Jump to ContentJump to Main Navigation

International Journal of Applied Mathematics and Computer Science

The Journal of University of Zielona Gora and Lubuskie Scientific Society

4 Issues per year

Over 100% increased IMPACT FACTOR 2012: 1.008
5-year IMPACT FACTOR: 1.146
Mathematical Citation Quotient 2012: 0.11

Open Access


Open Access

An operational Haar wavelet method for solving fractional Volterra integral equations

Habibollah Saeedi1, / Nasibeh Mollahasani1, / Mahmoud Moghadam1 / Gennady Chuev1,

Department of Mathematics, Faculty of Mathematics and Computer Science, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran1

Max Planck Institute for Mathematics in the Sciences (MIS), Inselstrasse 22, Leipzig 04103, Germany2

Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran3

Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region 142290, Russia4

This content is open access.
(CC BY-NC-ND 4.0)

Citation Information: International Journal of Applied Mathematics and Computer Science. Volume 21, Issue 3, Pages 535–547, ISSN (Print) 1641-876X, DOI: 10.2478/v10006-011-0042-x, September 2011

Publication History:
Published Online:

An operational Haar wavelet method for solving fractional Volterra integral equations

A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

Keywords: fractional Volterra integral equation; Abel integral equation; fractional calculus; Haar wavelet method; operational matrices

  • Abdalkhania, J. (1990). Numerical approach to the solution of Abel integral equations of the second kind with nonsmooth solution, Journal of Computational and Applied Mathematics 29(3): 249-255.[CrossRef]

  • Akansu, A.N. and Haddad, R.A. (1981). Multiresolution Signal Decomposition, Academic Press Inc., San Diego, CA.

  • Bagley, R.L. and Torvik, P.J. (1985). Fractional calculus in the transient analysis of viscoelastically damped structures, American Institute of Aeronautics and Astronautics Journal 23(6): 918-925.

  • Baillie, R.T. (1996). Long memory processes and fractional integration in econometrics, Journal of Econometrics 73(1): 5-59.

  • Baratella, P. and Orsi, A.P. (2004). New approach to the numerical solution of weakly singular Volterra integral equations, Journal of Computational and Applied Mathematics 163(2): 401-418.

  • Brunner, H. (1984). The numerical solution of integral equations with weakly singular kernels, in D.F. GriMths (Ed.), Numerical Analysis, Lecture Notes in Mathematics, Vol. 1066, Springer, Berlin, pp. 50-71.

  • Chen, C.F. and Hsiao, C.H. (1997). Haar wavelet method for solving lumped and distributed parameter systems, IEE Proceedings: Control Theory and Applications 144(1): 87-94.

  • Chena, W., Suna, H., Zhang, X. and Korôsak, D. (2010). Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications 59(5): 265-274.[Web of Science]

  • Chiodo, S., Chuev, G.N., Erofeeva, S.E., Fedorov, M.V., Russo, N. and Sicilia, E. (2007). Comparative study of electrostatic solvent response by RISM and PCM methods, International Journal of Quantum Chemistry 107: 265-274.[Web of Science]

  • Chow, T.S. (2005). Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Physics Letters A 342(1-2): 148-155.

  • Chuev, G.N., Fedorov, M.V. and Crain, J. (2007). Improved estimates for hydration free energy obtained by the reference interaction site model, Chemical Physics Letters 448: 198-202.[Web of Science]

  • Chuev, G.N., Fedorov, M.V., Chiodo, S., Russo, N. and Sicilia, E. (2008). Hydration of ionic species studied by the reference interaction site model with a repulsive bridge correction, Journal of Computational Chemistry 29(14): 2406-2415.[PubMed] [Web of Science] [CrossRef]

  • Chuev, G.N., Chiodo, S., Fedorov, M.V., Russo, N. and Sicilia, E. (2006). Quasilinear RISM-SCF approach for computing solvation free energy of molecular ions, Chemical Physics Letters 418: 485-489.

  • Dixon, J. (1985). On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with non-smooth solution, BIT 25(4): 624-634.

  • Hsiao, C.H. and Wu, S.P. (2007). Numerical solution of timevarying functional differential equations via Haar wavelets, Applied Mathematics and Computation 188(1): 1049-1058.[Web of Science]

  • Lepik, Ü. and Tamme, E. (2004). Application of the Haar wavelets for solution of linear integral equations, Dynamical Systems and Applications, Proceedings, Antalya, Turkey, pp. 494-507.

  • Lepik, Ü. (2009). Solving fractional integral equations by the Haar wavelet method, Applied Mathematics and Computation 214(2): 468-478.

  • Li, C. and Wang, Y. (2009). Numerical algorithm based on Adomian decomposition for fractional differential equations, Computers & Mathematics with Applications 57(10): 1672-1681.[CrossRef]

  • Magin, R.L. (2004). Fractional calculus in bioengineering. Part 2, Critical Reviews in Bioengineering 32: 105-193.

  • Mainardi, F. (1997). Fractional calculus: ‘Some basic problems in continuum and statistical mechanics’, in A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, NY.

  • Mandelbrot, B. (1967). Some noises with 1/f spectrum, a bridge between direct current and white noise, IEEE Transactions on Information Theory 13: 289-298.

  • Maleknejad, K. and Mirzaee, F. (2005). Using rationalized Haar wavelet for solving linear integral equations, Applied Mathematics and Computation 160(2): 579-587.

  • Meral, F.C., Royston, T.J. and Magin, R. (2010). Fractional calculus in viscoelasticity: An experimental study, Communications in Nonlinear Science and Numerical Simulation 15(4): 939-945.[Web of Science] [CrossRef]

  • Metzler, R. and Nonnenmacher, T.F. (2003). Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials, International Journal of Plasticity 19(7): 941-959.[CrossRef]

  • Miller, K. and Feldstein, A. (1971). Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM Journal on Mathematical Analysis 2: 242-258.[CrossRef]

  • Miller, K. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, NY.

  • Pandey, R.K., Singh, O.P. and Singh, V.K. (2009). Efficient algorithms to solve singular integral equations of Abel type, Computers and Mathematics with Applications 57(4): 664-676.

  • Podlubny, I. (1999). Fractional Differential Equations, Academic Press, New York, NY.

  • Strang, G. (1989). Wavelets and dilation equations, SIAM Review 31(4): 614-627.[CrossRef]

  • Vainikko, G. and Pedas, A. (1981). The properties of solutions of weakly singular integral equations, Journal of the AustralianMathematical Society, Series B: AppliedMathematics 22: 419-430.

  • Vetterli, M. and Kovacevic, J. (1995). Wavelets and Subband Coding, Prentice Hall, Englewood Cliffs, NJ.

  • Yousefi, S.A. (2006). Numerical solution of Abel's integral equation by using Legendre wavelets, Applied Mathematics and Computation 175(1): 574-580.

  • Zaman, K.B.M.Q. and Yu, J.C. (1995). Power spectral density of subsonic jetnoise, Journal of Sound and Vibration 98(4): 519-537.

Comments (0)

Please log in or register to comment.