ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Haar wavelet fractional derivative; pp. 55–64
PDF | 10.3176/proc.2022.1.05

Author
Carlo Cattani
Abstract

In this paper, the fundamental properties of fractional calculus are discussed with the aim of extending the definition of fractional operators by using wavelets. The Haar wavelet fractional operator is defined, in a more general form, independently on the kernel of the fractional integral.

References

1. Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and applications to heat transfer model. Therm. Sci., 2016, 20(2), 763–769.
https://doi.org/10.2298/TSCI160111018A

2. Bagley, R. On the equivalence of the Riemann-Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fract. Calc. Appl. Anal., 2007, 10(2), 123–126. 

3. Bulut, A., Oruç Ö. and Esen, A. Numerical solutions of fractional system of partial differential equations By Haar wavelets. CMES-Computer Modeling in Engineering & Sciences, 2015, 108(4), 263–284.

4. Caputo, M. Linear model of dissipation whose Q is almost frequency independent – II. Geophys. J. Int., 1967, 13(5), 529–539. 
https://doi.org/10.1111/j.1365-246X.1967.tb02303.x

5. Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl., 2015, 1(2), 73–85.

6. Cattani, C. Harmonic wavelets towards solution of nonlinear PDE. Comput. Math. Appl., 2005, 50(8–9), 1191–1210.
https://doi.org/10.1016/j.camwa.2005.07.001

7. Cattani, C. Connection coefficients of Shannon wavelets. Math. Model. Anal., 2006, 11(2), 117–132.
https://doi.org/10.3846/13926292.2006.9637307

8. Cattani, C. Shannon wavelet analysis. In Proceedings of the 7th International Conference on Computational Science – ICCS 2007. Lecture Notes in Computer Science, 4488, Part II,  Beijing, China, May 27–30, 2007 (Shi, Y., van Albada,  G. D., Dongarra, J. and Sloot, P. M. A., eds).  Springer, Berlin, Heidelberg, 2007, 982–989.

9. Cattani, C. Shannon wavelets theory. Math. Probl. Eng.,  2008, 2008, 164808. 
https://doi.org/10.1155/2008/164808

10. Cattani, C. Shannon wavelets for the solution of integrodifferential equations. Math. Probl. Eng., 2010, 2010, 408418.  
https://doi.org/10.1155/2010/408418

11. Cattani, C. Fractional calculus and Shannon wavelet. Math. Probl. Eng., 2012, 2012, 502812.
https://doi.org/10.1155/2012/502812

12. Cattani, C. Local fractional calculus on Shannon wavelet basis. In Fractional Dynamics (Cattani, C., Srivastava, H. and Yang, X. J., eds). De Gruyter, Krakow, 2015, ch. 1.
https://doi.org/10.1515/9783110472097

13. Cattani, C. and Guariglia, E. Fractional derivative of the Hurwitz ζ-function and chaotic decay to zero. J. King Saud Univ. Sci.,  2016, 28(1), 75–81.
https://doi.org/10.1016/j.jksus.2015.04.003

14. Cattani, C., Srivastava, H. M. and Yang, X.-J. Fractional Dynamics. De Gruyter Open, Warsaw, 2016.
https://doi.org/10.1515/9783110472097

15. Dalir, M. and Bashour, M. Applications of fractional calculus. Appl. Math. Sci., 2010, 4(21), 1021–1032. 

16. Daubechies, I. Ten lectures on waveletsCBMS-NSF Regional Conf. Ser. in Appl. Math. SIAM, Philadelphia, PA, 1992.
https://doi.org/10.1137/1.9781611970104

17. de Oliveira, E. C. and Tenreiro Machado, J. A. A review of definitions for fractional derivatives and integrals. Math. Probl. Eng., 2014, 2014, 238459.
https://doi.org/10.1155/2014/238459

18. Esen, A., Bulut, F. and Oruç, Ö. A unified approach for the numerical solution of time fractional Burgers´ type equations. Eur. Phys. J. Plus, 2016, 131, 116. 
https://doi.org/10.1140/epjp/i2016-16116-5

19. Hein, H. and Feklistova, L. Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets. Eng. Struct., 2011, 33(12), 3696–3701.
https://doi.org/10.1016/j.engstruct.2011.08.006

20. Hein, H. and Feklistova, L. Computationally efficient delamination detection in composite beams using Haar wavelets. Mech. Syst. Signal Process., 2011, 25(6), 2257–2270.
https://doi.org/10.1016/j.ymssp.2011.02.003

21. Kamata, M. and Nakamula, A. Riemann–Liouville integrals of fractional order and extended KP hierarchy. J. Phys. A: Math. Gen., 2002, 35(45), 9657–9670.
https://doi.org/10.1088/0305-4470/35/45/312

22. Lepik, Ü. Numerical solution of differential equations using Haar wavelets. Math.  Comput. Simul., 2005, 68(2), 127–143. 
https://doi.org/10.1016/j.matcom.2004.10.005

23. Lepik, Ü. Haar wavelet method for nonlinear integro-differential equations. Appl. Math. Comput., 2006, 176(1), 324–333.
https://doi.org/10.1016/j.amc.2005.09.021

24. Lepik, Ü. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput., 2007, 185(1), 695–704.
https://doi.org/10.1016/j.amc.2006.07.077

25. Lepik, Ü. Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci.,  2007, 56(1), 28–46.
https://doi.org/10.3176/phys.math.2007.1.03

26. Lepik, Ü. Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput., 2009, 214(2), 468–478.
https://doi.org/10.1016/j.amc.2009.04.015

27. Lepik, Ü. and Hein, H. Haar Wavelets: With Applications. Springer, Berlin, 2014.
https://doi.org/10.1007/978-3-319-04295-4

28. Li, C., Dao, X. and Guo, P. Fractional derivatives in complex planes. Nonlinear Anal. Theory Methods Appl. 2009, 71(5–6), 1857–1869.
https://doi.org/10.1016/j.na.2009.01.021

29. Liu, K., Hu, R.-J., Cattani, C., Xie, G.-N., Yang, X.-J. and Zhao, Y. Local fractional Z-transforms with applications to signals on Cantor sets. Abstr. Appl. Anal., 2014, 2014, 638648.
https://doi.org/10.1155/2014/638648

30. Majak, J., Pohlak, M. and Eerme, M. Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells. Mech. Compos. Mater., 2009, 45(6), 631–642.
https://doi.org/10.1007/s11029-010-9119-0

31. Majak, J., Shvartsman, B. S., Kirs, M., Pohlak, M. and Herranen, H. Convergence theorem for the Haar wavelet based discretization method. Compos. Struct., 2015, 126, 227–232.
https://doi.org/10.1016/j.compstruct.2015.02.050

32. Majak, J., Shvartsman, B. S., Karjust, K., Mikola, M., Haavajõe, A. and Pohlak, M. On the accuracy of the Haar wavelet discretization method. Compos. B. Eng., 2015, 80, 321–327.
https://doi.org/10.1016/j.compositesb.2015.06.008

33. Majak, J., Ratas, M., Karjust, K.  and Shvartsman, B. S. Higher order Haar wavelet method for solving differential equations. In Wavelet Theory (Modammady, S., ed.). IntechOpen, 2020.
https://doi.org/10.5772/intechopen.94520

34. Odibat, Z. M. and Shawagfeh, N. T. Generalized Taylor´s formula. Appl. Math. Comput., 2007, 186(1), 286–293.
https://doi.org/10.1016/j.amc.2006.07.102

35. Ortigueira, M. D. and Tenreiro Machado, J. A. What is a fractional derivative? J. Comput. Phys., 2015, 293, 4–13.
https://doi.org/10.1016/j.jcp.2014.07.019

36. Ortigueira M. D. and Tenreiro Machado, J. A. Which derivative? Fractal Fract., 2017, 1(1), 3.
https://doi.org/10.3390/fractalfract1010003

37. Oruç, Ö., Esen, A. and Bulut, F. A Haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng. Comput., 2019, 35(1), 75–86.
https://doi.org/10.1007/s00366-018-0584-8

38. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York, NY, 1998.

39. Ratas, M., Salupere, A. and Majak, J. Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids. Math. Model. Anal., 2021, 26(1), 147–169.
https://doi.org/10.3846/mma.2021.12920

40. Wang, X. Fractional geometric calculus: toward a unified mathematical language for physics and engineering. In Proceedings of The Fifth Symposium on Fractional Differentiation and its Applications (FDA12)Hohai University, Nanjing, China, May 14–17, 2012.

41. Yang, X.-J., Baleanu, D. and Srivastava, H. M. Local Fractional Integral Transforms and Their Applications.  Academic Press, New York, NY, 2015.
https://doi.org/10.1016/B978-0-12-804002-7.00002-4

42. Yang, X.-J., Gao, F., Terneiro Machado, J. A. and Baleanu, D. A new fractional derivative involving the normalized sinc function without singular kernel. 2017, arXiv:1701.05590.
https://doi.org/10.1140/epjst/e2018-00020-2

43. Zhao, Y., Baleanu, D., Cattani, C., Cheng, D.-F. and Yang, X.-J. Maxwell´s equations on Cantor sets: a local fractional ppproach. Adv. High Energy Phys., 2013, 2013, 686371.
https://doi.org/10.1155/2013/686371
 

Back to Issue