Abstract
We provide a numerical method to solve a certain class of fractional differential equations involving \(\psi \)-Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce \(\psi \)-shifted Legendre polynomial basis, and we derive an explicit formula for the \(\psi \)-fractional integral of \(\psi \)-shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.
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Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear. Sci. Numer. Simulat. 44, 460–481 (2017)
Almeida, R., Malinowska, A.B., Monteiro, M.T.T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods. Appl. Sci. 41, 336–352 (2018)
Baleanu, D., Bhrawy, A.H., Taha, T.M.: Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. 2013. https://doi.org/10.1155/2013/546502.
Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S.: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 293, 142–156 (2015)
Chen, W., Zhang, X.D., Korosak, D.: Investigation on fractional and fractal derivative relaxation–oscillation models. Int. J. Nonlinear. Sci. Numer. Simulat. 11(1), 3–9 (2010)
Debnath, L., Mikusiński, P.: Introduction to Hilbert Spaces with Applications, 2nd edn. Academic Press, Cambridge (1999)
Dimitrov, Y.: Higher-order numerical solutions of the fractional relaxation–oscillation equation using fractional integration (2016). arXiv:1603.08733
Doha, E.H., Ahmed, H.M., El-Soubhy, S.I.: Explicit formulae for the coefficients of integrated expansions of Laguerre and Hermite polynomials and their integrals. Integral Transform. Spec. Funct. 20, 491–503 (2009)
Doha, E.H., Bhrawy, A.H., Ezz-Eliden, S.S.: A Chebeshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364–2373 (2011)
Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)
Gulsu, M., Ozturk, Y., Anapali, A.: Numerical approach for solving fractional relaxation–oscillation equation. Appl. Math. Model. 37(8), 5927–5937 (2013)
Hamarsheh, M., Ismail, A., Odibat, Z.: Optimal homotopy asymptotic method for solving fractional relaxation–oscillation equation. J. Interpol. Appl. Sci. Comput. 2, 98–111 (2015)
Hwang, C., Shih, Y.: Laguerre operational matrices for fractional calculus and applications. Int. J. Control. 34, 577–584 (1981)
Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012.: Article ID 142 (2012)
Khosravian-Arab, H., Almeida, R.: Numerical solution for fractional variational problems using the Jacobi polynomials. Appl. Math. Model. 39(21), 6461–6470 (2015)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)
Lederman, R.R., Rokhlin, V.: On the analytical and numerical properties of the truncated Laplace transform. Part II. SIAM J. Numer. Anal 54(2), 665–687 (2016)
Luke, Y.: The Special Functions and Their Approximations, vol. 2. Academic Press, New York (1969)
Magin, R., Ortigueira, M.D., Podlubny, I., Trujillo, J.: On the fractional signals and systems. Signal Process. 91, 350–371 (2011)
Mainardi, F.: Fractional relaxation oscillation and fractional diffusion-wave phenomena. Chaos Solit. Fract. 7, 1461–1477 (1996)
Odibat, Z., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)
Paraskevopoulos, P.N.: Chebyshev series approach to system identification, analysis and control. J. Frankl. Inst. 316, 135–157 (1983)
Paraskevopoulos, P.N.: Legendre series approach to identification and analysis of linear systems. IEEE Trans. Autom. Control 30, 585–589 (1985)
Saadatmandi, A., Deghan, M.: A new operational matrix for solving fractional-order differential equation. Comput. Math. Appl. 59, 1326–1336 (2010)
Shah, F.A., Abass, R.: Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics. Int. J. Comput. Math. Sci. Eng. 6(2), 1–17 (2017)
Tofighi, A.: The intrinsic damping of the fractional oscillator. Phys. A 329, 29–34 (2003)
Acknowledgements
Ricardo Almeida was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013. Bessem Samet extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No RGP–237.
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Almeida, R., Jleli, M. & Samet, B. A numerical study of fractional relaxation–oscillation equations involving \(\psi \)-Caputo fractional derivative. RACSAM 113, 1873–1891 (2019). https://doi.org/10.1007/s13398-018-0590-0
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DOI: https://doi.org/10.1007/s13398-018-0590-0
Keywords
- \(\psi \)-Caputo fractional derivative
- \(\psi \)-Shifted
- Fractional relaxation–oscillation equation
- Convergence