Variational formulation for the stationary fractional advection dispersion equation

By Vincent J. Ervin, John Paul Roop. 2006

788 citations recorded by the source corpus.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/bb510bd2cecd435c79a59f2ec31d777f75a31d76

How researchers see this paper

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

In Mechanics of Materials

1. Fawang Liu
2. Mehdi Dehghan
3. Chuanju Xu
4. Pinghui Zhuang
5. Abbas Saadatmandi
6. G. Em Karniadakis
7. Mehrdad Lakestani
8. Qingxia Liu

In Modeling and Simulation

1. Mehdi Dehghan
2. Fawang Liu
3. G. Em Karniadakis
4. Saeed Kazem
5. Ali H. Bhrawy
6. Chuanju Xu
7. Mehrdad Lakestani
8. S. Irandoust‐pakchin

In Numerical Analysis

1. Fawang Liu
2. Mehdi Dehghan
3. G. Em Karniadakis
4. Mehrdad Lakestani
5. Ali H. Bhrawy
6. Abbas Saadatmandi
7. Jafar Saberi-Nadjafi
8. M. Izadkhah

Citations in the corpus

• Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation (2010, background): “In [10,11,41], Ervin and Roop presented a Galerkin finite element approximation for variational solution to the steady state fractional advection dispersion equations.”
• Fractional embedding of differential operators and Lagrangian systems (2006, methodology): “Following [ 23 ] we introduce the left and right fractional derivatives as well as convenient functional spaces on which we have the semi-group property.”
• 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].”
• Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem (2013, methodology): “The existing numerical methods includes finite difference [7,11,19], Galerkin finite element [4, 5, 15], finite difference/spectral method [9], time-space spectral method [8], and so on.”
• Exact Solution of Some Linear Fractional Differential Equations by Laplace Transform (2013, methodology): “During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integro-differential equations and dynamic systems containing fractional derivatives, such as Adomian’s decomposition method [9–13], He’s variatio”
• Local discontinuous Galerkin methods for fractional diffusion equations (2013, neutral): “Nevertheless, methods based on finite difference methods [23] and finite element formulations [10,11] have been developed and successfully applied.”
• Adaptive finite element method for fractional differential equations using hierarchical matrices (2016, background): “However, as the size of the problem increases, the time required to solve the final system of equations increases considerably due to the nonlocality of the fractional differential operators [43, 23, 10, 11, 30, 9, 4, 36, 37, 40, 26].”
• ADJOINT FRACTIONAL DIFFERENTIAL EXPRESSIONS AND OPERATORS (2007, methodology): “If k = 0, we obtain the known fact that left-sided and right-sided Riemann-Liouville fractional integrals are adjoint [7].”
• An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data (2015, neutral): “Further, we note that for u ∈ H β/ 2 (Ω) (see [5] and [8]) Further, we recall the fact that for u ∈ C ∞ 0 (Ω) and s ∈ (1 / 2 , 1) e By the density of C ∞ 0 (Ω) in H s (Ω) for s ∈ (1 / 2 , 1) we obtain for all u ∈ H β/ 2 (Ω) Likewise, the right sided case follows: Thus for u, v ∈ H β/ 2 (Ω), there…”
• Variational approach to solutions for a class of fractional Hamiltonian systems (2014, methodology): “Then, I ̨ 1 and I ̨ 1 are equivalent with equivalent seminorms and norms [23].”