Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation
Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation
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DOI：
10.4208/cicp.020709.221209a
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发表时间：
2010-09
影响因子：
3.7
通讯作者：
X. Li;Chuanju Xu;许传炬
中科院分区：
文献类型：
--
作者：
X. Li;Chuanju Xu;许传炬
抽象的。本文研究时空分数阶扩散方程的初边值问题及其数值解。同时考虑了分数导数的两种定义，即Riemann-Liouville定义和Caputo One。在这两种情况下，我们都证明了弱解的适定性。此外，基于所提出的弱公式，我们构造了一种有效的谱方法来数值逼近弱解。本文的主要工作有三个方面：第一，建立了时空分数阶扩散方程变分解的理论框架。我们找到了合适的函数空间和范数，使时空分数次扩散问题可以表示为一个椭圆弱问题，然后利用已有的椭圆问题理论证明了弱解的存在唯一性。其次，我们证明了在Riemann-Liouville定义的情况下，时空分数阶扩散方程的适定性不需要任何初始条件。这与卡普托定义的情况不同，在卡普托定义中，为了建立适定性，必须将初始条件积分到弱公式中。最后，由于弱形式，我们能够构造一种有效的数值方法来求解时空分数阶扩散问题。AMS科目分类：35S10、35A05、65M70、65M12
Abstract. In this paper, we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem. AMS subject classifications: 35S10, 35A05, 65M70, 65M12