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An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation
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作者 Wei Zeng Chuanju Xu 《Science China Mathematics》 SCIE CSCD 2024年第10期2387-2408,共22页
In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the init... In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity. 展开更多
关键词 subdiffusion equations variable transformation Ψ-Sobolev spaces WELL-POSEDNESS space-time Galerkin spectral method error estimate fast algorithm
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SHARP POINTWISE-IN-TIME ERROR ESTIMATE OF L1 SCHEME FOR NONLINEAR SUBDIFFUSION EQUATIONS
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作者 Dongfang Li Hongyu Qin Jiwei Zhang 《Journal of Computational Mathematics》 SCIE CSCD 2024年第3期662-678,共17页
An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity p... An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis. 展开更多
关键词 Sharp pointwise-in-time error estimate Ll scheme Nonlinear subdiffusion equations Non-smooth solutions
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Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System 被引量:3
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作者 Seakweng Vong Zhibo Wang 《Advances in Applied Mathematics and Mechanics》 SCIE 2014年第4期419-435,共17页
In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the... In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results. 展开更多
关键词 Fourth-order fractional subdiffusion equation compact difference scheme energy method STABILITY CONVERGENCE
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Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation
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作者 Na Zhang Weihua Deng Yujiang Wu 《Advances in Applied Mathematics and Mechanics》 SCIE 2012年第4期496-518,共23页
We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.Th... We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis. 展开更多
关键词 Modified subdiffusion equation finite difference method finite element method STABILITY convergence rate
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