Abstract
Owing to increasing applications of the fractional relaxation–oscillation equations across various scientific endeavours, a considerable amount of attention has been paid for solving these equations. Our endeavour is to develop an elegant numerical scheme based on Gegenbauer wavelets for solving the fractional-order relaxation–oscillation equations. To facilitate the narrative, the Gegenbauer wavelets are presented and the corresponding operational matrix of fractional-order integration is constructed via the block pulse functions. The prime features of the Gegenbauer wavelets and block pulse functions are then utilized to reduce the system at hand into a set of algebraic equations, solved by means of Newton method. The efficiency and accuracy of the proposed numerical scheme are demonstrated via several illustrative examples.
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References
Samadyar, N., Ordokhani, Y., Mirzaee, F.: Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion. Commun. Nonlinear Sci. Numer. Simul. 90, 105346 (2020)
Samadyar, N., Mirzaee, F.: Numerical scheme for solving singular fractional partial integro-differential equation via orthonormal Bernoulli polynomials. Int. J. Numer. Model. 32, e2652 (2019)
Mirzaee, F., Samadyar, N.: Implicit meshless method to solve 2D fractional stochastic Tricomi-type equation defined on irregular domain occurring in fractal transonic flow. Numer. Meth. Partial Diff. Eq. 37, 1781–1799 (2021)
Mirzaee, F., Samadyar, Nasrin: Application of hat basis functions for solving two-dimensional stochastic fractional integral equations. Comp. Appl. Math. 4(37), 4899–4916 (2018)
Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. ASME J. Appl. Mech. 51(2), 294–308 (1984)
Beyer, H., Kempfle, S.: Definition of physically consistent damping laws with fractional derivatives. Zeitsch. für Angewandte Math. Mechanik. 75(8), 623–635 (1995)
Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent. II. J. Roy. Austral. Soc. 13, 529–539 (1967)
Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solit. Fract. 7(9), 1461–1477 (1996)
Tofighi, A.: The intrinsic damping of the fractional oscillator. Phys. A. 329, 29–34 (2003)
Chen, W., Zhang, X., Korošak, D.: Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simulat. 11, 3–9 (2010)
Anjara, F., Solofoniaina, J.: Solution of general fractional oscillation relaxation equation by Adomians method. Gen. Math. Notes. 20(2), 1–11 (2014)
Yıldırıma, A., Momanib, S.: Series solutions of a fractional oscillator by means of the homotopy perturbation method. Int. J. Comput. Math. 87(5), 1072–1082 (2010)
Al-rabtah, A., Ertürk, V.S., Momani, S.: Solutions of a fractional oscillator by using differential transform method. Comput. Math. Appl. 59, 1356–1362 (2010)
Odibat, Z., Momani, S.: Application of variational iteration method to equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7, 271–291 (2006)
Yi, M.-X., Huang, J., Wei, J.-X.: Block pulse operational matrix method for solving fractional partial differential equation. Appl. Math. Comput. 221, 121–131 (2013)
Shah, F.A., Abass, R.: Haar wavelet operational matrix method for the numerical solution of fractional-order differential equations. Nonlinear Eng. 4(4), 203–213 (2015)
Shah, F.A., Abass, R.: Generalized wavelet collocation method for solving fractional relaxation-oscillation equation arising in fluid mechanics. Internat. J. Comput. Mater. Sci. Eng. 6(2), 1–17 (2017)
Shah, F.A., Abass, R., Debnath, L.: Numerical solution of fractional differential equations using Haar wavelet collocation method. Internat. J. Appl. Comput. Math. 3, 2423–2445 (2017)
Shah, F.A., Abass, R.: Solution of fractional oscillator equations using ultraspherical wavelets. Int. J. Geomet. Methods Mod. Phys. 6(5), 1950075 (2019)
Li, X.: Numerical solution of fractional differential equations using cubic \(B\)-spline wavelet collocation method. Commun. Nonlinear Sci. Numer. Simulat. 17, 3934–3946 (2012)
Gülsu, M., Öztürk, Y., Anapali, A.: Numerical approach for solving fractional relaxation-oscillation equation. Appl. Mathemat. Modell. 37(8), 5927–5937 (2013)
