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Numerical solution of nonlinear three-point boundary value problem on the positive half-line. (English) Zbl 1252.34021

Summary: We present an efficient numerical algorithm to solve the three-point boundary value problem on the half-line based on the reproducing kernel theorem. Considering the boundary conditions including a limit form, a new weighted reproducing kernel space is established to overcome the difficulty. By applying the reproducing property and the existence of an orthogonal basis in the weighted reproducing kernel space, the approximate solution is constructed by the orthogonal projection of the exact solution. Convergence is also discussed. We demonstrate the accuracy of the method by numerical experiments.

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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[1] Giovangigli, Nonadiabatic plane laminar flames and their singular limits, Siam Journal on Mathematical Analysis 21 pp 1305– (1990) · Zbl 0714.34042 · doi:10.1137/0521072
[2] Smooke, Solution of burner stabilized premixed laminar flames by boundary value methods, Journal of Computational Physics 48 pp 72– (1982) · Zbl 0492.65065 · doi:10.1016/0021-9991(82)90036-5
[3] Simon, Quenching of flame propagation with heat loss, Journal of Mathematical Chemistry 31 pp 313– (2002) · Zbl 1023.92049 · doi:10.1023/A:1020840222396
[4] Simon, Evans function analysis of the stability of non-adiabatic flames, Combustion Theory and Modelling 7 pp 545– (2003) · Zbl 1068.80531 · doi:10.1088/1364-7830/7/3/306
[5] Baxley, Existence and uniqueness of nonlinear boundary value problems on infinite intervals, Journal of Mathematical Analysis and Applications 147 pp 127– (1990) · Zbl 0719.34037 · doi:10.1016/0022-247X(90)90388-V
[6] Baily, The Mathematical Theory of Infectious Diseases (1975) · Zbl 0329.10020
[7] Britton, Reaction-Diffusion Equations and their Applications to Biology (1986) · Zbl 0602.92001
[8] O’Regan, Theory of Singular Boundary Value Problems (1994) · doi:10.1142/2352
[9] Agarwal, Infinite Interval Problems for Differential, Difference and Integral Equations (2001) · Zbl 0988.34002 · doi:10.1007/978-94-010-0718-4
[10] Murray, Mathematical Biology, Biomathematics Texts (1989)
[11] Przeradzki, Travelling waves for reaction-diffusion equations with time depending nonlinearities, Journal of Mathematical Analysis and Applications 281 pp 164– (2003) · Zbl 1032.35089 · doi:10.1016/S0022-247X(02)00632-7
[12] Djebali, A class of second order BVPs on infinite intervals, Electronic Journal of Qualitative Theory of Differential Equations 4 pp 1– (2006) · Zbl 1134.34018 · doi:10.14232/ejqtde.2006.1.4
[13] Djebali, Existence results for a class of BVPs on the positive half-line, Communications on Applied Nonlinear Analysis 14 pp 13– (2007) · Zbl 1129.34017
[14] Djebali, Multiple positive solutions for singular BVPs on the positive half-line, Computers Mathematics with Applications 55 pp 2940– (2008) · Zbl 1142.34316 · doi:10.1016/j.camwa.2007.11.023
[15] Yan, Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity, Journal of Computational and Applied Mathematics 197 pp 365– (2006) · Zbl 1116.34016 · doi:10.1016/j.cam.2005.11.010
[16] Lian, Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line, Journal of Mathematical Analysis and Applications 321 pp 781– (2006) · Zbl 1104.34020 · doi:10.1016/j.jmaa.2005.09.001
[17] Lian, Positive solutions for multi-point boundary value problem on the half-line, Journal of Mathematical Analysis and Applications 325 pp 1339– (2007) · Zbl 1110.34018 · doi:10.1016/j.jmaa.2006.02.075
[18] Lian, Solvability for second-order three-point boundary value problems on a half-line, Applied Mathematics Letters 19 pp 1000– (2006) · Zbl 1123.34307 · doi:10.1016/j.aml.2005.10.018
[19] Djebali, Multiple unbounded positive solutions for three-point BVPs with sign-changing nonlinearities on the positive half-Line, Acta Applicandae Mathematicae 109 pp 361– (2010) · Zbl 1195.34042 · doi:10.1007/s10440-008-9322-3
[20] Cui, Nonlinear Numerical Analysis in the Reproducing Kernel Space (2009) · Zbl 1165.65300
[21] Du, Constructive approximation of solution for fourth-order nonlinear boundary value problems, Mathematical Methods in the Applied Sciences 32 pp 723– (2009) · Zbl 1170.34015 · doi:10.1002/mma.1064
[22] Lin, A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space, Mathematical Methods in the Applied Sciences 34 pp 44– (2011) · Zbl 1206.65187 · doi:10.1002/mma.1327
[23] Lin, Numerical method for solving the nonlinear four-point boundary value problems, Communications in Nonlinear Science and Numerical Simulation 15 pp 3855– (2010) · Zbl 1222.65072 · doi:10.1016/j.cnsns.2010.02.013
[24] Lin, A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space, Applied Mathematics and Computation 218 pp 7362– (2012) · Zbl 1246.65122 · doi:10.1016/j.amc.2011.11.009
[25] Jiang, Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method, Communications in Nonlinear Science and Numerical Simulation 15 pp 1754– (2010) · Zbl 1222.65085 · doi:10.1016/j.cnsns.2009.07.022
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