An Improvement on Collocation Algorithm to Solve Initial Value Problems
Subject Areas : International Journal of Industrial Mathematics
مهران نیکآریا
1
,
سعید سرآبادان
2
1 - Department of Electrical Engineering and Information Technology, Iranian Research Organization for Science and Technology, Tehran, Iran.
2 - Department of Mathematics, Faculty of Science, Imam Hossein Comprehensive University, Tehran, Iran.
Keywords: High accuracy, IVP, CPU time, Bessel functions, Collocation Spectral algorithm,
Abstract :
In this paper an improved version of the collocation method is proposed to solve ordinary differential equations with initial conditions. Our proposed algorithm is described by applying it to some well-known IVPs. The results are compared with basic collocation algorithms to show the advantages, applicability and efficiency of the proposed method. Based on numerical results, the proposed algorithm has better accuracy and execution time.
[1] S. Abbasbandy, T. Hayat, Solution of the MHD Falkner-Skan flow by Homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 3591-3598.
[2] B. Abdollahi, E. Babolian, Theory of blockpulse functions in numerical solution of Fred holm integral equations of the second kind, International Journal of Industrial Mathematics 8 (2016) 157- 163.
[3] V. K. Baranwal, R. K. Pandey, M. P. Tripathi, O. Singh, An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on anew iterative method, Communications in Nonlinear Science and Numerical Simulation 17 (2012) 3906-3921.
[4] W. W. Bell, Special Functions For Scientists And Engineers, Published simultaneously, D. Van Nostrand Company, Canada, Ltd. 1967.
[5] J. P. Boyd, Chebyshev and Fourier spectral methods. 2nd ed, New York Dover, 2000.
[6] M. Dehghan, M. Abbaszadeh, A. Mohebbi, Legendre spectral element method for solving time fractional modified anomalous subdiffusion equation, Applied Mathematical Modelling 40 (2015) 3635-36541.
[7] M. Dehghan, M. Abbaszadeh, Two meshless procedures: moving kriging interpolation and element-free Galerkin for fractional PDEs, Applicable Analysis 1 (2016) 31-4.
[8] M. Dehghan, J. Manafian, A. Saadatmandi, Application of semi-analytical methods for solving the Rosenau-Hyman equation arising in the pattern formation in liquid drops, International Journal of Numerical Methods for Heat and Fluid Flow 22 (2012) 777-790.
[9] M. Dehghan, R. Salehi, The use of Homotopy analysis method to solve the timedependent nonlinear Eikonal partial differential equation, Zeitschrift fur Naturforschung A 66 (2011) 259-271.
[10] F. Fattahzadeh, Numerical solution of general nonlinear Fredholm-Volterra integral equations using Chebyshev approximation, International Journal of Industrial Mathematics 8 (2016) 81-86.
[11] B. Y. Guo, J. P. Yan, Legendre Gauss collocation method for initial value problems of second order ordinary differential equations, Applied Numerical Mathematics 56 (2009) 13861408.
[12] S. A. Hossayni, J. A. Rad, K. Parand, S. Abbasbandy, Application of the exact operational matrices for solving the emdenfowler equations, arising in astrophysics, International Journal of Industrial Mathematics 7(2015) 351-374.
[13] S. G. Hosseini, E. Babolian, S. Abbasbandy, A new algorithm for solving Van Der Pol equation based on piecewise spectral Adomian decomposition method, International Journal of Industrial Mathematics 8 (2016) 177-184.
[14] S. A. Khuri, A. Sayfy, A Laplace variational iteration strategy for the solution of differential equations, Applied Mathematics Letters 25 (2012) 2298-2305.
[15] E. Kreyszig, Introductory functional analysis with applications, John Wiley, New York, 2000.
[16] H. R. Marzban, S. Hoseini, M. Razzaghi, Solution of Volterras population model via block-pulse functions and Lagrangeinterpolating polynomials, Mathematical Methods in the Applied Sciences 32 (2009) 127- 134.
[17] K. Parand, M. Nikarya, J. A. Rad, Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocation method, Celestial Mechanics and Dynamical Astronomy 116 (2013) 97-107.
[18] K. Parand, M. Nikarya, New numerical method based on generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind. Journal, Nonlinear Engineering 8 (2019) 438-448
[19] K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Applied Mathematical Modelling 38 (2014) 4137-4147.
[20] K. Parand, J. A. Rad, M. Nikarya, A new numerical algorithm based on the first kind of modified Bessel function to solve population growth in a closed system, International Journal of Computer Mathematics 91 (2014) 1239-1254.
[21] K. Parand, M. Nikarya, Application of Bessel functions and jacobian free Newton method to solve time-fractional burger equation, Nonlinear Engineering 8 (2019) 688-694.
[22] K. Parand, S. A. Hossayni, J. A. Rad, Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model, Applied Mathematical Modelling 15 (2016) 993-1011.
[23] K. Parand, M. Nikarya, A novel method to solve nonlinear Klein-Gordon equation arising in quantum field theory based on Bessel functions and Jacobian free Newton-Krylov subspace methods, Communications in Theoretical Physics 69 (2018) 637.
[24] J. Shen, T. Tang, L. L. Wang, Spectral Methods Algorithms, Analysics and Applicattions, first edition, Springer, Berlin, 2001.
[25] H. Vosughi, E. Shivanian, S. Abbasbandy, A new analytical technique to solve Volterra’s integral equations, Mathematical Methods in The Applied Sciences 34 (2011) 1243-1253.
[26] G. Watson, A treatise on the theory of Bessel Functions, 2nd edition, Cambridge University Press, Cambridge (England), 1967.
[27] M. Wazwaz, A reliable study for extensions of the Bratu problem with boundary conditions, Mathematical Methods in The Applied Sciences 35 (2012) 845-856.
[28] A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Applied Mathematics and Computation 118 (2001) 287-310.
[29] J. P. Yan, B. Y. Guo, A collocation method for initial value problems of second-order odes by using Laguerre functions, Numerical Mathematics: Theory, Methods and Applications 4 (2011) 283-295.
[30] Y. Zhang, K. T. Chong, N. Kazantzis, A. G. Parlos, Discretization of nonlinear inputdriven dynamical systems using the Adomian Decomposition Method, Applied Mathematical Modelling 36 (2012) 5856-5875.