On the numerical Picard iterations with collocations for the initial value problem

Ernest Scheiber

doi:10.33993/jnaat481-1146

Authors

Ernest Scheiber retired, Romania

DOI:

https://doi.org/10.33993/jnaat481-1146

Keywords:

Picard iterations, initial value problem, collocation

Abstract views: 203

Abstract

Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term "numerical" emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed a convergence result is given. Finally some numerical experiments are reported.

Downloads

Download data is not yet available.

References

Bai X., Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems.PhD Dissertation, 2010, Texas A&M University.

Bobkov V.V., Faleichik B.V., Mandrik P.A., Repnikov V.I., Solving Stiff Problems Using Generalized Picard Iterations. AIP ConferenceProceeding 1168, 65 (2009), https://doi.org/10.1063/1.3241550 DOI: https://doi.org/10.1063/1.3241550

Causley F.M., Seal C.D., On the Convergence of Spectral Deferred Correction Methods. arXiv:1706.06245v1 [math.NA], 2017.

Faleichik B.V., Analytic Iterativ Processes and Numerical Algorithms for Stiff Problems. Computational Methods in Applied Mathematics, 8 (2008), no. 2, 116-129. https://doi.org/10.2478/cmam-2008-0008 DOI: https://doi.org/10.2478/cmam-2008-0008

Fukushima T., Picard Iteration Method, Chebyshev Polynomial Approximation, and Global Numerical Integration of Dynamical Motions. The Astronomical J.,113 (1997), no. 5, 1909-1914. https://doi.org/10.1086/118404 DOI: https://doi.org/10.1086/118404

Fukushima T., Vector Integration of Dynamical Motions by the Picard-Chebyshev Method. The Astronomical J.,113 (1997), no. 6, 2325-2328. https://doi.org/10.1086/118443 DOI: https://doi.org/10.1086/118443

Hairer E., Wanner G., Nørsett S., Solving Ordinary Differential Equations I Non-stiff Problems.Second Ed, Springer, Berlin, 1993.

Scheiber E., A multistep method to solve the initial value problem. The 4th Romanian-German Seminar on Approximation Theory and its Applications. Brasov, 2000 (ed. H.Gonska, D. Kacso, L. Beutel) Gerhard Mercator Universitat, Duisburg, 124-132.

Shampine L.F., Gordon M.K., Computer solution of ordinary differential equation. The initial value problem. W.H. Freeman and Company, San Francisco, 1975.

Stoer J., Bulirsch R., Introduction to Numerical Analysis. Springer-Verlag, New York, 1993. DOI: https://doi.org/10.1007/978-1-4757-2272-7

Trefethen N. L., Approximation Theory and Approximation Practice. SIAM, 2012.

***, http://mathfaculty.fullerton.edu/mathews/n2003/PicardIterationMod. html, 2017.