Finite difference/spectral approximations for the time-fractional diffusion equation

By Yumin Lin, Chuanju Xu. Journal of Computational Physics, 2007

1,815 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c053aad9a466fc24779b11079493398eab1aff3a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Changpin Li
2. Mingrong Cui
3. Ian L Turner
4. Fawang Liu
5. Fanhai Zeng
6. Ali H. Bhrawy
7. M. A. Zaky
8. Zhibo Wang

In Modeling and Simulation

1. Changpin Li
2. Ali H. Bhrawy
3. Fawang Liu
4. M. A. Zaky
5. Ian L Turner
6. Fanhai Zeng
7. Weihua Deng
8. Dumitru Băleanu

In Numerical Analysis

1. Changpin Li
2. Fawang Liu
3. Ali H. Bhrawy
4. Ian L Turner
5. M. A. Zaky
6. Fanhai Zeng
7. Vo Anh
8. Mingrong Cui

Citations in the corpus

• An inverse problem for a one-dimensional time-fractional diffusion problem (2012, methodology): “We note that the mathematical and numerical analysis of the direct problem of the timefractional diffusion has gained much attention in recent years; see [24, 16, 5, 12, 22] for a rather incomplete list.”
• Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, methodology): “1) using a finite difference/ spectral approximation [24] with time step Δt = 0.”
• Compact finite difference method for the fractional diffusion equation (2009, positive): “Another example is inspired by Example 1 given in paper [26], @uðx;tÞ @t 1⁄4 0D 1 c t @2u @x2 h i þ f ðx; tÞ; 0 < t 1; 0 < x < 1; uð0; tÞ 1⁄4 0; uð1; tÞ 1⁄4 0; 0 t 1; uðx;0Þ 1⁄4 0; 0 x 1: 8<>: ð25Þ”
• A backward problem for the time-fractional diffusion equation (2010, methodology): “As for the numerical methods for the initial/boundary value problem for fractional diffusion equations, see [18,19].”
• An implicit RBF meshless approach for time fractional diffusion equations (2011, neutral): “where the truncation error R̃k+1 satisfies [38] ∣∣R̃k+1 ∣∣ ≤ C ( t)2−α .”
• Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion (2013, positive): “Lin and Xu [14] proposed a numerical method based a finite difference scheme in time and Legendre spectral method in space, showed its unconditional stability, and provided error estimates.”
• A new difference scheme for the time fractional diffusion equation (2014, methodology): “Difference schemes of the increased order of approximation such as the compact difference scheme [14, 17, 18, 19] and spectral method [11, 20, 21] were applied to improve the spatial accuracy of fractional diffusion equations.”
• A unified Petrov-Galerkin spectral method for fractional PDEs (2015, background): “We demonstrate that this framework can reduce to hyperbolic FPDEs such as timeand space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost.”
• Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients (2015, methodology): “In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11–13], finite element methods [14, 15], and spectral methods [16].”
• A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems (2018, methodology): “Included are the well-known L1 formula [14, 17, 20, 24, 25], the fast L1 formula [16], the Alikhanov approximation [2, 12, 15], and their applications for multi-term and distributed-order Caputo derivatives (see Remark 5).”