Abstract
In this paper, the implicit midpoint method is presented for solving Riesz tempered fractional diffusion equation with a nonlinear source term, where the tempered fractional partial derivatives are evaluated by the modified second-order Lubich tempered difference operator. Stability and convergence analyses of the numerical method are given. The numerical experiments demonstrate that the proposed method is effective.
Highlights
IntroductionWe consider Riesz tempered fractional diffusion equation with a nonlinear source term
In this paper, we consider Riesz tempered fractional diffusion equation with a nonlinear source term∂u(x, t) ∂α,λu(x, t) ∂t = κ ∂|x|α + g x, t, u(x, t),(x, t) ∈ (a, b) × (0, T], (1.1)with the initial and boundary conditions u(x, 0) = φ(x), x ∈ [a, b], (1.2)u(a, t) = 0, u(b, t) = 0, t ∈ [0, T], (1.3)where 1 < α < 2, λ ≥ 0, the diffusion coefficient κ is a positive constant, φ(x) is a known sufficiently smooth function, g(x, t, u) satisfies the Lipschitz condition g(x, t, u) – g(x, t, υ) ≤ L|u – υ|, ∀u, υ ∈ R, (1.4) here
To discretize the Riemann–Liouville tempered fractional derivatives, we would introduce the modified second-order Lubich tempered difference operators δxα– and δxα+ at the point (xi, tn), which are defined as δxα–u(xi, tn)
Summary
We consider Riesz tempered fractional diffusion equation with a nonlinear source term. Yu et al [16] proposed a third-order difference scheme for one side Riemann–Liouville tempered fractional diffusion equation and given the stability and convergence analysis. 5. To discretize the Riemann–Liouville tempered fractional derivatives, we would introduce the modified second-order Lubich tempered difference operators δxα– and δxα+ at the point (xi, tn), which are defined as δxα–u(xi, tn). Remark 2.1 In [24], Li and Deng combined the Crank–Nicolson method with a tempered weighted and shifted Grünwald–Letnikov operator to propose a numerical method with the accuracy of O(τ 2 + h2) for tempered fractional diffusion equation with a linear source term, where the tempered weighted and shifted Grünwald–Letnikov operators with second-order accuracy are defined as δxα–u(xi, tn) γ1ehλ + γ2 + γ3e–hλ.
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