136
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A new method of solving the best approximate solution for a nonlinear fractional equation

, &
Pages 1702-1718 | Received 28 Apr 2022, Accepted 02 May 2023, Published online: 12 Jun 2023

References

  • A.H. Bhrawy and D. Baleanu, A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients, Rep. Math. Phys. 72 (2013), pp. 219–233.
  • C. Canuto, A. Quarteroni, M.Y. Hussaini, and T.A. Zang, Spectral Methods Fundamentals in Single Domains, Scientific Computation, Springer-Verlag, Berlin, 2006, pp. 283–287.
  • Z. Chen and W. Jiang, Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind, Appl. Math. Comput. 217 (2011), pp. 7790–7798.
  • Z. Chen, L.B. Wu, and Y.Z. Lin, Exact solution of a class of fractional integro-differential equations with the weakly singular kernel based on a new fractional reproducing kernel space, Math. Methods. Appl. Sci. 41 (2018), pp. 3841–3855.
  • M.G. Cui and Y.Z. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers, 2008.
  • M. Dehghan, M. Abbaszadeh, and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations, Eng. Anal. Bound. Elem. 50 (2015), pp. 412–434.
  • H. Du and Z. Chen, A new reproducing kernel method with higher convergence order for solving a Volterra–Fredholm integral equation, Appl. Math. Lett. 102 (2020), Article ID 106117.
  • S. Hosseinpour, A. Nazemi, and E. Tohidi, Müntz–Legendre spectral collocation method for solving delay fractional optimal control problems, J. Comput. Appl. Math. 351 (2019), pp. 344–363.
  • D.M. Hou, Y.M. Lin, and C.J. Xu, A Müntz-collocation spectral method for weakly singular Volterra integral equation. Available at https://doi.org/10.48550/arXiv.1904.09594
  • W. Jiang and N. Liu, A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model, Appl. Numer. Math. 119 (2017), pp. 18–32.
  • M. Lakestani, B.N. Saray, and M. Dehghan, Numerical solution for the weakly singular Fredholm integro differential equations using Legendre multiwavelets, J. Comput. Appl. Math. 235 (2011), pp. 3291–3303.
  • C. Li, F. Zeng, and F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal. 15 (2012), pp. 383–406.
  • Z.T. Liu and S.J. Lü, Galerkin spectral method for nonlinear time fractional cable equation with smooth and nonsmooth solutions, Appl. Math. Comput. 350 (2019), pp. 32–47.
  • Z.T. Liu, S.J. Lü, and F.W. Liu, Fully discrete spectral methods for solving time fractional nonlinear Sine-Gordon equation with smooth and non-smooth solutions, Appl. Math. Comput. 333 (2018), pp. 442–457.
  • M.A. Lynch, Large amplitude instability in finite difference approximates to the Klein–Gordon equation, Appl. Numer. Math. 31 (1999), pp. 173–182.
  • A. Mardani, M.R. Hooshmandasl, M.H. Heydari, and C. Cattani, A meshless method for solving the time fractional advection–diffusion equation with variable coefficients, Comput. Math. Appl. 75 (2018), pp. 122–133.
  • A. Mohebbi, M. Abbaszadeh, and M. Dehghan, A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys. 240 (2013), pp. 36–48.
  • A.M. Nagy, Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc-Chebyshev collocation method, Appl. Math. Comput. 310 (2017), pp. 139–148.
  • D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, in Proceedings of the 1968 23rd ACM National Conference, ACM, pp. 517–524.
  • L. Shi, Z. Chen, X.H. Ding, and Q. Ma, A new stable collocation method for solving a class of nonlinear fractional delay differential equations, Mathematical Methods in the Applied Sciences, 2018, pp. 1–16.
  • A.M. Wazwaz, New travelling wave solutions to the Boussinesq and Klein–Gordon equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 889–901.
  • L.L. Wei, Analysis of a new finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Math. Comput. 304 (2017), pp. 180–189.
  • L.L. Wei and Y.N. He, Analysis of the fractional Kawahara equation using an implicit fully discrete local discontinuous Galerkin method, Numer. Methods Partial Differ. Equ. 29 (2013), pp. 1441–1458.
  • L.L. Wei and Y.N. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model. 38 (2014), pp. 1511–1522.
  • Q. Xu and J.S. Hesthaven, Stable multi-domain spectral penalty methods for fractional partial differential equations, J. Comput. Phys. 257 (2014), pp. 241–258.
  • Y. Yang, Y.P. Chen, Yunqing Huang, and Huayi Wei, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, Comput. Math. Appl. 73 (2017), pp. 1218–1232.
  • X.G. Zhang and H. Du, A generalized collocation method in reproducing kernel space for solving a weakly singular Fredholm integro-differential equations, Appl. Numer. Math. 156 (2020), pp. 158–173.
  • M. Zheng, F. Liu, I. Turner, and V. Anh, A novel high order space–time spectral method for the time-fractional Fokker–Planck equation, SIAM J. Sci. Comput. 37 (2015), pp. A701–A724.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.