Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 181–202 | Cite as

A Wavelet Based Rationalized Approach for the Numerical Solution of Differential and Integral Equations

  • Aditya Kaushik
  • Geetika Gupta
  • Manju SharmaEmail author
  • Vishal Gupta
Original Research


A wavelet based rationalized method is presented for the numerical solution of differential, integral and integro-differential equations. Rationalized Haar functions are used to estimate the solution. Their fundamental properties are discussed. A rigorous convergence analysis is presented. The operational matrix of the product of two rationalized Haar functions is used to reduce the dynamical system to an algebraic system. A variety of model problems are taken into account so as to test the efficiency of the proposed method. The result so obtained are compared with the available exact solutions. In addition, proposed scheme is compared with some state of the art existing methods. It is found that the Haar wavelet operational matrix is the fastest. Moreover, the results obtained are mathematically simple and the desired accuracy of the solution is obtained using small number of grid points. The main advantages of the wavelet method are its simplicity, fast transformation, possibility of implementation of fast algorithms and low computational cost with high accuracy.


Wavelets Rationalized functions Numerical method Differential equations Integral equations 


  1. 1.
    Grossmann, A., Morlet, J.: Decomposition of Hardy Functions into square integrable wavelets of constant shape. SIAM J. Math. Anal 15(4), 723–736 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Shore, J.E.: On the Application of Haar Functions. Naval Research Laboratory, Washington DC (1973)CrossRefGoogle Scholar
  3. 3.
    Ohkita, M.: Evaluation of analytic functions by generalized definite integration. Math. Comput. Simul. 27, 511–517 (1985)CrossRefzbMATHGoogle Scholar
  4. 4.
    Haar, A., Haar, Alfred.: Zur Theorie de orthogonalen Funktionensysteme. Mathematische Annalen 69(3), 331–371 (1910)Google Scholar
  5. 5.
    Engel, J.: Density estimation with Haar Series. Stat. Probab. Lett. 9, 111–117 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Yuguo, H.E., Sun, Jigui: Complete quantum circuit of Haar wavelet based MRA. Chin. Sci. Bull. 50, 1796–1798 (2005)CrossRefGoogle Scholar
  7. 7.
    Porwik, P., Lisowska, A.: The Haar wavelet transform in digital image processing its status and achievements. Mach. Graph. Vis. 13, 79–98 (2004)zbMATHGoogle Scholar
  8. 8.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic, USA (2009)zbMATHGoogle Scholar
  9. 9.
    Alajbegovic, H., Zecic, D., Huskanovic, A.: Image Compression Using the Haar Wavelet Transform. TMT (2009)Google Scholar
  10. 10.
    Skobelev, S.P.: Application of extended boundary conditions and the Haar wavelets in the analysis of wave scattering of thin screens. J. Commun. Technol. Electron. 51(7), 748–758 (2006)CrossRefGoogle Scholar
  11. 11.
    Karimi, H.R., Lohmann, B.: Haar wavelet-based robust optimal control for vibration reduction of vehicle engine body system. Electr. Eng 89, 469–478 (2007)CrossRefGoogle Scholar
  12. 12.
    Kim, B.H., Park, T.: Application of multi-resolution analysis of wavelets to nondestructive damage evaluation: I. Theory. KSCE J. Civ. Eng. 9(6), 505–512 (2005)CrossRefGoogle Scholar
  13. 13.
    Tian, Y., Herzberg, A.M.: Estimation and optimal designs for linear Haar-wavelet models. Metrika 65, 311–324 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Murtagh, F.: The Haar wavelet transform of a dendrogram. J. Classif. 24, 3–32 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kopenkov, V.N.: Efficient algorithms of local discrete wavelet transform with Haar-like bases. Patt. Recognit. Image Anal. 18(4), 654–661 (2008)CrossRefGoogle Scholar
  16. 16.
    Subramani, P., Sahu, R., Verma, S.: Feature selection using Haar wavelet power spectrum. Bioinformatics 7, 432 (2006)Google Scholar
  17. 17.
    Meyer, Y., Roques, S.: Progress in Wavelet Analysis and Applications, pp. 9–18. Frontiers Publishers (1993)Google Scholar
  18. 18.
