The Application of Quartic Trigonometric B-spline for Solving Second Order Singular Boundary Value Problems

Tayyaba Akram, Muhammad Abbas, Ahmad Izani Ismail

Abstract


In this paper, a quartic trigonometric B-spline collocation approach is described and presented for the numerical solution of the second order singular boundary value problems. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. The unknown coefficients \(C_{i}\), \(i=-4,-3,\ldots,n-1\) are obtained through optimization. The maximum errors \((L_{\infty})\) and norm errors \((L_{2})\) are also computed for different space size steps to assess the performance of the proposed technique. The rate of convergence is discussed numerically to be of fourth-order. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The comparison shows that the quartic trigonometric B-spline method is superior as it yields more accurate solutions.

Keywords


Quartic trigonometric basis functions; Trigonometric collocation method; Singular boundary value problem

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References


M. Abbas, A.A. Majid, A.I.M. Ismail and A. Rashid, Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system, PLoS ONE 9 (1) (2014), e83265, DOI: 10.1371/journal.pone.0083265.

M. Abbas, A.A. Majid, A.I.M. Ismail and A. Rashid, Numerical method using cubic trigonometric B-spline technique for non-classical diffusion problem, Abstract and Applied Analysis 2014 (2014), Article ID 849682, 10 pages, DOI: 10.1155/2014/849682.

M. Abbas, A.A. Majid, A.I.M. Ismail and A. Rashid, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Applied Mathematics and Computation 239 (2014), 74 – 88, DOI: 10.1016/j.amc.2014.04.031.

N.N. Abd. Hamid, Splines for Linear Two-Point Boundary Value Problems, Master’s Thesis, Universiti Sains Malaysia (2010).

E.L. Albasiny and W.D. Hoskins, Cubic spline solutions to two-point boundary value problems, The Computer Journal 12 (2) (1969), 151 – 153, DOI: 10.1093/comjnl/12.2.151.

W.G. Bickley, Piecewise cubic interpolation and two-point boundary problems, The Computer Journal 11 (2) (1968), 206 – 208, DOI: 10.1093/comjnl/11.2.206.

N. Caglar and H. Caglar, B-spline solution of singular boundary value problems, Applied Mathematics and Computation 182 (2) (2006), 1509 – 1513, DOI: 10.1016/j.amc.2006.05.035.

H. Caglar, N. Caglar and M. Ozer, B-spline solution of non-linear singular boundary value problems arising in physiology, Chaos Solitons and Fractals 39 (3) (2009), 1232 – 1237, DOI: 10.1016/j.chaos.2007.06.007.

D.J. Fyfe, The use of cubic splines in the solution of two-point boundary value problems, The Computer Journal 12 (2) (1969), 188 – 192, DOI: 10.1093/camjnl/12.2.188.

J. Goh, A.A. Majid and A.I.M. Ismail, A quartic B-spline for second-order singular boundary value problems, Computers and Mathematics with Applications 64 (2012), 115 – 120, DOI: 10.106/j.camwa.2012.01.022.

Y. Gupta and M. Kumar, A computer based numerical method for singular boundary value problems, International Journal of Computer Applications 30 (1) (2011), 21 – 25.

N. Hamid, A.A. Majid and A.I.M. Ismail, Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two, World Academy of Science, Engineering and Technology 70 (2010), 798 – 803.

A.S. Heilat, N.N.A. Hamid and A.I. Ismail, Extended cubic B-spline method for solving a linear system of second-order boundary value problems, SpringerPlus 5 (2016), 1314, DOI: 10.1186/340064-016-2936-4.

M.K. Kadalbajoo and V.K. Aggarwal, Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Applied Mathematics and Computation 160 (2005), 851 – 863, DOI: 1016/j.amc.2003.12.2004.

M.K. Kadalbajoo and V. Kumar, B-spline method for a class of singular two-point boundary value problems using optimal grid, Applied Mathematics and Computation 188 (2) (2007), 1856 – 1869, DOI: 10.1016/j.amc.2006.11.050.

P. Koch, T. Lyche, M. Neamtu and L. Schumaker, Control curves and knot insertion for trigonometric splines, Advances in Computational Mathematics 3 (1995), 405 – 424.

M. Kumar and N. Singh, A collection of computational techniques for solving singular boundary value problems, Advances in Engineering Software 40 (2009), 288 – 297, DOI: 10.1016/j.advengsoft.2008.04.010.

M. Kumar, A difference method for singular two-point boundary value problems, Applied Mathematics and Computation 146 (2-3) (2003), 879 – 884, DOI: 10.1016/S0096-3003(02)00646-X.

S. Liu, Modified hierarchy basis for solving singular boundary value problems, Journal of Mathematical Analysis and Application 325 (2007), 1240 – 1256, DOI: 10.1016/j.jmma.2006.02.043.

A.S.V. Ravi Kanth and Y.N. Reddy, Cubic spline for a class of singular two-point boundary value problems, Applied Mathematics and Computation 170 (2)(2005), 733 – 740, DOI: 10.10.16/j.amc.2004.12.049.

A.S.V. Ravi Kanth and Y.N. Reddy, Higher order finite difference method for a class of singular boundary value problems, Applied Mathematics and Computation 155 (1) (2004), 249 – 258, DOI: 10.1016/S0096-3003(03)00774-4.

I.J. Schoenberg, On trigonometric spline interpolation, J.Math. Mech. 13 (1964), 795 – 825.

L.F. Shampine, Singular boundary value problems for ordinary differential equations, Applied Mathematics and Computation 138 (1) (2003), 99 – 112, DOI: 10.1016/S0096-3003(02)00111-X.

M.N. Suardi, N.Z.F.M. Radzuan and J. Sulaiman, Cubic B-spline solution for two-point boundary value problem with AOR iterative method, Journal of Physics: Conf. Series 890 (2017), 012015, DOI: 10.1088/1742-6596/890/1/012015.

G. Walz, Identities for trigonometric B-splines with an application to curve design, BIT Numerical Mathematics 37 (1997), 189 – 201, DOI: 10.1007/BF02510180.

S.M. Zin, A.A. Majid, A.I. Md. Ismail and M. Abbas, Application of hybrid cubic B-spline collocation approach for solving a generalized nonlinear Klien-Gordon equation, Mathematical Problems in Engineering 2014 (2014), Article ID 108560, 10 pages, DOI: 10.1155/2014/108560.

S.M. Zin, M. Abbas, A.A. Majid and A.I.M. Ismail, A new trigonometric spline approach to numerical solution of generalized nonlinear Klien-Gordon equation, PLOS ONE 9 (5) (2014), e95774, DOI: 10.1371/journal.pone.0095774.




DOI: http://dx.doi.org/10.26713%2Fcma.v9i3.832

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