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A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. (English) Zbl 1215.34014

The authors study a numerical algorithm for solving fourth-order multi-point boundary value problems

u (4) + i=0 3 a i (x)u (i) (x)=f(x),0x1,u(ξ 1 )=b 1 ,u ' (ξ 1 )=b 2 ,u '' (ξ 1 )=b 3 ,u(ξ 2 )-u(ξ 3 )=b 4 ,

where a i C[a,b], 0<ξ 1 <ξ 2 <ξ 3 <1 and b i (i=1,2,3,4) are real numbers. They present an algorithm for solving the above problems based on the reproducing kernel method. The characteristic feature of this method is that a global approximation can be established on the whole solution domain and the convergence is uniform.

MSC:
34A45Theoretical approximation of solutions of ODE
34B05Linear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
47B32Operators in reproducing-kernel Hilbert spaces
References:
[1]Henderson, J.; Kunkel, C. J.: Uniqueness of solution of linear nonlocal boundary value problems, Appl. math. Lett. 21, 1053-1056 (2008) · Zbl 1158.34309 · doi:10.1016/j.aml.2006.06.024
[2]Moshiinsky, M.: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. soc. Mat. mexicana 7, 1-25 (1950)
[3]Timoshenko, S.: Theory of elastic stability, (1961)
[4]Eloe, P. W.; Henderson, Johnny: Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations, J. math. Anal. appl. 331, 240-247 (2007)
[5]Graef, John R.; Webb, J. R. L.: Third order boundary value problems with nonlocal boundary conditions, Nonlinear anal. 71, 1542-1551 (2009) · Zbl 1189.34034 · doi:10.1016/j.na.2008.12.047
[6]Liu, B.: Solvability of multi-point boundary value problem at resonance (IV), Appl. math. Comput. 143, 275-299 (2003) · Zbl 1071.34014 · doi:10.1016/S0096-3003(02)00361-2
[7]Feng, W.; Webb, J. R. L.: Solvability of m-point boundary value problems with nonlinear growth, J. math. Anal. appl. 212, 467-480 (1997) · Zbl 0883.34020 · doi:10.1006/jmaa.1997.5520
[8]Geng, F. Z.: Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. math. Comput. 215, 2095-2102 (2009) · Zbl 1178.65085 · doi:10.1016/j.amc.2009.08.002
[9]Lin, Y. Z.; Lin, J. N.: Numerical algorithm about a class of linear nonlocal boundary value problems, Appl. math. Lett. 23, 997-1002 (2010) · Zbl 1201.65130 · doi:10.1016/j.aml.2010.04.025
[10]Cui, M. G.; Lin, Y. Z.: Nonlinear numerical analysis in reproducing kernel space, (2009)
[11]Berlinet, A.; Thomas-Agnan, Christine: Reproducing kernel Hilbert space in probability and statistics, (2004)
[12]Cui, M. G.; Geng, F. Z.: Solving singular two-point boundary value problem in reproducing kernel space, J. comput. Appl. math. 205, 6-15 (2007) · Zbl 1149.65057 · doi:10.1016/j.cam.2006.04.037
[13]Geng, F. Z.: A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Appl. math. Comput. 213, 163-169 (2009) · Zbl 1166.65358 · doi:10.1016/j.amc.2009.02.053
[14]Geng, F. Z.; Cui, M. G.: Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl. math. Comput. 192, 389-398 (2007) · Zbl 1193.34017 · doi:10.1016/j.amc.2007.03.016
[15]Geng, F. Z.; Cui, M. G.: Solving a nonlinear system of second order boundary value problems, J. math. Anal. appl. 327, 1167-1181 (2007) · Zbl 1113.34009 · doi:10.1016/j.jmaa.2006.05.011