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Article
Unconditional optimal error bounds of the fast nonuniform Alikhanov scheme for a nonlinear time-fractional biharmonic equation
In this work, we will consider the two-dimensional nonlinear time-fractional biharmonic equation with weakly singular solutions. By introducing an intermediate variable, the original problem is transformed int...
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Article
\(\alpha \) -Robust Error Analysis of Two Nonuniform Schemes for Subdiffusion Equations with Variable-Order Derivatives
In this paper, we will consider the variable-order subdiffusion initial-boundary value problem with weakly singular solutions. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in time, two eff...
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Article
Optimal error analysis of the Alikhanov formula for a time-fractional Schrödinger equation
In this paper, we have developed a fully discrete Alikhanov finite element method to solve the time-fractional Schrödinger equation with non-smooth solution. The proposed scheme uses the Alikhanov formula on g...
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Article
A Sharp \(\alpha \) -Robust \(L^\infty (H^1)\) Error Bound for a Time-Fractional Allen-Cahn Problem Discretised by the Alikhanov \(L2-1_\sigma \) Scheme and a Standard FEM
A time-fractional Allen-Cahn initial-boundary value problem is considered, where the bounded spatial domain \(\Omega \) ...
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Article
\(\beta \) -Robust Superconvergent Analysis of a Finite Element Method for the Distributed Order Time-Fractional Diffusion Equation
A distributed order time fractional diffusion equation whose solution has a weak singularity near the initial time \(t = 0\) ...
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Article
α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation
An initial-boundary value problem of the form D ...
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Article
Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation
A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of ℝ d \(\mathbb {R}^{d}\) for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are der...
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Article
Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem
A time-fractional initial-boundary value problem is considered, where the differential equation has a sum of fractional derivatives of different orders, and the spatial domain lies in \({\mathbb {R}}^d\)Rd with $...
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Article
Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem
A reaction-diffusion problem with a Caputo time derivative of order \(\alpha \in (0,1)\) ...
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Article
Open AccessA partially penalty immersed Crouzeix-Raviart finite element method for interface problems
The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we ...
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Chapter and Conference Paper
Local Discontinuous Galerkin Methods for Reaction-Diffusion Systems on Unstructured Triangular Meshes
In this paper, on two-dimension unstructured meshes, a fully-discrete scheme is presented for the reaction-diffusion systems, which are often used as mathematical models for many biological, physical and chemi...
