Generalization of Chebyshev wavelet collocation method to the rth-order differential equations

Abstract

Chebyshev wavelets operational matrices play an important role for the numeric solution of \textit{r}th order differential equations. In this study, operational matrices of \textit{rth} integration of Chebyshev wavelets are presented and a general procedure of these matrices is  correspondingly given. Disadvantages of Chebyshev wavelets methods is eliminated for \textit{r}th integration of $\Psi (t)$. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. Algebraic equation system has been obtained by using the Chebyshev collocation points and solved. The proposed method has been applied to the three nonlinear boundary value problems using quasilinearization technique. Numerical examples showed the applicability and accuracy.

Keywords

• Chebyshev wavelets,collocation method,nonlinear differential equations,quasilinearization technique

Kaynakça

1. [1] C. Hwang, Y.P. Shih, Laguerre series direct method for variational problems, J. Optim. Theory Appl. 39 (1983) 143-149.
2. [2] R.Y. Chang, M.L. Wang, Shifted Legendre direct method for variational problems series, J. Optim. Theory Appl. 39 (1983) 299- 307.
3. [3] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for variational problems, Int. J. Syst. Sci. 16 (1985) 855-861.
4. [4] M. Razzaghi, M. Razzaghi, Fourier series direct method for variational problems, Int. J. Control 48 (1988) 887-895.
5. [5] 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.
6. [6] 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.
7. [7] F.C. Chen and C.H. Hsiao, AWalsh series direct method for solving variational problems, J. Franklin Inst. 300 (1975), pp. 265-280.
8. [8] 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.

