Analysis in the Reproducing Kernel Space (2008)
[6]	DOI: 10.1016/j.na.2006.06.012 · Zbl 1113.35102
[7]	DOI: 10.1016/S0096-3003(02)00954-2 · Zbl 1038.65088
[8]	DOI: 10.1016/j.apnum.2004.02.002 · Zbl 1063.65079
[9]	DOI: 10.1002/num.20019 · Zbl 1059.65072
[10]	DOI: 10.1002/num.20071 · Zbl 1084.65099
[11]	Dehghan M., Int. J. Nonlinear Sci. Numer. Simul. 7 pp 447– (2006) · Zbl 06942228
[12]	DOI: 10.1016/j.chaos.2005.11.010 · Zbl 1139.35352
[13]	DOI: 10.1002/num.20299 · Zbl 1142.65080
[14]	DOI: 10.1155/JAMSA.2005.13 · Zbl 1077.74019
[15]	DOI: 10.1016/j.na.2005.12.005 · Zbl 1105.35044
[16]	DOI: 10.1016/j.apm.2008.03.006 · Zbl 1168.65403