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Proceedings of the Estonian Academy of Sciences. Physics. Mathematics

Application of the Haar wavelet transform to solving integral and differential equations; pp. 28–46

Full article in PDF format | 10.3176/phys.math.2007.1.03

Ülo Lepik


A survey on the use of the Haar wavelet method for solving nonlinear integral and differential equations is presented. This approach is applicable to different kinds of integral equations (Fredholm, Volterra, and integro-differential equations). Application to partial differential equations is exemplified by solving the sine-Gordon equation. All these problems are solved with the aid of collocation techniques.

Computer simulation is carried out for problems the exact solution of which is known. This allows us to estimate the precision of the obtained numerical results. High accuracy of the results even in the case of a small number of collocation points is observed.


1. Daubechies, I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 1988, 41, 909–996.

2. Chen, M.-Q., Hwang, C. and Shin, Y.-P. The computation of wavelet–Galerkin approximation on a bounded interval. Int. J. Numer. Methods Eng., 1996, 39, 2921–2944.<2921::AID-NME983>3.0.CO;2-D

3. Dahmen, W., Kurdila, A. J. and Oswald, P. (eds). Multiscale Wavelet Methods for Partial Differential Equations. Academic Press, San Diego, 1997.

4. Strang, G. and Nguyen, T. Wavelets and Filter Banks. Wellesley–Cambridge Press, Wellesley (MA), 1997.

5. Jameson, L. A wavelet-optimized, very much order adaptive grid and order numerical method. SIAM J. Sci. Comput., 1998, 19, 1980–2013.

6. Stanković, R. S. and Falkowski, B. J. The Haar wavelet transform: its status and achievements. Comput. Electr. Eng., 2003, 29, 25–44.

7. Cattani, C. Haar wavelet splines. J. Interdisciplinary Math., 2001, 4, 35–47.

8. Cattani, C. Haar wavelets based technique in evolution problems. Proc. Estonian Acad. Sci. Phys. Math., 2004, 53, 45–65.

9. Chen, C. F. and Hsiao, C.-H. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl., 1997, 144, 87–94.

10. Chen, C. F. and Hsiao, C.-H. Wavelet approach to optimising dynamic systems. IEE Proc. Control Theory Appl., 1997, 146, 213–219.

11. Hsiao, C.-H. and Wang, W.-J. State analysis of time-varying singular bilinear systems via Haar wavelets. Math. Comput. Simul., 2000, 52, 11–20.

12. Hsiao, C.-H. and Wang, W.-J. State analysis of time-varying singular nonlinear systems via Haar wavelets. Math. Comput. Simul., 1999, 51, 91–100.

13. Hsiao, C.-H. and Wang, W.-J. Haar wavelet approach to nonlinear stiff systems. Math. Comput. Simul., 2001, 57, 347–353.

14. Hsiao, C.-H. Haar wavelet direct method for solving variational problems. Math. Comput. Simul., 2004, 64, 569–585.

15. Hsiao, G.-C. and Rathsfeld, A. Wavelet collocation methods for a first kind boundary integral equation in acoustic scattering. Adv. Comput. Math., 2002, 17, 281–308.

16. Alpert, B., Beylkin, G., Coifman, R. and Rokhlin, V. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 1993, 14, 159–184.

17. Vainikko, G., Kivinukk, A. and Lippus, J. Fast solvers of integral equations of the second kind: wavelet methods. J. Complexity, 2005, 21, 243–273.

18. Yousefi, S. and Razzaghi, M. Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations. Math. Comput. Simul., 2005, 70, 1–8.

19. Maleknejad, K., Aghazadeh, N. and Molapourasl, F. Numerical solution of Fredholm integral equation of the first kind with collocation method and estimation of error bound. Appl. Math. Comput., 2006, 179, 352–359.

20. Xiao, J.-Y., Wen, L.-H. and Zhang, D. Solving second kind Fredholm integral equation by periodic wavelet Galerkin method. Appl. Math. Comput., 2006, 175, 508–518.

21. Kaneko, H., Noren, R. D. and Novaprateep, B. Wavelet applications to the Petrov–Galerkin method for Hammerstein equations. Appl. Numer. Math., 2003, 45, 255–273.

22. Maleknejad, K. and Derili, H. The collocation method for Hammerstein equations by Daubechies wavelets. Appl. Math. Comput., 2006, 172, 846–864.

23. Avudainayagam, A. and Vano, C. Wavelet–Galerkin method for integro-differential equations. Appl. Numer. Math., 2000, 32, 247–254.

24. Fedorov, M. V. and Chyev, G. N. Wavelet method for solving integral equations of simple liquids. J. Mol. Liq., 2005, 120, 159–162.

25. Liang, X.-Z., Liu, M.-C. and Che, X.-J. Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets. J. Comput. Appl. Math., 2001, 136, 149–161.

26. Mahmoudi, Y. Wavelet Galerkin method for numerical solution of nonlinear integral equation. Appl. Math. Comput., 2005, 167, 1119–1129.

27. Khellat, F. and Yousefi, S. A. The linear Legendre mother wavelets operational matrix of integration and its application. J. Franklin Inst., 2006, 143, 181–190.

28. Maleknejad, K. and Lotfi, T. Expansion method for linear integral equations by cardinal B-spline wavelet and Shannon wavelet bases to obtain Galerkin system. Appl. Math. Comput., 2006, 175, 347–355.

29. Maleknejad, K. and Karami, M. Numerical solution of non-linear Fredholm integral equations by using multiwavelets in the Petrov–Galerkin method. Appl. Math. Comput., 2005, 168, 102–110.

30. Lepik, Ü. and Tamme, E. Application of the Haar wavelets for solution of linear integral equations. In 5–10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings. 2005, 395–407.

31. Lepik, Ü. and Tamme, E. Solution of nonlinear integral equations via the Haar wavelet method. Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 17–27.

32. Lepik, Ü. Haar wavelet method for non-linear integro-differential equations. Appl. Math. Comput., 2006, 176, 324–333.

33. Maleknejad, K. and Mirzaee, F. Using rationalized Haar wavelet for solving linear integral equations. Appl. Math. Comput., 2005, 160, 579–587.

34. Lepik, Ü. Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul., 2003, 68, 127–143.

35. Lepik, Ü. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput. (accepted).

36. Newland, D. E. An Introduction to Random Vibrations, Spectral and Wavelet Analysis. Longman Scientific and Technical, New York, 1993.

37. Kim, B. H., Kim, H. and Park, T. Nondestructive damage evaluation of plates using the multiresolution analysis of two-dimensional Haar wavelet. J. Sound Vibration, 2006, 292, 82–104.

38. Kumar, B. V. R. and Mehra, M. Wavelet based preconditioners for sparse linear systems. Appl. Math. Comput., 2005, 171, 203–224.

39. Cattani, C. Wavelet analysis of dynamical systems. Electron. Commun. (Kiev), 2002, 17, 115–124.

40. Forinash, K. and Willis, C. R. Nonlinear response of the sine-Gordon breather to an a.c. driver. Physica D, 2001, 149, 95–106.


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