Skip to main content
SpringerLink
Log in

A novel operational vector for solving the general form of distributed order fractional differential equations in the time domain based on the second kind Chebyshev wavelets

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The main aim of this research study is to present a new and efficient numerical method based on the second kind Chebyshev wavelets for solving the general form of distributed order fractional differential equations in the time domain with the Caputo fractional derivatives. For the first time, based on the second kind Chebyshev wavelets, an exact formula for the operational vector of the Riemann-Liouville fractional integral operator is derived by using the unit step function and Laplace transform method. Applying this operational vector via the collocation method in our approach provides an approximate solution by converting the problem under consideration into a system of algebraic equations which can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, five test problems are considered to confirm the reliability and effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Abd-Elhameed, W.M., Doha, E.H., Youssri, Y.H.: New spectral second kind Chebyshev wavelets algorithm for solving linear and nonlinear second-order differential equations involving singular and Bratu type equations. Abstr. Appl. Anal. 2013, 1–10 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Amin, R., Shah, K., Asif, M., Khan, I., Ullah, F.: An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet. J. Comput. Appl. Math. 381, 113028 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanackovic, T.M.: A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech. 159, 77–86 (2002)

    Article  MATH  Google Scholar 

  4. Atanackovic, T.M., Budincevic, M., Pilipovic, S.: On a fractional distributed order oscillator. J. Phys. A, Math. Gen. 38(30), 6703–6713 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atanackovic, T.M., Pilipovic, S., Zorica, D.: Time distributed order diffusion-wave equation, II. Applications of the Laplace and Fourier transformations. Proc. R. Soc. A 465, 1893–1917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atanackovic, T.M., Pilipovic, S., Zorica, D.: Distributed order fractional wave equation on a finite domain. Stress relaxation in a rod. Int. J. Eng. Sci 49(2), 175–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Atanackovic, T.M., Pilipovic, S., Zorica, D.: Distributed-order fractional wave equation on a finite domain: creep and forced oscillations of a rod. Contin. Mech. Thermodyn. 23(4), 305–318 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations-part i. Int. J. Appl. Math. 2, 865–882 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations-part II. Int. J. Appl. Math. 2, 965–988 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44, 141–183 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bisheh-Niasar, M., Saadatmandi, A., Akrami-Arani, M.: A new family of high-order difference schemes for the solution of second order boundary value problems. Iran. J. Math. Chem. 9(3), 187–199 (2018)

    MATH  Google Scholar 

  12. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2006)

    Book  MATH  Google Scholar 

  13. Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar, V.Y.: Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6, 259–279 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Chui, C.K., Wavelets, A: Mathematical Tool for Signal Analysis. SIAM, Philadelphia (1997)

    Google Scholar 

  15. Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equation. J. Comput. Appl. Math. 225(1), 96–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Esmaeilbeigi, M., Mirzaee, F., Moazami, D.: Radial basis functions method for solving three-dimensional linear Fredholm integral equations on the cubic domains. Iran. J. Numer. Anal. Optim. 7(2), 15–38 (2017)

    MATH  Google Scholar 

  17. Esmaeili, S.H., Shamsi, M., Luchkob, Y.: Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput. Math. Appl. 62, 918–929 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gu, J.S., Jiang, W.S.: The Haar wavelets operational matrix of integration. Int. J. Syst. Sci. 27, 623–628 (1996)

    Article  MATH  Google Scholar 

  19. Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed Order Dynamic System Stability, Simulation and Perspective. Springer, London (2012)

    Book  MATH  Google Scholar 

  20. Jibenja, N., Yuttanan, B., Razzaghi, M.: An efficient method for numerical solutions of distributed order fractional differential equations. J. Comput. Nonlinear Dynam. 13, 1–11 (2018)

    Article  Google Scholar 

  21. Katsikadelis, J.T.: Fractional distributed order oscillator: a numerical solution. J. Serb. Soc. Comput. Mech. 6, 148–159 (2012)

    Google Scholar 

  22. Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khalil, H., Shah, K., Khan, R.A.: Approximate solution of boundary value problems using shifted Legendre polynomials. Appl. Comput. Math. 16 (3), 1–15 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov-Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39(3), A1003–A1037 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, J., Liu, F., Feng, L., Turner, I.: A novel finite volume method for the Riesz space distributed-order diffusion equation. Comput. Math. Appl. 74, 772–783 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Y., Sheng, H., Chen, Y.Q.: On distributed order integrator/differentiator. Signal Process 91, 1079–1084 (2011)

    Article  MATH  Google Scholar 

  27. Mainardi, F., Mura, A., Gorenflo, R., Stojanovic, M.: The two form of fractional relaxation of distributed order. J. Vib. Control 13, 1249–1268 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Maleknejad, K., Rashidinia, J., Eftekhari, T.: A new and efficient numerical method based on shifted fractional-order Jacobi operational matrices for solving some classes of two-dimensional nonlinear fractional integral equations, Numerical Methods for Partial Differential Equations (2021). https://doi.org/10.1002/num.22762

