An operational Haar wavelet method for solving fractional Volterra integral equations

Habibollah Saeedi; Nasibeh Mollahasani; Mahmoud Mohseni Moghadam; Gennady N. Chuev

International Journal of Applied Mathematics and Computer Science (2011)

  • Volume: 21, Issue: 3, page 535-547
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Habibollah Saeedi, et al. "An operational Haar wavelet method for solving fractional Volterra integral equations." International Journal of Applied Mathematics and Computer Science 21.3 (2011): 535-547. <http://eudml.org/doc/208068>.

@article{HabibollahSaeedi2011,
abstract = {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.},
author = {Habibollah Saeedi, Nasibeh Mollahasani, Mahmoud Mohseni Moghadam, Gennady N. Chuev},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {fractional Volterra integral equation; Abel integral equation; fractional calculus; Haar wavelet method; operational matrices; error bound; numerical examples},
language = {eng},
number = {3},
pages = {535-547},
title = {An operational Haar wavelet method for solving fractional Volterra integral equations},
url = {http://eudml.org/doc/208068},
volume = {21},
year = {2011},
}

TY - JOUR
AU - Habibollah Saeedi
AU - Nasibeh Mollahasani
AU - Mahmoud Mohseni Moghadam
AU - Gennady N. Chuev
TI - An operational Haar wavelet method for solving fractional Volterra integral equations
JO - International Journal of Applied Mathematics and Computer Science
PY - 2011
VL - 21
IS - 3
SP - 535
EP - 547
AB - 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.
LA - eng
KW - fractional Volterra integral equation; Abel integral equation; fractional calculus; Haar wavelet method; operational matrices; error bound; numerical examples
UR - http://eudml.org/doc/208068
ER -

References

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  1. 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. 
  2. Akansu, A.N. and Haddad, R.A. (1981). Multiresolution Signal Decomposition, Academic Press Inc., San Diego, CA. Zbl0947.94001
  3. 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. Zbl0562.73071
  4. Baillie, R.T. (1996). Long memory processes and fractional integration in econometrics, Journal of Econometrics 73(1): 5-59. Zbl0854.62099
  5. 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. Zbl1038.65144
  6. 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. Zbl0543.65090
  7. 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. Zbl0880.93014
  8. Chena, W., Suna, H., Zhang, X. and Korŏsak, D. (2010). Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications 59(5): 265-274. 
  9. 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. 
  10. Chow, T.S. (2005). Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Physics Letters A 342(1-2): 148-155. 
  11. 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. 
  12. 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. 
  13. 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. 
  14. 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. Zbl0584.65091
  15. 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. Zbl1118.65077
  16. 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. 
  17. Lepik, Ü. (2009). Solving fractional integral equations by the Haar wavelet method, Applied Mathematics and Computation 214(2): 468-478. Zbl1170.65106
  18. Li, C. and Wang, Y. (2009). Numerical algorithm based on Adomian decomposition for fractional differential equations, Computers & Mathematics with Applications 57(10): 1672-1681. Zbl1186.65110
  19. Magin, R.L. (2004). Fractional calculus in bioengineering. Part 2, Critical Reviews in Bioengineering 32: 105-193. 
  20. 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. Zbl0917.73004
  21. 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. Zbl0148.40507
  22. Maleknejad, K. and Mirzaee, F. (2005). Using rationalized Haar wavelet for solving linear integral equations, Applied Mathematics and Computation 160(2): 579-587. Zbl1067.65150
  23. 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. Zbl1221.74012
  24. 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. Zbl1090.74673
  25. 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. Zbl0217.15602
  26. Miller, K. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, NY. Zbl0789.26002
  27. 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. Zbl1165.45303
  28. Podlubny, I. (1999). Fractional Differential Equations, Academic Press, New York, NY. Zbl0924.34008
  29. Strang, G. (1989). Wavelets and dilation equations, SIAM Review 31(4): 614-627. Zbl0683.42030
  30. Vainikko, G. and Pedas, A. (1981). The properties of solutions of weakly singular integral equations, Journal of the Australian Mathematical Society, Series B: Applied Mathematics 22: 419-430. Zbl0475.65085
  31. Vetterli, M. and Kovacevic, J. (1995). Wavelets and Subband Coding, Prentice Hall, Englewood Cliffs, NJ. Zbl0885.94002
  32. Yousefi, S.A. (2006). Numerical solution of Abel's integral equation by using Legendre wavelets, Applied Mathematics and Computation 175(1): 574-580. Zbl1088.65124
  33. 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. 

Citations in EuDML Documents

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  1. M. Kirs, M. Mikola, A. Haavajõe, E. Õunapuu, B. Shvartsman, J. Majak, Haar wavelet method for vibration analysis of nanobeams
  2. Babak Shiri, Sedaghat Shahmorad, Gholamreza Hojjati, Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type
  3. Przemysław Śliwiński, Zygmunt Hasiewicz, Paweł Wachel, A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets
  4. Rafał Stanisławski, Krzysztof J. Latawiec, Normalized finite fractional differences: Computational and accuracy breakthroughs

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