    Li, B., Chen, X.: Wavelet based numerical analysis: a review and classification. Finite Elem. Anal. Des. 81, 14–31 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ohkita, M., Kobayashi, Y.: An application of rationalized Haar functions to solution of linear differential equations. IEEE Trans. Circ. Syst. 33(9), 853–862 (1986)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ohkita, M., Kobayashi, Y.: An Application of rationalized Haar functions to solution of linear partial differential equations. Math. Comput. Simul. 30, 419–428 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, C.F., Hsiao, C.H.: Wavelet approach to optimizing dynamic systems. IEE Proc. Control Theory Appl. 146(2), 213–219 (1999)CrossRefGoogle Scholar
  22. 22.
    Hsiao, C.H., Wang, W.J.: Optimalcontrol of linear time-varying systems via Haar wavelets. J. Optim. Theory Appl. 103(3), 641–655 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hsiao, C.H., Wang, W.J.: State analysis of time-varying singular non-linear systems via Haar wavelets. Math. Comput. Simul. 51, 91–100 (1999)CrossRefGoogle Scholar
  24. 24.
    Hsiao, C.H., Wang, W.J.: State analysis and optimal control of time-varying discrete systems via Haar wavelets. J. Optim. Theory Appl. 103(3), 623–640 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Karimi, H.R.: A computational method for optimal control problem of time-varying state-delayed systems by Haar wavelets. Int. J. Comput. Math. 83(2), 235–246 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hsiao, C.H., Wang, W.J.: State analysis of time-varying singular bilinear systems via Haar wavelets. Math. Comput. Simul. 52, 11–20 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hsiao, C.H., Wang, W.J.: State analysis and parameter estimation of bilinear systems via Haar Wavelets. IEEE Trans. Circ. Syst. I Fund. Theory Appl. 47(2), 246–250 (2000)CrossRefGoogle Scholar
  28. 28.
    Hsiao, C.H., Wang, W.J.: Haar wavelet approach to non-linear stiff systems. Math. Comput. Simul. 57, 347–353 (2001)CrossRefzbMATHGoogle Scholar
  29. 29.
    Hsiao, C.H.: Haar wavelet approach to linear stiff systems. Math. Comput. Simul. 64, 561–567 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lepik, U.: Haar wavelet method for solving stiff differential equations. Math. Model. Anal. 14(4), 461–481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Karimi, H.R., Maralani, P.J., Moshiri, B., Lohmann, B.: Numerically efficient approximations to the optimal control of linear singularly perturbed systems based on Haar wavelets. Int. J. Comput. Math. 82(4), 495–507 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Karimi, H.R., Moshiri, B., Lohmann, B., Maralani, P.J.: Haar wavelet-based approach for optimal control of second-order linear systems in time domain. J. Dyn. Control Syst. 11(2), 237–252 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Qasem, A.F.: Numerical solution for linear parabolic reaction-double diffusivity system using the operational matrices of the Haar wavelets method. J. Comp. Math. 5(1), 177–195 (2008)Google Scholar
  34. 34.
    Razzaghi, M., Ordokhani, Y.: An application of rationalized Haar functions for variational problems. Appl. Math. Comput. 122, 353–364 (2001)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Razzaghi, M., Ordokhani, Y.: Solution for a classical problem in the calculus of variations via rationalized Haar functions. Kybernetika 37(5), 575–583 (2001)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hsiao, C.H.: Haar wavelet direct method for solving variational problems. Mathematics and Computers in Simulation 64, 569–585 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lepik, U.: Solution of optimal control problems via Haar wavelets. Int. J. Pure Appl. Math. 55(1), 81–94 (2009)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Dai, R., Cochran, J.E.: Wavelet collocation method for optimal control problems. J. Optim. Theory Appl. 143, 265–278 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Marzban, H.R., Razzaghi, M.: Rationalized Haar approach for non-linear constrained optimal control problems. Appl. Math. Model. 34, 174–183 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. XVI, 909–996 (1988)Google Scholar
  41. 41.
    Ryzhik, L.: Lecture Notes : The Haar functions and the Brownian motion. (2012)
  42. 42.
    Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of \(L^{2}({\mathbb{R}})\). Trans. Am. Math. Soc. 315(1), 69–87 (1989)zbMATHGoogle Scholar
  43. 43.