Hamarsheh, M., Ismail, A., Odibat, Z.: Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation. J. Interpol. App. Sci. Comput. 2, 98–111 (2015)
Srivastava, H.M., Shah, F.A., Irfan, M.: A generalized wavelet quasilinearization method for solving population growth model of fractional order. Math. Meth. Appl. Sci. (2020). https://doi.org/10.1002/mma.6542
Shah, F.A., Irfan, M.: A computational wavelet method for solving dual-phase-lag model of bioheat transfer during hyperthermia treatment. Comput. Math. Methods. (2020). https://doi.org/10.1002/cmm4.1095
Debnath, L., Shah, F.A.: Lectuer Notes on Wavelet Transforms. Birkhäuser, Boston (2017)
Debnath, L., Shah, F.A.: Wavelet Transforms and Their Applications. Birkhäuser, New York (2015)
Lepik, U., Hein, H.: Haar Wavelets with Applications. Springer, New York (2014)
Samadyar, N., Mirzaee, F.: Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itǒ-Volterra integral equations of Abel type. Int. J. Numer. Model. 1(33), e2688 (2020)
Mirzaee, F., Solhi, E., Samadyar, N.: Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra-Fredholm integral equations. Appl. Numer. Math. 161, 275–285 (2021)
Mirzaee, F., Samadyar, N.: Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itŏ-Volterra integral equations. Multi. Model. Mater. Struct. 3(15), 575–598 (2019)
Mirzaee, F., Samadyar, N.: On the numerical solution of stochastic quadratic integral equations via operational matrix method. Math. Meth. Appl. Sci. 41, 4465–4479 (2018)
Ozdemir, N., Secer, A., Bayram, M.: The Gegenbauer wavelets-based computational methods for the coupled system of Burgers’ equations with time-fractional derivative. Mathematics 7, 486 (2019). https://doi.org/10.3390/math7060486
Secer, A., Ozdemir, N.: An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation. Adv. Difference Eqs. (2019). https://doi.org/10.1186/s13662-019-2297-8
Kumar, S., Pandey, P., Das, S.: Gegenbauer wavelet operational matrixmethod for solving variable-order non-linear reaction-diffusion and Galilei invariant advection-diffusion equations. Comput. Appl. Math. 88, 162 (2019). https://doi.org/10.1007/s40314-019-0952-z
Elhameed, W.M., Youssri, Y.H.: New ultraspherical wavelets spectral solutions for fractional Riccati differential equations. Abstr. Appl. Anal. 2014, 8 (2014)
Rehman, M., Saeed, U.: Gegenbauer wavelets operational matrix method for fractional differential equations. J. Korean Math. Soc. 52(5), 1069–1096 (2015)
Celik, I.: Generalization of Gegenbauer wavelet collocation method to the Ggeneralized Kuramoto-Sivashinsky equation. Int. J. Appl. Comput. Math. 4, 111 (2018). https://doi.org/10.1007/s40819-018-0546-2
Celik, I.: Gegenbauer wavelet collocation method for the extended Fisher-Kolmogorov equation in two dimensions. Math. Meth. Appl. Sci. 16, 1–14 (2020). https://doi.org/10.1002/mma.6300
Srivastava, H.M., Shah, F.A., Abass, R.: An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation. Russian J. Math. Phys. 26(1), 77–93 (2019)
Lizorkin, P.I.: Fractional integration and differentiation. Encyclopedia of Mathematics EMS Press, (2001)
Miller, K.S., Kenneth, S., Bertram, R.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, Hoboken (1993)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies. Elsevier (North-Holland) Science Publishers, Amsterdam (2006)
Mirzaee, F., Alipour, S., Samadyar, N.: Numerical solution based on hybrid of block-pulse and parabolic functions for solving a system of nonlinear stochastic Itŏ-Volterra integral equations of fractional order. J. Comput. Appl. Math. 349, 157–171 (2019)
Chen, C.F., Hsiao, C.H.: Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl. 144, 87–94 (1997)
Kai, D., Guido, W.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997)
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Nisar, K.S., Shah, F.A. A numerical scheme based on Gegenbauer wavelets for solving a class of relaxation–oscillation equations of fractional order. Math Sci 17, 233–245 (2023). https://doi.org/10.1007/s40096-022-00465-1
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DOI: https://doi.org/10.1007/s40096-022-00465-1
Keywords
- Gegenbauer wavelet
- Gegenbauer polynomial
- Relaxation–oscillation equation
- Operational matrices
- Block pulse functions