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A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In the current study, a new technique for solving various kinds of fractional partial differential equations based upon Genocchi hybrid functions (GHFs) is projected. Genocchi hybrid collocation technique by utilizing delay operational matrix, integer and the fractional operational matrix of the derivative is introduced. We calculate the delay operational matrix without error, this matrix is a reason to achieve the approximate solution with high accuracy. These operational matrices are applied to convert the proposed equations to a set of algebraic equations with unknown GHFs coefficients. Also, we investigate the error analysis based on Sobolev space. At last, several examples are examined with graphs and tables to demonstrates the efficiency and accuracy of the numerical plan.

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References

  1. Abazari, R., Ganji, M.: Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay. Int. J. Comput. Math. 88(8), 1749–1762 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Abu Arqub, O., Al-Smadi, M.: Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer. Methods Partial Differ. Equ. 34(5), 1577–1597 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Abu Arqub, O.: Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Partial Differ. Equ. 34(5), 1759–1780 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Al-Smadi, M., Arqub, O.A.: Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates. Appl. Math. Comput. 342, 280–294 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Araci, S., Acikgoz, M., Sen, E.: On the extended Kim’s p-adic q-deformed fermionic integrals in the p-adic integer ring. J. Number Theory 133(10), 3348–3361 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Araci, S.: Novel identities for q-Genocchi Numbers and Polynomials. J. Funct. Space Appl. 2012, Article ID 214961. https://doi.org/10.1155/2012/214961

  7. Araci, S., Acikgoz, M., Bagdasaryan, A., Sen, E.: The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials. arXiv:1312.7838 (2013)

  8. Araci, S.: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 233, 599–607 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Araci, S., Sen, E., Acikgoz, M.: Theorems on Genocchi polynomials of higher order arising from Genocchi basis. Taiwan J. Math. 18(2), 473–482 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Arqub, O.A., Odibat, Z., Al-Smadi, M.: Numerical solutions of time-fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates. Nonlinear Dyn. 94(3), 1819–1834 (2018)

    MATH  Google Scholar 

  11. Aziz, I., Amin, R.: Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl. Math. Model. 40, 10286–10299 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(201), 201–210 (1983)

    MATH  Google Scholar 

  13. Bayad, A., Kim, T.: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(2), 247–53 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Benson, D., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, Levy motion and the made tracer tests. Transp. Porous Media 42(1–2), 211–240 (2001)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  16. Chen, Y., Wu, Y., Cui, Y., Wang, Z., Jin, D.: Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. J. Comput. Sci. 1(3), 146–149 (2010)

    Google Scholar 

  17. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336, 433–453 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Hatano, Y., Hatano, N.: Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resour. Res. 34(5), 1027–1033 (1998)

    Google Scholar 

  19. Hofling, F., Franosch, T.: Anomalous transport in the crowded world of biological cells. Rep. Progr. Phys. 76(4), 046602 (2013)

    MathSciNet  Google Scholar 

  20. Isah, A., Phang, C.: New Operational Matrix of Derivative for Solving Non-linear Fractional Differential Equations via Genocchi Polynomials. Journal of King Saud University-Science, Riyadh (2017)

  21. Isah, A., Phang, C.: Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Eng. 9(4), 2123–2128 (2018)

    Google Scholar 

  22. Isah, A., Phang, C., Phang, P.: Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations. Int. J. Differ. Equ. 2017 (2017)

  23. Karatay, I., Kale, N., Bayramoglu, S.R.: A new difference scheme for time fractional heat equations based on Crank–Nicholson method. Fract. Calc. Appl. Anal. 16(4), 893–910 (2013)

    MATH  Google Scholar 

  24. Keller, A.A.: Contribution of the delay differential equations to the complex economic macrodynamics. WSEAS Trans. Syst. 9(4), 358–371 (2010)

    Google Scholar 

  25. Li, B., Luo, H., Xie, X.: A time-spectral algorithm for fractional wave problems. J. Sci. Comput. 77(2), 1164–1184 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Loh, J.R., Phang, C., Isah, A.: New operational matrix via Genocchi polynomials for solving Fredholm–Volterra fractional integro-differential equations. Adv. Math. Phys. 2017, Article ID 3821870. https://doi.org/10.1155/2017/3821870

  27. Marzban, H.R., Razzaghi, M.: Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series. J. Vib. Control 11, 1455–1468 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Marzban, H.R., Razzaghi, M.: Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series. J. Sound Vib. 292, 954–963 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Marzban, H.R., Razzaghi, M.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions. Math. Probl. Eng. 6, 85–97 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Marzban, H.R., Razzaghi, M.: Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials. J. Franklin Inst. 341, 279–293 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for nonlinear constrained optimal control problems. Commun. Nonlinear. Sci. Numer. Simul. 17, 1831–1843 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for optimal control of systems described by integro-differential equations. Appl. Math. Model. 37, 3355–3368 (2013)