  29. Maleknejad, K., Rashidinia, J., Eftekhari, T.: Numerical solutions of distributed order fractional differential equations in the time domain using the müntz-legendre wavelets approach. Numer. Methods Partial Differ. Equ. 37(1), 707–731 (2021). https://doi.org/10.1002/num.22548

    Article  Google Scholar 

  30. Maleknejad, K., Rashidinia, J., Eftekhari, T.: Existence, uniqueness, and numerical analysis of solutions for some classes of two-dimensional nonlinear fractional integral equations in a Banach space. Comput. Appl. Math. 39(4), 1–22 (2020). https://doi.org/10.1007/s40314-020-01322-4

    Article  MATH  Google Scholar 

  31. Maleknejad, K., Rashidinia, J., Eftekhari, T.: Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations. Comput. Appl. Math. 39(2), 1–34 (2020). https://doi.org/10.1007/s40314-020-1126-8

    Article  MathSciNet  MATH  Google Scholar 

  32. Maleknejad, K., Rashidinia, J., Eftekhari, T.: Numerical solution of three-dimensional Volterra-Fredholm integral equations of the first and second kinds based on Bernstein’s approximation. Appl. Math. Comput. 339, 272–285 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Mashayekhi, S., Razzaghi, M.: Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 315, 169–181 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mashoof, M., Refahi Shekhani, A.H.: Simulating the solution of the distributed order fractional differential equations by block-pulse wavelets, UPB Scientific Bulletin, Series A:. Appl. Math. Phys. 79, 193–206 (2017)

    Google Scholar 

  35. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  36. Ming, Q., Hwang, C., Shih, Y.P.: The computation of wavelet-Galerkin approximation on a bounded interval. Int. J. Numer. Meth. Eng. 39, 2921–2944 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Podlubny, I., Skovranek, T., Vinagre Jara, B.M., Petras, I., Verbitsky, V., Chen, Y.Q.: Matrix Approach to Discrete Fractional Calculus-III: Non-Equidistant Grids, Variable Step Length and Distributed Orders. Phil. Trans. R. Soc. A 371(1990), 20120153 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pourbabaee, M., Saadatmandi, A.: A novel Legendre operational matrix for distributed order fractional differential equations. Appl. Math. Comput. 361, 215–231 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Rahimkhani, P., Ordokhani, Y., Babolian, E.: Mü ntz-legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer Algor. 77, 1283–1305 (2018)

    Article  MATH  Google Scholar 

  40. Rashidinia, J., Eftekhari, T., Maleknejad, K.: Numerical solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations using shifted Jacobi operational matrices via collocation method. J. King Saud Univ. Sci. 33(1), 1–11 (2021). https://doi.org/10.1016/j.jksus.2020.101244

    Article  Google Scholar 

  41. Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. ASME Appl. Mech. Rev. 50, 15–67 (1997)

    Article  Google Scholar 

  42. Saadatmandi, A., Khani, A., Azizi, M.R.: A sinc-Gauss-Jacobi collocation method for solving Volterra’s population growth model with fractional order. Tbil. Math. J. 11(2), 123–137 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Sedaghat, S., Nemati, S., Ordokhani, Y.: Application of the hybrid functions to solve neutral delay functional differential equations. Int. J. Comput. Math. 94(3), 503–514 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. Semary, M.S., Hassan, H.N., Radwan, A.G.: Modified methods for solving two classes of distributed order linear fractional differential equations. Appl. Math. Comput. 323, 106–119 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Su, N., Nelson, P.N., Connor, S.: The distributed-order fractional diffusion-wave equation of groundwater flow: theory and application to pumping and slug tests. J. Hydrol. 529, 1262–1273 (2015)

    Article  Google Scholar 

  46. Trung Duong, P.L., Kwok, E., Lee, M.: Deterministic analysis of distributed order systems using operational matrix. Appl. Math. Model. 40, 1929–1940 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, Y., Fan, Q.: The second kind Chebyshev wavelet method for solving fractional differential equations. Appl. Math. Comput. 218, 8592–8601 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Yuttanan, B., Razzaghi, M.: Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model. https://doi.org/10.1016/j.apm.2019.01.013 (2019)

  49. Zak, M.A., Tenreiro Machado, J.A.: On the formulation and numerical simulation of distributed order fractional optimal control. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Iran University of Science & Technology (IUST), Narmak, Tehran, 16846 13114, Iran

    Jalil Rashidinia, Tahereh Eftekhari & Khosrow Maleknejad

Authors
  1. Jalil Rashidinia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tahereh Eftekhari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khosrow Maleknejad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Rashidinia.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rashidinia, J., Eftekhari, T. & Maleknejad, K. A novel operational vector for solving the general form of distributed order fractional differential equations in the time domain based on the second kind Chebyshev wavelets. Numer Algor 88, 1617–1639 (2021). https://doi.org/10.1007/s11075-021-01088-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01088-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation