Numerical Solution of Differential Equations by Using Chebyshev Wavelet Collocation Method

Year 2013, Volume: 10 Issue: 2, - , 01.11.2013

İbrahim Celik

Abstract

In this article, a new method known as the Chebyshev
wavelet collocation method is presented for the solution of second-order linear
ordinary differential equations (ODEs). The method is based on the
approximation of the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, an algebraic equation
system has been obtained and solved. Hence the implicit forms of the
approximate solution of second-order linear ordinary differential equations
have been obtained. This present method has been
applied to the Bessel differential equation of order zero and the Lane–Emden
equation. These calculations demonstrate
that the accuracy of the Chebyshev wavelet collocation method is quite high
even in the case of a small number of grid points. The present method is a very
reliable, simple, fast, computationally efficient, flexible, and convenient
alternative method.

Keywords

Chebyshev wavelet, collocation, Legendre wavelet, approximate solution

Numerical Solution of Differential Equations by Using Chebyshev Wavelet Collocation Method

Abstract

In this article, a new method known as the Chebyshev wavelet collocation method is presented for the solution of second-order linear ordinary differential equations (ODEs). The method is based on the approximation of the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, an algebraic equation system has been obtained and solved. Hence the implicit forms of the approximate solution of second-order linear ordinary differential equations have been obtained. This present method has been applied to the Bessel differential equation of order zero and the Lane–Emden equation. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite high even in the case of a small number of grid points. The present method is a very reliable, simple, fast, computationally efficient, flexible, and convenient alternative method.

Keywords

Chebyshev wavelet, collocation, Legendre wavelet, approximate solution

References

• C. Hwang, Y.P. Shih, Laguerre series direct method for variational problems, J. Optim. Theory Appl. 39 (1983) 143–149.
• R.Y. Chang, M.L. Wang, Shifted Legendre direct method for variational problems series, J. Optim. Theory Appl. 39 (1983) 299–307.
• I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for variational problems, Int. J. Syst. Sci. 16 (1985) 855–861.
• M. Razzaghi, M. Razzaghi, Fourier series direct method for variational problems, Int. J. Control 48 (1988) 887–895.
• C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEEE Proc. Control Theory Appl. 144 (1997) 87–93.
• K. Maleknejad, M. T. Kajani, Y. Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets, Kybernetes, Int. J. Syst. Math. 32 (2003) 1530–1539.
• F.C. Chen and C.H. Hsiao, A Walsh series direct method for solving variational problems, J. Franklin Inst. 300 (1975), pp. 265–280.
• F.C. Chen, Y.T. Tsay, and T.T. Wu, Walsh operational matrices for fractional calculus and their application to distributed parameter system, J. Franklin Inst. 503 (1977), pp. 267–284.
• I.R. Howang and Y.P. Sheh, Solution of integral equations via Legendre polynomials, J. Comput. Electrical Engineering 9 (1982), pp. 123–129.
• J.S. Gu andW.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996), pp. 623–628.
• M. T. Kajani and A. H. Vencheh, Solving linear integro-differential equation with Legendre wavelet, Int. J. Comput. Math. 81 (6) (2004), pp. 719–726.
• M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and Computers in Simulation 53 (2000) 185-192.
• M. Razzaghi, S. Yousefi, Legendre wavelets operational matrix of integration, Int. J. Syst. Sci. 32 (4) (2001) 495–502.
• E. Babolian, F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188 (2007) 417-426.
• M. T. Kajania, A. H. Vencheha and M. Ghasemib, The Chebyshev wavelets operational matrix of integration and product operation matrix, Int. J. of Comput. Math. 86/7 (2009) 1118–1125
• L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
• Y. Ordokhani, S. Davaei far, Approximate Solutions of Differential Equations by Using Bernstein Polynomials, ISRN Applied Mathematics, Vol.2011 (2011), Article ID 787694, 15 pages