    Boggess, A., Narcowich, F.J.: A first course of wavelets with Fourier Analysis. Wiley, USA (2009)zbMATHGoogle Scholar
  44. 44.
    Ruch, D.K., Van Fleet, P.J.: Wavelet Theory—an Elementary Approach with Applications. Wiley, USA (2009)CrossRefzbMATHGoogle Scholar
  45. 45.
    Chen, C.F., Hsiao, C.H.: Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl. 144(1), 87–94 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Lepik, U.: Exploring vibrations of cracked beams by the Haar wavelet method. Estonian J. Eng. 18(1), 58–75 (2012)CrossRefGoogle Scholar
  47. 47.
    Hsiao, C.H.: State analysis of linear time delayed systems via Haar wavelets. Math. Comput. Simul. 44, 457–470 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Hariharan, G., Kannan, K.: Haar wavelet method for solving some nonlinear Parabolic equations. J. Math. Chem. 48, 1044–1061 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lepik, U.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Chang, P., Piau, P.: Haar wavelet matrices designation in numerical solution of ordinary differential equations. IAENG Int. J. Appl. Mathe. 38, 3 (2008)Google Scholar
  51. 51.
    Mishra, V., Kaur, H., Mittal, R.C.: Haar wavelet algorithm for solving certain differential, integral and integro-differential equations. Int. J. Appl. Math. Mech. 8(6), 69–82 (2012)Google Scholar
  52. 52.
    Lepik, U., Tamme, E.: Applications of the Haar wavelets for solution of Linear Integral Equations, Dynamical Systems and Applications, Proceedings, Antalya, Turkey (2004), pp. 494–507Google Scholar
  53. 53.
    Sunmonu, A.: Implementation of wavelet solutions to second order differential equations with maple. Appl. Math. Sci. 6(127), 6311–6326 (2012)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  55. 55.
    Kaur, H., Mittal, R.C., Mishra, V.: Haar wavelet quasi-linearization approach for solving Lane Emden Equations. Int. J. Math. Comput. Appl. Res. 2(4), 47–60 (2012)Google Scholar
  56. 56.
    Kaur, H., Mittal, R.C., Mishra, V.: Haar wavelet quasi-linearization approach for solving nonlinear boundary value problems. Am. J. Comput. Math. 1, 176–182 (2011)CrossRefGoogle Scholar
  57. 57.
    Lepik, U., Tamme, E.: Solution of non-linear Fredholm integral equations via the Haar wavelet method. Proc. Estonian Acad. Sci. Phys. Math. 56(4), 17–27 (2007)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Aziz, I., Islam, S.: New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets. J. Comput. Appl. Math. 239, 333–345 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Derili, H., Sohrabi, S., Arzhang, A.: Two-dimensional wavelets for numerical solution of integral equations. Math. Sci. 6(5) (2012)Google Scholar
  60. 60.
    Lepik, U.: Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci Phys. Math 56(1), 28–46 (2007)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Shi, Z., Liu, T., Gao, B.: Haar wavelet method for solving wave equation. In: International Conference on Computer Application and System Modeling, pp. 561–564 (2010)Google Scholar
  62. 62.
    Hariharan, G., Kannan, K.: A comparative study of a Haar wavelet method and a restrictive Taylor’s series method for solving convection-diffusion equations. Int. J. Comput. Methods Eng. Sci. Mech. 11(4), 173–184 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Ismail, H.N.A., Elbarbary, E.M.E., Salem, G.S.E.: Restrictive Taylor as approximation for solving convection diffusion equation. Appl. Math. Comput. 147, 355–363 (2004)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Erlebacher, G., Hussaini, M.Y., Jameson, L.M.: Wavelets: Theory and Application. Oxford University, Oxford (1996)zbMATHGoogle Scholar
  65. 65.
    Hernandez, E., Weiss, G.: A First Course of Wavelets. CRC Press (1996)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  • Aditya Kaushik
    • 1
  • Geetika Gupta
    • 2
  • Manju Sharma
    • 3
    • 4
    Email author
  • Vishal Gupta
    • 5
  1. 1.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia
  2. 2.Department of MathematicsMCM DAV CollegeChandigarhIndia
  3. 3.Department of MathematicsKVA DAV CollegeKarnalIndia
  4. 4.Department of MathematicsPanjab UniversityChandigarhIndia
  5. 5.Department of MathematicsMaharishi Markandeshwar University, MullanaAmbalaIndia

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