    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)

    MathSciNet  MATH  Google Scholar 

  34. Matar, M.M.: Existence of solution involving Genocchi numbers for nonlocal anti-periodic boundary value problem of arbitrary fractional order. RACSAM 112(4), 945–956 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Phang, C., Ismail, N.F., Isah, A., Loh, J.R.: A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration. J. Vib. Control 24(14), 3036–3048 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Pimenov, V.G., Hendy, A.S.: A Numerical solution for a class of time fractional diffusion equations with delay. Int. J. Appl. Math. Comput. Sci. 27(3), 477–488 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Polyanin, A.D., Zhurov, A.I.: Functional constraints method for constructing exact solutions to delay reaction–diffusion equations and more complex nonlinear equations. Commun. Nonlinear. Sci. Numer. Simul. 19(3), 417–430 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times returns in high frequency financial data: an empirical study. Physica A 314(–4), 749–755 (2002)

    MATH  Google Scholar 

  39. Ren, J., Long, X., Mao, S., Zhang, J.: Super convergence of finite element approximations for the fractional diffusion-wave equation. J. Sci. Comput. 72(3), 917–935 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Saadatmandi, A., Dehghan, M., Azizi, M.R.: The Sinc-Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4125–4136 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Sadeghi Roshan, S., Jafari, H., Baleanu, D.: Solving FDEs with Caputo–Fabrizio derivative by operational matrix based on Genocchi polynomials. Math. Methods Appl. Sci. 41(18), 9134–9141 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Sakar, M.G., Uludag, F., Erdogan, F.: Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl. Math. Model. 40(13–14), 6639–6649 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Sarwar, S., Alkhalaf, S., Iqbal, S., Zahid, M.A.: A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations. Comput. Math. Appl. 70, 942–953 (2015)

    MathSciNet  Google Scholar 

  44. Singh, B.K., Kumar, P.: Fractional variational iteration method for solving fractional partial differential equations with proportional delay. Int. J. Differ. Equ. 2017, Article ID 5206380. https://doi.org/10.1155/2017/5206380

  45. Solodushkin, S.I., Yumanovaa, I.F., De Staelen, R.H.: First order partial differential equations with time delay and retardation of a state variable. J. Comput. Appl. Math. 289, 322–330 (2015)

    MathSciNet  MATH  Google Scholar 

  46. Srivastava, H.M., Kurt, B., Simsek, Y.: Some families of Genocchi type polynomials and their interpolation functions. Integr. Transf. Spec. F 23(12), 919–938 (2012)

    MathSciNet  MATH  Google Scholar 

  47. Tanthanuch, J.: Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4978–4987 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Wang, X.T., Li, Y.M.: Numerical solutions of integro differential systems by hybrid of general blockpulse functions and the second Chebyshev polynomials. Appl. Math. Comput. 209, 266–272 (2009)

    MathSciNet  Google Scholar 

  49. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)

    MATH  Google Scholar 

  50. Zhang, Z.B., Sun, Z.Z.: A linearized compact difference scheme for a class of nonlinear delay partial differential equations. Appl. Math. Model. 37(3), 742–752 (2013)

    MathSciNet  MATH  Google Scholar 

  51. Zhou, F., Xu, X.: The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Appl. Math. Comput. 280, 11–29 (2016)

    MathSciNet  MATH  Google Scholar 

  52. Zubik-Kowal, B.: Chebyshev pseudospectral method and waveform relaxation for differential and differential–functional parabolic equations. Appl. Numer. Math. 34(2–3), 309–328 (2000)

    MathSciNet  MATH  Google Scholar 

  53. Zubik-Kowal, B., Jackiewicz, Z.: Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Appl. Numer. Math. 56(3–4), 433–443 (2006)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We express our sincere thanks to the anonymous referees for valuable suggestions that improved the final manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Haniye Dehestani & Yadollah Ordokhani

  2. Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

    Mohsen Razzaghi

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  1. Haniye Dehestani
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  2. Yadollah Ordokhani
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  3. Mohsen Razzaghi
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Correspondence to Yadollah Ordokhani.

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Dehestani, H., Ordokhani, Y. & Razzaghi, M. A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions. RACSAM 113, 3297–3321 (2019). https://doi.org/10.1007/s13398-019-00694-5

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