A hybrid collocation method based on combining the third kind Chebyshev polynomials and block-pulse functions for solving higher-order initial value problems

Saeid Jahangiri, khosrow maleknejad, Majid Tavassoli Kajani


The purpose of this paper is to propose a new collocation method for solving linear and nonlinear differential equations of high order as well as solving the differential equation on a very large interval. The new collocation method is based on a hybrid method combining the functions of the third kind Chebyshev polynomials and Block-Pulse functions uses. In the proposed method, the large interval of the problems is divided to small sub-intervals and in each sub-interval, collocation method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution of the differential equation on each sub-intervals. The proposed method is more accurate than the previous methods. Numerical examples show the capability and effciency of the presented method compared to existing methods.


Block-puls functions; collocation method; Chebyshev polynomials; higher-order initial value problems (HOIVPs); hybrid collocation method.

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Abdel-Halim Hassan, I.H. (2004). Differential transformation

technique for solving higher-order initial value problems. Applied

Mathematics and Computation, 154(2):299-311.

Allison, C. (1970). The numerical solution of coupled differential

equations arising from the schrodinger equation. Journal of Computational Physics. 6(3):378-391.

Awoyemi, D.O. (2003). A P-Stable linear multistep method for solving

general third order ordinary differential equations. International journal

of Computer Mathemathics, 80(8):985-991.

Awoyemi, D.O. & Idowu, O.M. (2005). A class hybrid collocation

methods for third order ordinary differential equations. International

journal of Computer Mathemathics, 82:1287-1293.

Berstein, R.B., Dalgarno, A., Massey, H. & Percival, J.C. (1963).

Thermal scattering of atoms by homonuclear diatomic molecules.

Proceedings of the Royal society of London A, 274. 1359:427-442.

Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. (2006).

Spectral Methods: Fundamentals in Single Domains, Scientific

Computation. Springer-Berlin. Germany, Pp. 91.

Guo, B. (2000). Jacobi approximations in certain Hilbert spaces

and their applications to singular differential equations. Journal of

Approximation Theory, 243(2):373-408.

Guo, B.Y. & Wang, L.l. (2004). Jacobi approximations in nonuniformly

Jacobiweighted Sobolev spaces. Journal of Approximation

Theory, 128(1):1-41.

Fehlberge, E. (1969). Low-order classical Runge-kutta formulas with

stepsize control and their application to some heat transfer problems.

NASA Technical Report, 14: Pp. 315.

Hsiao, C.H. (2009). Hybrid function method for solving Fredholm and

Volterra integral equations of the second kind. Journal of Computational

and Applied Mathematics, 230(1):59-68.

Jiang, Z.H. & Schaufelberger, W. (1991). Block pulse functions and

their applications in control systems. Springer-Verlag Berlin Heidelberg

NewYork, 37. Pp. 3.

John C. Mason & David Handscomb, D. (2003). Chebyshev

polynomials. CRC Press LLC, 43. Pp. 68.

Karimi Dizicheh, A., Ismail, F., Tavassoli Kajani, M. & Maleki, M.

(2013). A legendre Wavelet Spectral Collocation Method for Solving

Oscillatory Initial Value Problems. Journal of Mathematics, 33: Article

ID 591-636. http://dx.doi.org/10.1155/2013/591636

Kayode, S.J. (2008). An efficient zerostable numerical method for

fourth order differential equations. International Journal of Mathematics

and Mathematical Sciences, 176: Article ID 364021. http://dx.doi.


Kayode, S.J. & Awoyemi, D.O. (2010). A multiderivative collocation

method for 5th order ordinary differential equations. Journal of

Mathematics and Statistics, 6(1):60-63.

Lakestani, M. & Dehghan, M. (2010). Numerical solution of Riccati

equation using the cubic B-spline scaling functions and Chebyshev

cardinal functions. Computer Physics Communications, 181(5):957-

Marzban, H.R. & Razzaghi, M. (2003). Hybrid functions approach

for linearly constrained quadratic optimal control problems. Applied

Mathematical Modelling, 27(6):471-485.

Maleknejad, K. & Tavassoli Kajani, M. (2003). Solving integrodifferential

equation by using hybrid Legendre and Block-Pulse

functions. International Journal of Applied Mathematics, 11(1):67-76.

Maleknejad, K., Basirat, B. & Hashemizadeh, E. (2011). Hybrid

Legendre polynomials and Block-Pulse functions approach for

nonlinear VolterraFredholm integro-differential equations. Computers

and Mathematics with Applications, 61(9):2821-2828.

Olabode, B.T. (2009). A six-step scheme for the solution of fourth order ordinary differential equations. The Pacific Journal of Science

and Technology, 10(1):143-148.

Tavassoli Kajani, M., Ghasemi Tabatabaei, F. & Maleki, M. (2012).

Rational second kind Chebyshev approximation for solving some

physical problems on semi-infinite intervals. Kuwait Journal of Science

& Engineering, 39:15-29.

Vanden Berghe, G., De Meyer, H., Van Daele, M. & Van Hecke, T.

(2000). Exponentially Runge-Kutta methods. Journal of Computational

and Appled Mathematics, 125(2):107-115.

Waeleh, N., Majid, Z.A. & Ismail, F. (2011). A new algorithm for

solving higher order IVPs of ODEs. Applied Mathematical Sciences,


Waeleh, N., Majid, Z.A., Ismail, F. & Suleiman, M. (2012). Numerical

solution of higher order ordinary differential equations by direct block

code. Journal of Mathematics and Statistics, 8(1):77-81.

Yzba?i, ?. (2011). A numerical approach for solving a class of

the nonlinear Lane-Emden type equations arising in astrophysics.

Mathematical Methods in The Applied Sciences, 34(18):2218-2230.

Yzba?i, ?. & ?ahin, N. (2012). On the solutions of a class of nonlinear

ordinary differential equations by the Bessel polynomials. Journal of

Numerical Mathematics, 20(1):55-79.

Yzba?i, ?. (2012). A numerical approximation based on the Bessel

functions of first kind for solutions of Riccati type differentialdifference

equations. Computers & Mathematics with Applications,


Yzba?i, ?. (2013). Numerical solutions of fractional Riccati type

differential equations by means of the Bernstein polynomials. Applied

Mathematics and Computations, 219(11):6328-6343.


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