Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula
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Cited by 2 publications
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High-order schemes based on extrapolation for semilinear fractional differential equation
By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order $$\alpha \in (1,2).$$ α ∈ ( 1 , 2 ) . The error has the asymptotic expansion $$ \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{5-\alpha } + \cdots \big ) + \big ( d_{2}^{*} \tau ^{4} + d_{3}^{*} \tau ^{6} + d_{4}^{*} \tau ^{8} + \cdots \big ) $$ ( d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯ ) + ( d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯ ) at any fixed time $$t_{N}= T, N \in {\mathbb {Z}}^{+}$$ t N = T , N ∈ Z + , where $$d_{i}, i=3, 4,\ldots $$ d i , i = 3 , 4 , … and $$d_{i}^{*}, i=2, 3,\ldots $$ d i ∗ , i = 2 , 3 , … denote some suitable constants and $$\tau = T/N$$ τ = T / N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.
High-order schemes based on extrapolation for semilinear fractional differential equation
By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order $$\alpha \in (1,2).$$ α ∈ ( 1 , 2 ) . The error has the asymptotic expansion $$ \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{5-\alpha } + \cdots \big ) + \big ( d_{2}^{*} \tau ^{4} + d_{3}^{*} \tau ^{6} + d_{4}^{*} \tau ^{8} + \cdots \big ) $$ ( d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯ ) + ( d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯ ) at any fixed time $$t_{N}= T, N \in {\mathbb {Z}}^{+}$$ t N = T , N ∈ Z + , where $$d_{i}, i=3, 4,\ldots $$ d i , i = 3 , 4 , … and $$d_{i}^{*}, i=2, 3,\ldots $$ d i ∗ , i = 2 , 3 , … denote some suitable constants and $$\tau = T/N$$ τ = T / N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.
Unconditionally Stable and Convergent Difference Scheme for Superdiffusion with Extrapolation
Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order $$\alpha \in (1, 2)$$ α ∈ ( 1 , 2 ) and the error is shown to have the asymptotic expansion $$ \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{5-\alpha } + \cdots \big ) + \big ( d_{2}^{*} \tau ^{4} + d_{3}^{*} \tau ^{6} + d_{4}^{*} \tau ^{8} + \cdots \big ) $$ ( d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯ ) + ( d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯ ) at any fixed time, where $$\tau $$ τ denotes the step size and $$d_{l}, l=3, 4, \dots $$ d l , l = 3 , 4 , ⋯ and $$d_{l}^{*}, l\,=\,2, 3, \dots $$ d l ∗ , l = 2 , 3 , ⋯ are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order $$O (\tau ^{3- \alpha }), \alpha \in (1, 2)$$ O ( τ 3 - α ) , α ∈ ( 1 , 2 ) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are $$O ( \tau ^{4- \alpha })$$ O ( τ 4 - α ) and $$O ( \tau ^{2(3- \alpha )}), \alpha \in (1, 2)$$ O ( τ 2 ( 3 - α ) ) , α ∈ ( 1 , 2 ) , respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.
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