9. [9] I.R. Howang and Y.P. Sheh, Solution of integral equations via Legendre polynomials, J. Comput. Electrical Engineering 9 (1982), pp. 123-129.
10. [10] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, PA, 1992.
11. [11] C. Cattani, Fractional calculus and shannon wavelets,Mathematical Problems in Engineering, vol. 2012(2012), Article ID 502812, 25.
12. [12] C. Cattani, Shannon wavelets for the solution of integro-differential equations, Mathematical Problems in Engineering, vol. 2010(2010), Article ID 408418, 22.
13. [13] M. Karabacak, M. Yigider. Shannon wavelet transform for solving fractional differential algebraic equations numerically.. International Journal of Engineering and Applied Sciences 3(12), (2016), 56-61.
14. [14] C. Cattani and A. Kudreyko, Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Applied Mathematics and Computation, 215/12 (2010), 4164-4171.
15. [15] M. H. Heydari, M. R. Hooshmandasl, F.M.M. Ghaini, and F. Fereidouni, Two-dimensionallegendre wavelets forsolving fractional poisson equation with dirichlet boundary conditions, Engineering Analysis with Boundary Elements, 37(2013) 1331-1338.
16. [16] M. H. Heydari,M. R. Hooshmandasl, F.M.Maalek Ghaini, and F.Mohammadi,Wavelet collocation method for solving multiorder fractional differential equations, Journal of Applied Mathematics, vol. 2012(2012), Article ID 542401, 19.
17. [17] M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, and C. Cattani, Wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 19/1 (2014), 37-48.
18. [18] J.S. Gu and W.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996), pp. 623-628.
19. [19] U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Applied Mathematics and Computation, 185 (2007) 695-704.
20. [20] W. Geng, Y. Chen, Y. Li, D. Wang, Wavelet method for nonlinear partial differential equations of fractional order, Computer and information science, 4/5 (2011) 28-35.
21. [21] G. Hariharan, K. Kannan, K.R. Sharma, Haar wavelet method for solving Fisher’s equation, Applied Mathematics and Computation, 211 (2009) 284-292.
22. [22] G. Hariharan and K. Kannan, Haar wavelet method for solving FitzHugh-Nagumo equation, Int. Journal of Mathematical and Statistical Sciences, 2:2 (2010) 59-63.
23. [23] G.Hariharan, K.Kannan, A Comparative Study of a Haar Wavelet Method and a Restrictive Taylor’s Series Method for Solving Convection-diffusion equations, Int. J. Comput. Methods in Eng. Sci. Mechanics, 11: 4 (2010) 173 -184.
24. [24] G.Hariharan, An efficient wavelet analysis method to film-pore diffusion model arising in mathematical chemistry, The Journal of Membrane Biology,, 247/4 (2014) 339-343.
25. [25] G.Hariharan, K.Kannan, Haar wavelet method for solving nonlinear parabolic equations, J.Math.Chem. 48 (2010) 1044-1061.
26. [26] H. Kaur, R. C. Mittal, V. Mishra, Haar wavelet quasilinearization approach for solving nonlinear boundary value problems, Amer. J. Comput. Math., 1, (2011) 176-182.
27. [27] M Karabacak, E C¸ elik. The Numerical Solution of Fractional Differential-Algebraic Equations (FDAEs) by Haar Wavelet Functions, Int. J. Eng. Appl. Sci. 2(10) (2015), 65-71.
28. [28]˙I. Celik, Haar wavelet method for solving generalized Burgers-Huxley equation, Arab Journal of Mathematical Sciences, 18 (2012)25-37.
29. [29]˙I. Celik, Haar wavelet approximation for magnetohydrodynamic flow equations AppliedMathematicalModelling, 37 (2013) 3894- 3902. [30] 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.
30. [31] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and Computers in Simulation 53 (2000) 185-192.
31. [32] M. Razzaghi, S. Yousefi, Legendre wavelets operational matrix of integration, Int. J. Syst. Sci. 32 (4) (2001) 495-502.
32. [33] E. Babolian, F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188 (2007) 417-426.
33. [34] E.Babolian, and F. Fattahzadeh, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation 188.1 (2007): 1016-1022.
34. [35] 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.
35. [36] M. Karabacak, M Yi˘gider, The Legendre wavelets method for approximate solution of fractional differential-algebraic equations, International Journal of Applied Mathematics and Statistics 57(4) (2018), 161-173
36. [37] H. Adibi, P. Assari, Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind, Mathematical Problems in Engineering V 2010, Article ID 138408, p 17.
37. [38] Y.X. Wang, Q.B. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218 (2012), 8592-8601.
38. [39] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and M. Li, Chebyshev wavelets method for solution of nonlinear fractional integrodifferential equations in a large interval, Advances in Math. Physics, vol. 2013(2013), Article ID 482083, 12.
39. [40] M. R. Hooshmandasl, M. H. Heydari, and F. M. M. Ghaini, Numerical solution of the one dimensional heat equation by using chebyshev wavelets method, Applied and Computational Mathematics, 1/6 (2012), 1-7, 2012.
40. [41] C. Yang, J, Hou, Chebyshev wavelets method for solving Bratu’s problem, Boundary Value Problems (2013) 2013:142.
41. [42] G.Hariharan, An efficient wavelet based approximation method to water quality assessment model in a uniform channel, Ain Shams Engineering Journal, 5/2 (2014), 525-532.
42. [43]˙I. Celik, Numerical Solution of Differential Equations by Using Chebyshev Wavelet Collocation Method, Cankaya University Journal of Science and Engineering, 10(2) (2013) 169-184.
43. [44]˙I. Celik, Chebyshev Wavelet collocation method for solving generalized Burgers-Huxley equation, Math. Meth. Appl. Sci., 39 (2016) 366-377.
44. [45]˙I. Celik, Free vibration of non-uniform Euler-Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method, Applied Mathematical Modelling 54 (2018): 268-280.
45. [46] L. Fox, I.B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968.
46. [47] K.N.S. K. Viswanadham, P. M. Krishna, R. S. Koneru, Numerical solutions of fourth order boundary value problems by Galerkin method with Quintic B-splines, Int. J. Nonli. Sci. 10 (2010) 222-230.
47. [48] S. Momani, K. Moadi, A reliable algorithm for solving fourth-order boundary value problems, J. Appl. Math. Comput. 22(3), (2006) 185-197.
48. [49] R.P. Agarwall, G. Akrivis, Boundary value problems occurring in plate deflection theoryJ. Comput. Appl. Math., 8 (1982), 145- 154.
49. [50] M.M. Chawla, C.P. Katti, Finite difference methods for two-point boundary value problems involving high order differential equations, BIT Numer. Math., 19 (1979), 27-33.
50. [51] J. Rashidinia, M. Ghasemia, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math., 235 (2011), 2325-2342.
51. [52] E.H. Twizell, A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems, Comp. Meth. Appl. Mech. Eng., 62 (1987), 293-303.
52. [53] E.H. Twizell, S.I.A. Tirmizi, Multiderivative methods for nonlinear beam problems, Commun. Appl. Numer. Meth., 4 (1988), 43-50.
53. [54] Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H. New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems. J. Egypt. Math. Soc., 24/2, (2015), 319-327.