Higher order orthogonal spline collocation method for periodic boundary value problems

Nitisha Pandey; Reena  Jain

doi:10.5269/bspm.78800

Nitisha Pandey VIT BHOPAL
Reena Jain

Resumo

This work thoroughly investigates various high-accuracy collocation methods for solving periodic
boundary value problems (PBVPs), highlighting their efficiency and rapid convergence. The study demon
strates that collocation techniques can achieve high-order accuracy while reducing the computational resources
required, making them a powerful alternative to traditional methods. Numerical experiments are conducted
to validate the effectiveness of our proposed approach and confirmed sixth-order accuracy in both, the solution
and its derivative. This finding is supported by error analysis and convergence rates for periodic boundary
conditions in second-order differential equations. During the comparison of our method to the finite difference
method, we found that our approach is superior. Specifically, increasing the number of grid points signifcantly reduces the error while maintaining a consistent order of convergence. Overall, the results under score the effectiveness of collocation methods in addressing PBVPs for differential equations and complex boundary
conditions.

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Referências

P. Benner, S. Chuiko, and A. Zuyev, A periodic boundary value problem with switchings under nonlinear perturbations, Boundary value problem, 2023(1), (2023), https://rdcu.be/euXsh.

X. Wang, and B. Zhu., On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses, Advances in Continuous and Discrete Models, 2022(1), (2022), https://rdcu.be/euXtn.

T. Xue, X. Fan, H. Cao, and L.Fu., A periodic boundary value problem of fractional differential equation involving p(t)-Laplacian operator, Mathematical Biosciences and Engineering, 20(3), 4421–4436, (2022), https://doi.org/10.3934/mbe.2023205.

W. Liu, L. Liu, and Y. Wu., Positive solutions of a singular boundary value problem for systems of second-order differential equations, Appl. Math. Comput., 208(2), 511-519, (2009), https://doi.org/10.1016/j.amc.2008.12.019.

Y. Zhou, and Y. Xu.,Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations, J Math. Anal. Appl., 320(2), 578-590, (2006), https://doi.org/10.1016/j.jmaa.2005.07.014.

B. Liu, L. Liu, and Y. Wu., Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem, Nonlinear Anal, Theory Methods Appl., 72, 3337-3345, (2010), https://doi.org/10.1016/j.na.2009.12.014.

Q. Yao., Positive solutions of nonlinear second-order periodic boundary value problems, Appl. Math. Lett., 20(5), 583-590, (2007), 10.1016/j.aml.2006.08.003.

M. Al-Smadi, O.A. Arqub, and Ah. E.Ajou., A numerical Iterative method for solving systems of first-order periodic boundary value problems, J. Appl. Math., 2014(1), 1-10, (2014), http://dx.doi.org/10.1155/2014/135465.

O.A. Arqub., Reproducing kernel algorithm for the analytical-numerical solutions of nonlinear systems of singular periodic boundary value problems, Math. Probl. Eng., 2015(1), 1-13, (2015), http://dx.doi.org/10.1155/2015/518406.

S. K. Bhal, P.Danumjaya, and G.Fairweather, The Crank–Nicolson orthogonal spline collocation method for onedimensional parabolic problems with interfaces, Journal of Computational and Applied Mathematics, 383, 113119, (2021), https://doi.org/10.1016/j.cam.2020.113119.

S. K. Bhala, P. Danumjaya, and G. Fairweather, High-order orthogonal spline collocation methods for twopoint boundary value problems with interfaces, Mathematics and Computers in Simulation, 174, 102-122, (2020), https://doi.org/10.1016/j.matcom.2020.03.001.

K. Wright., Chebyshev collocation methods for ordinary differential equations, Comput. J., 6(4), 358–365, (1964), https://doi.org/10.1093/comjnl/6.4.358.

M. F. Kaspshitskaya and A. Yu. Luchka, The collocation method, USSR Computational Mathematics and Mathematical Physics, 8(5), 19-39, (1968), https://doi.org/10.1016/0041-5553(68)90123-7.

N.Sharp and M.Trummer, A spectral collocation method for system of singularly perturbed boundary value problems, Procedia Computer Science, 108, 725–734, (2017), https://doi.org/10.1016/j.procs.2017.05.012.