Asymptotic expansions and approximations for the Caputo derivative
- PDF / 586,198 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 100 Downloads / 227 Views
Asymptotic expansions and approximations for the Caputo derivative Yuri Dimitrov1 · Radan Miryanov2 · Venelin Todorov3,4
Received: 27 December 2017 / Revised: 25 February 2018 / Accepted: 10 May 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
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 − α 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. Keywords Binomial coefficient · Asymptotic expansion · Approximation of the Caputo derivative · Numerical solution Mathematics Subject Classification 11B65 · 34A07 · 34E05 · 65D30
Communicated by José Tenreiro Machado.
B
Yuri Dimitrov [email protected] Radan Miryanov [email protected] Venelin Todorov [email protected]; [email protected]
1
Department of Mathematics and Physics, University of Forestry, 1756 Sofia, Bulgaria
2
Department of Statistics and Applied Mathematics, University of Economics, 9002 Varna, Bulgaria
3
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str, bl. 8, 1113 Sofia, Bulgaria
4
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str, bl. 25A, 1113 Sofia, Bulgaria
123
Y. Dimitrov et al.
1 Introduction The Caputo and Riemann–Liouville fractional derivatives are the two main approaches for generalizing the integer order derivatives. When 0 < α < 1 the Caputo and Riemann– Liouville derivatives with a lower limit at the point zero are defined as x y (t) 1 y (α) (x) = D α y(x) = dt, Γ (1 − α) 0 (x − t)α x y(t) d 1 D αR L y(x) = dt. Γ (1 − α) dx 0 (x − t)α The Caputo and Riemann–Liouville derivatives are related as D αR L y(x) = D α y(x) +
y(0) . Γ (1 − α)x α
The Caputo derivative is a suitable choice for a fractional derivative in fractional differential equations. Fractional differential equations is a growing field of mathematics with applications in finance, bioengineering, control theory, quantum mechanics (Magin 2004; Wang and Xu 2007; Monje et al. 2010; Zhang et al. 2016). The finite difference schemes for numerical solution of fractional differential equations involve approximations for the fractional derivative. Let h = x/N and yβ = y(βh) for 0 ≤ β ≤ N . The Grünwald–Letnikov difference approximation is a first-order approximation of the Riemann–Liouville derivative, when y ∈ C 1 [0, x] and it is a first-order approximation of the Caputo derivative when the function y satisfies the condition y(0) = 0: AGL N [y(x)] =
N −1 1 k α (−1) y(x − kh) = y (α) (x) + O(h). hα k k=0
When the function
Data Loading...
