Abstract
The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.
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References
del Cartea, A., Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374(2), 749–763 (2007)
Srivastava, V.K., Kumar, S., Awasthi, M.K., Singh, B.K.: Two-dimensional time fractional-order biological population model and its analytical solution. Egypt. J. Basic Appl. Sci. 1(1), 71–76 (2014)
Dimitrov, Y.: A second order approximation for the Caputo fractional derivative. J. Fractional Calc. Appl. 7(2), 175–195 (2016)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)
Dimitrov, Y.: Three-point approximation for the Caputo fractional derivative. Commun. Appl. Math. Comput. 31(4), 413–442 (2017)
Dimitrov, Y.: Approximations for the Caputo derivative (I). J. Fractional Calc. Appl. 9(1), 35–63 (2018)
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52(3), 1418–1438 (2014)
Ding, H., Li, C.: High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J. Sci. Comput. 71(2), 759–784 (2017)
Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
Ren, L., Wang, Y.-M.: A fourth-order extrapolated compact difference method for time-fractional convection-reaction-diffusion equations with spatially variable coefficients. Appl. Math. Comput. 312, 1–22 (2017)
Jin, B., Zhou, Z.: A finite element method with singularity reconstruction for fractional boundary value problems. ESIM: M2AN 49, 1261–1283 (2015)
Quintana-Murillo, J., Yuste, S.B.: A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations. Eur. Phys. J. Special Top. 222(8), 1987–1998 (2013)
Zeng, F., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Methods Appl. Mech. Eng. 327, 478–502 (2017)
Zhang, Y.-N., Sun, Z.-Z., Liao, H.-L.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer (2010)
Diethelm, K., Siegmund, S., Tuan, H.T.: Asymptotic behavior of solutions of linear multi-order fractional differential systems. Fractional Calc. Appl. Anal. 20(5), 1165–1195 (2017)
Dimitrov, Y.: Numerical approximations for fractional differential equations. J. Fractional Calc. Appl. 5(3S), 1–45 (2014)
Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36(1), 197–221 (2016)
Li, C., Chen, A., Ye, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230(9), 3352–3368 (2011)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Bastero, J., Galve, F., Peña, A., Romano, M.: Inequalities for the gamma function and estimates for the volume of sections of Bnp. Proc. A.M.S., 130(1), 183–192 (2001)
Acknowledgements
This work was supported by the Bulgarian Academy of Sciences through the Program for Career Development of Young Scientists, Grant DFNP-17-88/2017, Project “Efficient Numerical Methods with an Improved Rate of Convergence for Applied Computational Problems”, by the Bulgarian National Fund of Science under Project DN 12/4-2017 “Advanced Analytical and Numerical Methods for Nonlinear Differential Equations with Applications in Finance and Environmental Pollution” and by the Bulgarian National Fund of Science under Project DN 12/5-2017 "Efficient Stochastic Methods and Algorithms for Large-Scale Computational Problems”.
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Dimitrov, Y., Dimov, I., Todorov, V. (2019). Numerical Solutions of Ordinary Fractional Differential Equations with Singularities. In: Georgiev, K., Todorov, M., Georgiev, I. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol 793. Springer, Cham. https://doi.org/10.1007/978-3-319-97277-0_7
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DOI: https://doi.org/10.1007/978-3-319-97277-0_7
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