Abstract
In this paper, a meshless method with ridge basis functions for solving the time fractional two-flow domain model problem is proposed. The method uses the L1 approximation formula based on piecewise linear interpolation to discretize the Caputo time fractional derivative \( (0 < \alpha < 1) \), and by means of the ridge basis function to construct the approximation function, and uses the collocation method to discretize the governing equation. The existence and uniqueness of the numerical solution are analyzed. The error between the proposed method and the finite difference method is compared by numerical examples; then the affecting factors of the calculation accuracy are discussed. The results show that the proposed method is feasible and simple.
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Ghehsareh, H.R., Bateni, S.H., Zaghian, A.: A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation. Eng. Anal. Bound. Elem. 61, 52–60 (2015)
Ghehsareh, H.R., Zaghian, A., Raei, M.: A local weak form meshless method to simulate a variable order time-fractional mobile–immobile transport model. Eng. Anal. Bound. Elem. 90, 63–75 (2018)
Liu, T.X., Liu, G.: Research progress of meshless method. J. Mech. Eng. 38, 7–12 (2002)
Qiao, Y., Zhai, S., Feng, X.: RBF-FD method for the high dimensional time fractional convection–diffusion equation. Int. Commun. Heat Mass Transf. 89, 230–240 (2017)
Shu, H.M., Huang, C.Q., Li, C.W.: A novel meshless method based on ridge basis function. J. China Univ. Pet. 32, 108–113 (2008)
Salehi, R.: A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numer. Algorithms 74, 1145–1168 (2017)
Sun, Z.Z., Gao, G.H.: Finite Difference Method for Fractional Differential Equations. Science Press, Beijing (2015)
Tian, S.L., Fang, B.Y., Wang, Z.G.: Approximation of two-point boundary value problems for differential games based on ridge basis function. J. Shandong Univ. (Sci. Ed.) 46, 38–41 (2011)
Wang, Z.G., Qin, X.Q., Guo, W., et al.: Meshless method with ridge basis functions. Appl. Math. Comput. 217, 1870–1886 (2010)
Zhang, L.W.: Error estimates for interpolation with ridge basis function. J. Fudan Univ. (Nat. Sci. Ed.) 44, 301–306 (2005)
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Qin, X., Li, K. & Hu, G. Meshless method with ridge basis functions for time fractional two-flow domain model. Math Sci 14, 375–385 (2020). https://doi.org/10.1007/s40096-020-00348-3
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DOI: https://doi.org/10.1007/s40096-020-00348-3