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Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations

  • Xiaoyong Xu EMAIL logo and Fengying Zhou
Published/Copyright: October 4, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 5

Abstract

In the present paper, an efficient method based on a new kind of Chebyshev wavelet together with Picard technique is developed for solving fractional nonlinear differential equations with initial and boundary conditions. The new orthonormal Chebyshev wavelet basis is constructed from a class of orthogonal polynomials called the fifth-kind Chebyshev polynomials. The convergence analysis and error estimation of the proposed Chebyshev wavelet expansion are studied. An exact formula for the Riemann-Liouville fractional integral of the Chebyshev wavelet is derived. Picard iteration is used to convert the fractional nonlinear differential equations into a fractional recurrence relation and then the proposed Chebyshev wavelet collocation method is applied on the converted problem. Several test problems are given to illustrate the performance and effectiveness of the proposed method and compared with the existing work in the literature.

Keywords: Chebyshev wavelet; convergence analysis; fractional differential equations; Picard iteration; Riemann–Liouville fractional integral
Corresponding author: Xiaoyong Xu, School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China, E-mail:

Funding source: Doctoral Scientific Research Foundation of East China University of Technology

Award Identifier / Grant number: DHBK2019213

Funding source: Natural Science Foundation of Jiangxi Province

Award Identifier / Grant number: 2020BABL201006

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 41962019

Acknowledgement

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by the National Natural Science Foundation of China (Grant No.41962019), the Natural Science Foundation of Jiangxi Province of China (Grant No.2020BABL201006) and the Doctoral Scientific Research Foundation of East China University of Technology (Grant No. DHBK2019213).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-11-02
Accepted: 2022-09-18
Published Online: 2022-10-04

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
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  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
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  9. Lie symmetry analysis for two-phase flow with mass transfer
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  11. Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
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https://doi.org/10.1515/ijnsns-2021-0413
Keywords for this article
Chebyshev wavelet; convergence analysis; fractional differential equations; Picard iteration; Riemann–Liouville fractional integral

Articles in the same Issue

  1. Frontmatter
  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
  4. Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
  5. A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
  6. Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
  8. A class of piecewise fractional functional differential equations with impulsive
  9. Lie symmetry analysis for two-phase flow with mass transfer
  10. Asymptotic behavior for stochastic plate equations with memory in unbounded domains
  11. Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
  12. A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
  13. Ground state solutions of Schrödinger system with fractional p-Laplacian
  14. Bifurcation analysis of a new stochastic traffic flow model
  15. An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
  16. Hilfer fractional stochastic evolution equations on infinite interval
  17. Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
  18. Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
  19. Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
  20. Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law
  21. Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1 < r < 2
  22. Shehu transform on time-fractional Schrödinger equations – an analytical approach
  23. A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
  24. Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
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