Numerical methods for solving the multi-term time-fractional wave-diffusion equation
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277 citations
Cites background or methods or result from "Numerical methods for solving the m..."
...Consider the following power law wave equation [30]:...
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...To demonstrate the accuracy of the proposed method, in Table 3 we compare maximum absolute errors of the present method at two choices of Jacobi parameters by selecting a few terms of the shifted Jacobi polynomial expansion N = M = 4, 6, 8, 10 together with the results obtained by using fractional predictor–corrector method FPCM-1 [30] and FPCM-2 [30], for y = 1....
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...5) h−1 FPCM-2 [30] FPCM-1 [30] 4 2....
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...We also compare the results given from our scheme and those reported in the literature such as difference scheme with spline function [73], implicit difference approximation [18] and fractional predictor–corrector method [30]....
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...The power law wave equation [30] is used to model sound wave propagation in anisotropic media that exhibits frequency dependent attenuation α(ω)....
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235 citations
Cites methods from "Numerical methods for solving the m..."
...We also refer to [24] for a numerical scheme based on a fractional predictor–corrector method for the multi-term time fractional wave-diffusion equation....
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229 citations
Cites background from "Numerical methods for solving the m..."
...The notion of fractional derivatives has been rapidly extended to many fractional partial differential equations (FPDEs), such as the fractional Burgers equation [30], the fractional Fokker–Planck equation [1], the fractional advection-diffusion equation [10], and fractional-order multiterm equations [22]....
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186 citations
References
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"Numerical methods for solving the m..." refers methods in this paper
...Generalized fractional partial differential equations have been used for describing important physical phenomena (see [1, 2, 10, 11, 14, 23, 24, 27])....
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3,828 citations
"Numerical methods for solving the m..." refers background in this paper
...1) where x and t are the space and time variables, k is an arbitrary positive constant, f(x, t) is a sufficiently smooth function, 0 < α ≤ 2 and Dα t is a Caputo fractional derivative of order α defined as [28]...
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...The Riemann-Liouville fractional derivative ∂ t p = 0D y t p of order y (0 ≤ m − 1 < y < m) is defined as (see [28]):...
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...The relationship between the Caputo fractional derivative and the Riemann-Liouville fractional derivative (see [28]) is...
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3,047 citations
"Numerical methods for solving the m..." refers methods in this paper
...Using a similar technique in [7], we can prove the result....
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