An hp-version spectral collocation method for multi-term nonlinear fractional initial value problems with variable-order fractional derivatives

References

• Z. Ding, A. Xiao, and M. Li, Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients, J. Comput. Appl. Math. 233 (2010), pp. 1905–1914. doi: 10.1016/j.cam.2009.09.027
   Web of Science ®Google Scholar
• A.A. El-Sayed and P. Agarwal, Numerical solution of multiterm variable-order fractional differential equations via shifted legendre polynomials, Math. Meth. Appl. Sci. 42 (2019), pp. 3978–3991. doi: 10.1002/mma.5627
   Web of Science ®Google Scholar
• A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.
   Google Scholar
• C.H. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput. 45 (1985), pp. 463–469. doi: 10.1090/S0025-5718-1985-0804935-7
   Web of Science ®Google Scholar
• C.H. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the first kind, IMA J. Numer. Anal. 7 (1987), pp. 97–106. doi: 10.1093/imanum/7.1.97
   Web of Science ®Google Scholar
• D. Li, H. Liao, W. Sun, J. Wang, and J. Zhang, Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems, Commun. Comput. Phys. 24 (2018), pp. 86–103.
   Web of Science ®Google Scholar
• D. Li, C. Wu, and Z. Zhang, Linearized galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput. 80 (2019), pp. 403–419. doi: 10.1007/s10915-019-00943-0
   Web of Science ®Google Scholar
• M. Li, C. Huang, and N. Wang, Galerkin finite element method for the nonlinear fractional Ginzburg-Landau equation, Appl. Numer. Math. 118 (2017), pp. 131–149. doi: 10.1016/j.apnum.2017.03.003
   Web of Science ®Google Scholar
• X. Li and B. Wu, A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations, J. Comput. Appl. Math. 311 (2017), pp. 387–393. doi: 10.1016/j.cam.2016.08.010
   Web of Science ®Google Scholar
• X. Li, H. Li, and B. Wu, A new numerical method for variable order fractional functional differential equations, Appl. Math. Lett. 68 (2017), pp. 80–86. doi: 10.1016/j.aml.2017.01.001
   Web of Science ®Google Scholar
• H. Liang and M. Stynes, Collocation methods for general Caputo two-point boundary value problems, J. Sci. Comput. 76 (2018), pp. 390–425. doi: 10.1007/s10915-017-0622-5
   Web of Science ®Google Scholar
• H. Liang and M. Stynes, Collocation methods for general Riemann-Liouville twopoint boundary value problems, Adv. Comput. Math. 45 (2019), pp. 897–928. doi: 10.1007/s10444-018-9645-1
   Web of Science ®Google Scholar
• X. Ma and C. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput. 219 (2013), pp. 6750–6760.
   Web of Science ®Google Scholar
• X. Ma and C. Huang, Spectral collocation method for linear fractional integrodifferential equations, Appl. Math. Model. 38 (2014), pp. 1434–1448. doi: 10.1016/j.apm.2013.08.013
   Web of Science ®Google Scholar
• W. McLean, K Mustapha, R. Ali, and O.M. Knio, Regularity theory for time-fractional advection-diffusion-reaction equations, Comput. Math. Appl. 79 (2020), pp. 947–961. doi: 10.1016/j.camwa.2019.08.008
   Web of Science ®Google Scholar
• B.P. Moghaddam and J.A.T Machado, Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput. 71 (2017), pp. 1351–1374. doi: 10.1007/s10915-016-0343-1
   Web of Science ®Google Scholar
• B.P. Moghaddam, J.A.T Machado, and H. Behforooz, An integro quadratic spline approach for a class of variable-order fractional initial value problems, Chaos Solit. Fract. 102 (2017), pp. 354–360. doi: 10.1016/j.chaos.2017.03.065
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
   Google Scholar
• S. Samko, Fractional integration and differentiation of variable order: an overview, Nonlinear Dynam.71 (2013), pp. 653–662. doi: 10.1007/s11071-012-0485-0
   Web of Science ®Google Scholar
• S. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. F 1 (1993), pp. 277–300. doi: 10.1080/10652469308819027
   Google Scholar
• J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin, 2011. Ser Comput Math 41.
   Google Scholar
• C. Sheng, Z. Wang, and B. Guo, A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal. 52 (2014), pp. 1953–1980. doi: 10.1137/130915200
   Web of Science ®Google Scholar
• H. Sun, W. Chen, H. Wei, and Y. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Special Topics 193 (2011), pp. 185–192. doi: 10.1140/epjst/e2011-01390-6
   Web of Science ®Google Scholar
• D. Tavares, R. Almeida, and D. Torres, Caputo derivatives of fractional variable order: numerical approximations, Nonlinear Sci. Numer. Simul. 35 (2016), pp. 69–87. doi: 10.1016/j.cnsns.2015.10.027
   Web of Science ®Google Scholar
• D. Valério and J. Costa, Variable-order fractional derivatives and their numerical approximations, Signal Process. 91 (2011), pp. 470–483. doi: 10.1016/j.sigpro.2010.04.006
   Web of Science ®Google Scholar
• C. Wang, Z. Wang, and H. Jia, An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels, J. Sci. Comput. 72 (2017), pp. 647–678. doi: 10.1007/s10915-017-0373-3
   Web of Science ®Google Scholar
• C. Wang, Z. Wang, and L. Wang, A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput. 76 (2018), pp. 166–188. doi: 10.1007/s10915-017-0616-3
   Web of Science ®Google Scholar
• D. Wang, A. Xiao, and W. Yang, Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput. 257 (2015), pp. 241–251.
   Web of Science ®Google Scholar
• Z. Wang, Y. Guo, and L. Yi, An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Compu.t 86 (2017), pp. 2285–2324. doi: 10.1090/mcom/3183
   Web of Science ®Google Scholar
• M.A. Zaky, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math. 145 (2019), pp. 429–457. doi: 10.1016/j.apnum.2019.05.008
   Web of Science ®Google Scholar
• M.A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math. 357 (2019), pp. 103–122. doi: 10.1016/j.cam.2019.01.046
   Web of Science ®Google Scholar
• M.A. Zaky and I.G. Ameen, On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems, Comp. Appl. Math. 38 (2019), UNSP 144. doi: 10.1007/s40314-019-0922-5
   Google Scholar
• M.A. Zaky, E.H. Doha, and J.A.T. Machado, A spectral framework for fractional variational problems based on fractional Jacobi functions, Appl. Numer. Math. 132 (2018), pp. 51–72. doi: 10.1016/j.apnum.2018.05.009
   Web of Science ®Google Scholar
• S. Zhang, The uniqueness result of solutions to initial value problems of differential equations of variable-order, RACSAM 112 (2018), pp. 407–423. doi: 10.1007/s13398-017-0389-4
   Google Scholar
• T. Zhang and H. Liang, Multistep collocation approximations to solutions of first-kind Volterra integral equations, Appl. Numer. Math. 130 (2018), pp. 171–183. doi: 10.1016/j.apnum.2018.04.005
   Web of Science ®Google Scholar
• Y. Zhao, S. Sun, Z. Han, and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), pp. 1312–1324. doi: 10.1016/j.camwa.2011.03.041
   Google Scholar
• Y. Zhao, S. Sun, Z. Han, and M. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011), pp. 6950–6958.
   Web of Science ®Google Scholar