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Asymptotic expansions and approximations for the Caputo derivative

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Abstract

In this paper, we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order \(2-\alpha \) and second-order approximations of the Caputo derivative by modifying the weights of the shifted Grünwald–Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Grünwald–Letnikov approximation is obtained which allows second-order numerical solutions of fractional differential equations with arbitrary values of the solutions and their first derivatives at the initial point.

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References

  • Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424–438

    Article  MathSciNet  MATH  Google Scholar 

  • Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA (2015) New operational matrices for solving fractional differential equations on the half-line. PLoS One 10(5):e0126620

    Article  Google Scholar 

  • Chen M, Deng W (2014) Fourth order accurate scheme for the space fractional diffusion equations. SIAM J Numer Anal 52(3):1418–1438

    Article  MathSciNet  MATH  Google Scholar 

  • Diethelm K, Siegmund S, Tuan HT (2017) Asymptotic behavior of solutions of linear multi-order fractional differential systems. Fract Calc Appl Anal 20(5):1165–1195

    Article  MathSciNet  MATH  Google Scholar 

  • Dimitrov Y (2014) Numerical approximations for fractional differential equations. J Fract Calc Appl 5(3S):1–45

    MathSciNet  Google Scholar 

  • Dimitrov Y (2016) Second-order approximation for the Caputo derivative. J Fract Calc Appl 7(2):175–195

    MathSciNet  Google Scholar 

  • Dimitrov Y, Miryanov R, Todorov V (2017) Quadrature formulas and Taylor series of secant and tangent. Econ Comput Sci 4:23–40

    Google Scholar 

  • Dimitrov Y (2018) Approximations for the Caputo derivative (I). J Fract Calc Appl 9(1):35–63

    MathSciNet  Google Scholar 

  • Ding H, Li C (2016) High-order algorithms for Riesz derivative and their applications (III). Fract Calc Appl Anal 19(1):19–55

    Article  MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2017) High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J Sci Comput 71(2):759–784

    Article  MathSciNet  MATH  Google Scholar 

  • El-Borai MM, El-Sayed WG, Jawad AM (2015) Adomian decomposition method for solving fractional differential equations. Int Res J Eng Technol 2(6):295–306

    Google Scholar 

  • Elezović N (2005) Asymptotic expansions of gamma and related functions, binomial coefficients, inequalities and means. J Math Inequal 9(4):1001–1054

    MathSciNet  MATH  Google Scholar 

  • Ertürk VS, Momani S (2008) Solving systems of fractional differential equations using differential transform method. J Comput Appl Math 215:142–151

    Article  MathSciNet  MATH  Google Scholar 

  • Ezz-Eldien SS, Hafez RM, Bhrawy AH, Baleanu D, El-Kalaawy AA (2017) New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. J Optim Theory Appl 174(1):295–320

    Article  MathSciNet  MATH  Google Scholar 

  • Gao GH, Sun ZZ, Zhang HW (2014) A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J Comput Phys 259:33–50

    Article  MathSciNet  MATH  Google Scholar 

  • Gao GH, Sun HW, Sun ZZ (2015) Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J Comput Phys 280:510–528

    Article  MathSciNet  MATH  Google Scholar 

  • Jin B, Lazarov R, Zhou Z (2016) An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J Numer Anal 36(1):197–221

    MathSciNet  MATH  Google Scholar 

  • Li C, Chen A, Ye J (2011) Numerical approaches to fractional calculus and fractional ordinary differential equation. J Comput Phys 230(9):3352–3368

    Article  MathSciNet  MATH  Google Scholar 

  • Lin Y, Xu C (2007) Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225:1533–1552

    Article  MathSciNet  MATH  Google Scholar 

  • Lubich C (1986) Discretized fractional calculus. SIAM J Math Anal 17(3):704–719

    Article  MathSciNet  MATH  Google Scholar 

  • Magin RL (2004) Fractional calculus in bioengineering. Crit Rev Biomed Eng 32(1):1–104

    Article  Google Scholar 

  • Monje CA, Chen YQ, Vinagre BM, Xue D, Feliu-Batlle V (2010) Fractional-order systems and controls: fundamentals and applications. Springer Science & Business Media, New York

