Numerical Solution of 2-point Boundary Value Problem by Subdivision Scheme
DOI:
https://doi.org/10.26713/cma.v10i1.980Keywords:
Subdivision scheme, Boundary value problem, Convergence, StabilityAbstract
A numerical approximating collocation algorithm is formulated that is based on binary 6-point approximating subdivision scheme to generate the curves. It is examined that the scheme is generating more smooth continuous solutions of the problems. Numerical example is given to illustrate the algorithm with its graphically representation.Downloads
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C. Beccari, G. Casciola and L. Romani, Interpolatory subdivision curves with local shape control, WSCG'2006, January 30 – February 3, 2006, Plzen, Czech Republic, WSCG2006 Full Papers Proceedings, Václav Skala-UNION Agency, https://dspace5.zcu.cz/bitstream/11025/6634/1/Beccari.pdf (2006).
C. Beccari, G. Casciola and L. Romani, A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics, Computer Aided Geometric Design 24(1) (2007), 1 – 9.
A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Memoirs of the American Mathematical Society 93(453) (1991), vi + 186.
C. De Boor, Cutting corners always works, Computer Aided Geometric Design 4(1-2) (1987), 125 – 131.
C. De Boor and C. De Boor, A Practical Guide to Splines, Vol. 27, p. 325, New York, Springer-Verlag (1978).
G. De Rham, Sur une courbe plane, Journal de Mathematiques Pures et Appliquees 35 (1956), 25 – 42.
S. Dubuc and J.-L. Merrien, A 4-point Hermite subdivision scheme, Mathematical Methods for Curves and Surfaces: Oslo 2000, T. Lyche and L.L. Schumaker (eds.), pp. 113–122, Vanderbilt University Press, Nashville, TN.
S.T. Ejaz, G. Mustafa and F. Khan, Subdivision schemes based collocation algorithms for solution of fourth order boundary value problems, Mathematical Problems in Engineering 2015(2015), Article ID 240138, 18 pages.
A. Ghaffar, G. Mustafa and K. Qin, Unification and application of 3-point approximating subdivision schemes of varying arity, Open Journal of Applied Sciences 2(4) (2012), 48 – 52.
A. Ghaffar and G. Mustafa, A family of even-point ternary approximating schemes, ISRN Applied Mathematics 2012(2012), Article ID 197383, 14 pages.
M.F. Hassan and N.A. Dodgson, Ternary and three-point univariate subdivision schemes, Curve and Surface Fitting Saint-Malo 2002, A. Cohen, J.-L. Merrien and L.L. Schumaker (eds.), pp. 199 – 208, Nashboro Press, Brentwood, TN (2002).
M.K. Jena, P. Shunmugaraj and P.C. Das, A non-stationary subdivision scheme for curve interpolation, Anziam J. 44(E) (2003), 216 – 235.
J. Kozak and M. Krajnc, Geometric interpolation by planar cubic polynomial curves, Computer Aided Geometric Design 24(2) (2007), 67 – 78.
D. Levin, Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes, Advances in Computational Mathematics 11(1) (1999), 41 – 54.
S.A. Manan, A. Ghaffar, M. Rizwan, G. Rahman and G. Kanwal, A subdivision approach to the approximate solution of 3rd order boundary value problem, Communications in Mathematics and Applications 9(4) (2018), 499 – 512, DOI: 10.26713/cma.v9i4.835.
G. Mustafa, F. Khan and A. Ghaffar, The m-point approximating subdivision scheme, Lobachevskii Journal of Mathematics 30(2) (2009), 138 – 145.
G. Mustafa, A. Ghaffar and M. Aslam, A subdivision-regularization framework for preventing over fitting of data by a model, Applications and Applied Mathematics: An International Journal 8(1) (2013), 178 – 190.
G. Mustafa and S.T. Ejaz, Numerical solution of two-point boundary value problems by interpolating subdivision schemes, Abstract and Applied Analysis 2014 (2014), Article ID 721314, 13 pages, DOI: 10.1155/2014/721314.
G. Mustafa, M. Abbas, S. T. Ejaz, A. I. M. Ismail and F. Khan, A numerical approach based on subdivision schemes for solving non-linear fourth order boundary value problems, Journal of Computational Analysis and Applications 23(1) (2017), 607 – 623.
R. Qu, Curve and surface interpolation by recursive subdivision algorithms, Computer Aided Drafting, Design and Manufacturing 4(2) (1994), 28 – 39.
R. Qu and R. P. Agarwal, A cross difference approach to the analysis of subdivision algorithms, Neural, Parallel and Scientific Computations 3(3) (1995), 393 – 416.
R. Qu and R. P. Agarwal, Solving two point boundary value problems by interpolatory subdivision algorithms, International Journal of Computer Mathematics 60(3-4) (1996), 279 – 294, DOI: 10.1080/00207169608804492.
G. Wang and C. Deng, On the degree elevation of B-spline curves a and corner cutting, Computer Aided Geometric Design 24(2) (2007), 90 – 98, DOI: 10.1016/j.cagd.2006.10.004.
H. Zheng, Z. Ye, Z. Chen and H. Zhao, A controllable ternary interpolatory subdivision scheme, International Journal of CAD/CAM 5(1) (2005), 29 – 38.
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