低扩散分数阶微分方程的有限差分方法-计算数学专业论文.docxVIP

南京航空航天大学硕士学位论文 南京航空航天大学硕士学位论文 摘 要 分数阶微分方程在科学工程领域中有着大量的应用，这促使我们不断寻求稳定、高效且 快速的求解方法，其中有限差分方法是求解微分方程定解问题运用最广泛的数值方法。本文 主要探讨了低扩散分数阶微分方程的有限差分方法。 第一章简要介绍了分数阶导数的定义、在工程科学领域的应用实例以及研究现状。 第二章讨论了低反应扩散方程，利用紧算子构造了高阶的差分格式，用矩阵方法证明了 差分解的存在唯一性，并且用 Fourier 方法分析了格式的稳定性，同时证明了格式的收敛性。 数值试验表明了此方法的有效性。 第三章对于二维低扩散方程做了初步研究，分别给出了隐格式、紧格式，讨论了差分解 的唯一性、稳定性，并用 Lax 相容性定理给出格式的收敛性。为提高计算速度、节省存储空 间又构造了交替方向隐格式（ADI 格式），并分析了格式的稳定性、收敛性。每个格式均给 出了相应的数值实验验证了理论分析的正确性和可行性。最后比较了三个格式的误差和计算 时间，证实了 ADI 格式的快速有效性。 关键词：低反应扩散方程，二维低扩散方程，有限差分方法，Fourier 方法，紧格式，ADI 格式 i 低扩散分数阶微分方程的有限差分方法 低扩散分数阶微分方程的有限差分方法 Abstract Fractional differential equations have been applied extensively in science and engineering, which encourages us to seek for stable, efficient and rapid numerical algorithms. This paper studies subdiffusion fractional differential equations by finite difference methods. In Chapter 1, we briefly introduce three definations of fractional derivatives. Many fractional differential models are generated. Present studies on fractional equations are also discussed. In Chapter 2, a fractional reaction-subdiffusion equation in one space variable is explored. A high order compact finite difference scheme is presented, and its uniqueness is proved by the matrix method. We analyze the stability of the scheme by the Fourier method, and prove the convergence simultaneously. Numerical experiments are carried out to verify the efficiency of the scheme. In Chapter 3, we do some pilot study for a two dimensional subdiffusion equation. Firstly we provide an implicit scheme and a compact scheme respectively. Then we discuss the sovability, stability of the two schemes and analyze the convergence by the Laxs Equivalence Theorem. In order to enhance the computation speed, we construct an Alternating Direction Implicit (ADI) scheme and analyze it’s stability and cenvergence. Numerical experiments are provided to verify the theoretical validity and feasibility. In the end, we provide the comparison table of the errors and computer time to ve