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A stable minimal search method for solving multi-order fractional differential equations based on reproducing kernel space

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Abstract

In this paper, a stable minimal search method based on reproducing kernel space is proposed for solving multi-order fractional differential equations. The existence and uniqueness of solution of the considered equation is proved and the smoothness of the solution is studied. Based on orthonormal bases, we give smooth transformation and a method for obtaining the ε-approximate solution by searching the minimum value. Subsequently, error estimation and stability analysis of the method are obtained. The final several numerical experiments are presented to illustrate the correctness of the theory and the effectiveness of the method.

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References

  1. Sahu, P.K., Ray, S.S.: Legendre spectral collocation method for the solution of the model describing biological species living together. J. Comput. Appl. Math. 47-55, 296 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Pandey, R.K., Sharma, S., Kumar, K.: Collocation method for Generalized Abel’s integral equations. J. Comput. Appl. Math. 118-128, 302 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Shan, Y.Y., Liu, W.J., Wu, B.Y.: Space-time Legendre-Gauss-Lobatto collocation method for two-dimensional generalized sine-Gordon equation, Appl. Numer. Math. 92-107, 122 (2017)

    MATH  Google Scholar 

  4. Li, H.C., Jiang, C.L., Lv, Z.Q.: A Galerkin energy-preserving method for two dimensional nonlinear Schrödinger equation, Appl. Math. Comput. 16-27, 324 (2018)

    Google Scholar 

  5. Zhang, H., Jiang, X.Y., Wang, C., Fan, W.P.: Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation, Numer. Algor. 79, 337–356 (2018)

    MATH  Google Scholar 

  6. An, J., Shen, J., Zhang, Z.M.: The spectral-Galerkin approximation of nonlinear eigenvalue problems. Appl. Numer. Math. 1-15, 131 (2018)

    MathSciNet  Google Scholar 

  7. Bassett, B., Kiedrowski, B.: Meshless local Petrov-Galerkin solution of the neutron transport equation with streamline-upwind Petrov-Galerkin stabilization. J. Comput. Phys. 1-59, 377 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Gupta, A.K., Ray, S.: On the solution of time-fractional KdV-Burgers equation using Petrov-Galerkin method for propagation of long wave in shallow water. Chaos Solitons Fractals 376-380, 116 (2018)

    MathSciNet  Google Scholar 

  9. Rashidinia, J., Mohmedi, E.: Convergence analysis of tau scheme for the fractional reaction-diffusion equation. E. Eur. Phys. J. Plus 1-14, 133 (2018)

    Google Scholar 

  10. Engheta, N: On fractional calculus and fractional multipoles in electromagnetism. Antennas Propag. 554-566, 44 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Lederman, C., Roquejoffre, J.M., Wolanski, N.: Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames. Ann. di Matem. 173-239, 183 (2004)

    MATH  Google Scholar 

  12. Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 803-806, 124 (2002)

    Google Scholar 

  13. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model. 35, 5662–5672 (2011)

    Article  MathSciNet  Google Scholar 

  14. Hesameddini, E., Rahimi, A., Asadollahifard, E: On the convergence of a new reliable algorithm for solving multi-order fractional differential equations. Commun. Nonlinear. Sci. Numer. Simulat. 154-164, 34 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Damarla, S.K., Kundu, M.: Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices. Appl. Math. Comput. 189-203, 263 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

  17. Diethelm, K.: The analysis of fractional differential equations, lecture notes in mathematics (2004)

  18. Chen, Z.Y., Wu, B., Xu, Y.S.: Multilevel augmentation methods for differential equations, Adv. Cpmput. Math. 213-238, 24 (2006)

    Google Scholar 

  19. Cui, M.G., Wu, B.Y.: Numerical Analysis in Reproducing Kernel Space, Scientific Press, Peking (2004)

  20. Yan, Q.J.: Numerical Analysis, 2nd ed., Beihang University Press (2001)

  21. Xia D.X.: Real Variable Function and Functional Analysis, Yan Shaozhong, Higher Education Press (2010)

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Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11701124) and the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF. 201717).

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, People’s Republic of China

    Longbin Wu, Zhong Chen & Xiaohua Ding

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  1. Longbin Wu
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  2. Zhong Chen
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  3. Xiaohua Ding
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Correspondence to Zhong Chen.

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Wu, L., Chen, Z. & Ding, X. A stable minimal search method for solving multi-order fractional differential equations based on reproducing kernel space. Numer Algor 87, 1707–1727 (2021). https://doi.org/10.1007/s11075-020-01026-0

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  • DOI: https://doi.org/10.1007/s11075-020-01026-0

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