    Book  MATH  Google Scholar 

  • Pedas A, Tamme E (2014) Numerical solution of nonlinear fractional differential equations by spline collocation methods. J Comput Appl Math 255:216–230

    Article  MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Fract Differ Equ. Academic Press, San Diego

    Google Scholar 

  • Ren L, Wang YM (2017) A fourth-order extrapolated compact difference method for time-fractional convection–reaction-diffusion equations with spatially variable coefficients. Appl Math Comput 312:1–22

    Article  MathSciNet  Google Scholar 

  • Tadjeran C, Meerschaert MM, Scheffer HP (2006) A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 213:205–213

    Article  MathSciNet  MATH  Google Scholar 

  • Tian W, Zhou H, Deng W (2015) A class of second order difference approximations for solving space fractional diffusion equations. Math Comput 84:1703–1727

    Article  MathSciNet  MATH  Google Scholar 

  • Tricomi F, Erdélyi A (1951) The asymptotic expansion of a ratio of gamma functions. Pac J Math 1(1):133–142

    Article  MathSciNet  MATH  Google Scholar 

  • Vong S, Wang Z (2014) High order difference schemes for a time-fractional differential equation with Neumann boundary conditions. East Asian J Appl Math 4(3):222–241

    Article  MathSciNet  MATH  Google Scholar 

  • Wang S, Xu M (2007) Generalized fractional Schrödinger equation with space-time fractional derivatives. J Math Phys 48:043502

    Article  MathSciNet  MATH  Google Scholar 

  • Yan Y, Pal K, Ford NJ (2014) Higher order numerical methods for solving fractional differential equations. BIT Numer Math 54:555–584

    Article  MathSciNet  MATH  Google Scholar 

  • Zayernouri M, Karniadakis GE (2014) Exponentially accurate spectral and spectral element methods for fractional ODEs. J Comput Phys 257:460–480

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Liu F, Turner I, Chen S (2016) The numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option. Appl Math Model Appl Math Model 40(1112):5819–5834

    Article  MathSciNet  Google Scholar 

  • Zheng M, Liu F, Liu Q, Burrage K, Simpson MJ (2017) Numerical solution of the time fractional reaction diffusion equation with a moving boundary. J Comput Phys 338:493–510

    Article  MathSciNet  Google Scholar 

  • Zhuang P, Liu F (2006) Implicit difference approximation for the time fractional diffusion equation. J Appl Math Comput 22(3):87–99

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The third author is supported by the Bulgarian Academy of Sciences through the Program for Career Development of Young Scientists, Grant DFNP-17-88/28.07.2017, Project Efficient Numerical Methods with an Improved Rate of Convergence for Applied Computational Problems, by the Bulgarian National Fund of Science under Project DN 12/5-2017, Project Efficient Stochastic Methods and Algorithms for Large-Scale Problems, and Project DN 12/4-2017, Project Advanced Analytical and Numerical Methods for Nonlinear Differential Equations with Applications in Finance and Environmental Pollution.

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Authors and Affiliations

  1. Department of Mathematics and Physics, University of Forestry, 1756, Sofia, Bulgaria

    Yuri Dimitrov

  2. Department of Statistics and Applied Mathematics, University of Economics, 9002, Varna, Bulgaria

    Radan Miryanov

  3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str, bl. 8, 1113, Sofia, Bulgaria

    Venelin Todorov

  4. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str, bl. 25A, 1113, Sofia, Bulgaria

    Venelin Todorov

Authors
  1. Yuri Dimitrov
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  2. Radan Miryanov
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  3. Venelin Todorov
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Correspondence to Yuri Dimitrov.

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Communicated by José Tenreiro Machado.

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Dimitrov, Y., Miryanov, R. & Todorov, V. Asymptotic expansions and approximations for the Caputo derivative. Comp. Appl. Math. 37, 5476–5499 (2018). https://doi.org/10.1007/s40314-018-0641-3

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  • DOI: https://doi.org/10.1007/s40314-018-0641-3

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