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Generalized Gegenbauer–Humbert wavelets for solving fractional partial differential equations

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Abstract

This article develops a method based on the generalized Gegenbauer–Humbert wavelets in concert with their operational matrices of fractional integration to deal with the fractional partial differential equations and find the approximate solutions of it. The goal is to show that the proposed method is appropriate for boundary and initial-boundary problems even though it is generalized form. The convergence of the method under study is investigated. The numerical results gained by the proposed method are considered and compared with other methods, to establish the effectiveness and accuracy.

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References

  1. Lederman C, Roquejoffre JM, Wolanski N (2004) Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames. Ann Di Mat 183:173–239

    Article  MathSciNet  MATH  Google Scholar 

  2. Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. Fract Fract Calc Contin Mech, pp 291–384

  3. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  4. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier Science B.V, Amsterdam

    MATH  Google Scholar 

  5. Bohannan G (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14:1487–1498

    Article  MathSciNet  Google Scholar 

  6. Jiang Y, Ma J (2011) High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math 235:3285–3290

    Article  MathSciNet  MATH  Google Scholar 

  7. Momani S, Al-Khaled K (2005) Numerical solutions for systems of fractional differential equations by the decomposition method. Appl Math Comput 162:1351–1365

    Article  MathSciNet  MATH  Google Scholar 

  8. Kumar D, Singh J, Kumar S (2015) Numerical computation of fractional multi- dimensional diffusion equations by using a modified homotopy perturbation method. J Assoc Arab Univ Basic Appl Sci 17:20–26

    Google Scholar 

  9. Luchko Y, Gorenflo R (1999) An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam 24:207–233

    MathSciNet  MATH  Google Scholar 

  10. Kazem S (2013) Exact solution of some linear fractional differential equations by Laplace transform. Int J Nonlinear Sci 16:3–11

    MathSciNet  MATH  Google Scholar 

  11. Yang AM, Zhang YZ, Long Y (2013) Fourier transforms to heat-conduction in a semi-infinite fractal bar. Therm Sci 17(3):707–713

    Article  Google Scholar 

  12. Razzaghi M, Yousefi S (2001) Legendre wavelets operational matrix of integration. Int J Syst Sci 32(4):495–502

    Article  MathSciNet  MATH  Google Scholar 

  13. Mittal RC, Pandit S (2019) New ccale-3 Haar wavelets algorithm for numerical simulation of second order ordinary differential equations. Proc Natl Acad Sci India Sect A Phys Sci 89:799–808

    Article  Google Scholar 

  14. Pandit S, Sharma S (2021) Sensitivity analysis of emerging parameters in the presence of thermal radiation on magnetohydrodynamic nanofluids via wavelets. Eng Comput, pp 1–10

  15. Pandit S, Sharma S (2020) Wavelet strategy for flow and heat transfer in CNT-water based fluid with asymmetric variable rectangular porous channel. Eng Comput, pp 1–11

  16. Neamaty A, Agheli B, Darzi R (2013) Solving fractional partial differential equation by using wavelet operational method. J Math Comput Sci 7:230–240

    Article  Google Scholar 

  17. Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput Phys Commun 188:59–67

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiwari R (2012) A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput Phys Commun 183:2413–2423

    Article  MathSciNet  MATH  Google Scholar 

  19. Rahimkhani P, Ordokhani Y, Babolian E (2017) Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J Comput Appl Math 309:493–510

    Article  MathSciNet  MATH  Google Scholar 

  20. Rahimkhani P, Ordokhani Y (2019) A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions. Numer Methods Partial Differ Equ 35:34–59

    Article  MathSciNet  MATH  Google Scholar 

  21. Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl Math Comput 234:267–276

    Article  MathSciNet  MATH  Google Scholar 

  22. Chohan MI, Shah K (2019) On a computational method for non-integer order partial differential equations in two dimensions. Eur J Pure Appl Math 12:39–57

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Firoozjaee MA, Yousefi SA (2018) A numerical approach for fractional partial differential equations by using Ritz approximation. Appl Math Comput 338:711–721

    Article  MathSciNet  MATH  Google Scholar 

  25. Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216:2276–2285

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang L, Ma Y, Meng Z (2014) Haar wavelet method for solving fractional partial differential equations numerically. Appl Math Comput 227:66–76

    Article  MathSciNet  MATH  Google Scholar 

  27. Rehman MU, Khan RA (2011) The Legendre wavelet method for solving fractional differential equations. Commun Nonlinear Sci 16:4163–4173

    Article  MathSciNet  MATH  Google Scholar 

  28. Heydari MH, Hooshmandasl MR, Ghaini FMM, Cattani C (2015) Wavelets method for the time fractional diffusion-wave equation. Phys Lett A 379:71–76

    Article  MathSciNet  MATH  Google Scholar 

  29. Keshavarz E, Ordokhani Y, Razzaghi M (2014) Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Model 38:6038–6051

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Alkhalissi JHS, Emiroğlu I, Seçer A, Bayram M (2020) The generalized Gegenbauer-Humberts wavelet for solving fractional differential equations. In: Society of Thermal Engineers of Serbia

  32. Saeed U (2015) Wavelet quasilinearization methods for fractional differential equations. School of Natural Sciences, National University of Sciences and Technology Pakistan, Karachi

    Google Scholar 

  33. Rahimkhani P, Ordokhani Y, Babolian E (2016) Fractional-order Bernoulli wavelets and their applications. Appl Math Model 40:8087–8107

    Article  MathSciNet  MATH  Google Scholar 

  34. Nemati A, Yousefi SA (2017) A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix. IMA J Math Control Inf 34:1079–1097

    MathSciNet  MATH  Google Scholar 

  35. Tian-Xiao He (2011) Characterizations of orthogonal generalized Gegenbauer-Humbert polynomials and orthogonal Sheffer-type polynomial. J Comput Anal Appl 13(4):701–723

    MathSciNet  MATH  Google Scholar 

  36. Srivastava HM, Khan WA, Haroon H (2019) Some expansions for a class of generalized Humbert matrix polynomials. RACSAM 113:3619–3634

    Article  MathSciNet  MATH  Google Scholar 

  37. Kilicman A, Al Zhour ZAA (2007) Kronecker operational matrices for fractional calculus and some applications. Appl Math Comput 187:250–265

    Article  MathSciNet  MATH  Google Scholar 

  38. Singh H, Singh CS (2018) Stable numerical solutions of fractional partial differential equations using Legendre scaling functions operational matrix. Ain Shams Eng J 9:717–725

    Article  Google Scholar 

  39. Izadkhah MM, Saberi-Nadjafi J (2014) Gegenbauer spectral method for time-fractional convection-diffusion equations with variable coefficients. Math Methods Appl Sci 38:3183–3194

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey

    Jumana H. S. Alkhalissi, Ibrahim Emiroglu, Aydin Secer & Fatih Tasci

  2. Department of Computer Engineering, Biruni University, Istanbul, Turkey

    Mustafa Bayram

Authors
  1. Jumana H. S. Alkhalissi
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  2. Ibrahim Emiroglu
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  3. Mustafa Bayram
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  4. Aydin Secer
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  5. Fatih Tasci
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Correspondence to Mustafa Bayram.

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Alkhalissi, J.H.S., Emiroglu, I., Bayram, M. et al. Generalized Gegenbauer–Humbert wavelets for solving fractional partial differential equations. Engineering with Computers 39, 1363–1374 (2023). https://doi.org/10.1007/s00366-021-01532-2

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  • DOI: https://doi.org/10.1007/s00366-021-01532-2

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