【作者】 王宇；

【导师】 李常品；

【作者基本信息】 上海大学 ， 计算数学， 2024， 硕士

【摘要】 近几十年来,分数阶微积分已广泛应用于自然科学和工程技术等领域.相较于经典微积分,分数阶微积分具有可积奇异性和非局部性,能更准确地描述具有记忆或遗传特性的材料、过程以及具有长距离相互作用的模型等复杂过程.本文主要研究时空分数阶扩散方程和扩散波方程的数值求解方法.然而,在现阶段有关于时空分数阶扩散方程和扩散波方程的数值研究中,时间方向的导数主要采用Caputo导数,其解具有代数渐近性.相比之下,本研究的模型采用Caputo-Hadamard导数作为时间方向的导数,对应发展方程的解具有对数渐近性,能够更准确地描述Lomnitz对数蠕变律以及超慢过程等.本论文的创新成果可以归纳为以下两个方面:第一,基于α∈（0,1）阶Caputo-Hadamard导数的离散公式,分别构造了空间带有Riesz导数的一维时空分数阶扩散方程以及带有分数阶Laplace算子的二维时空分数阶扩散方程的全离散格式;第二,应用α∈（1,2）阶Caputo-Hadamard导数的离散公式,提出了一维时空分数阶扩散波方程和二维时空分数阶扩散波方程的全离散格式,其中空间导数分别采用Riesz导数和分数阶Laplace算子.具体的研究内容如下:第二章介绍了Caputo-Hadamard导数的几种数值逼近公式,对于阶α∈（0,1）的情形,有L1、L1-2和L2-1_σ公式;对于阶α∈（1,2）的情形,有H2N2、L2₁公式.此外,还介绍了一维Riesz导数的加权位移Gr（?）nwald-Letnikov公式和二维分数阶Laplace算子的分数阶中心差分公式,并给出了这些逼近公式的系数性质.这些内容将成为后续章节中时空分数阶扩散方程和扩散波方程的全离散格式的建立和分析重要基础.第三章研究了Caputo-Hadamard型时空分数阶扩散方程的数值格式.分别应用L1、L1-2和L2-1_σ公式近似Caputo-Hadamard导数,应用加权位移Gr（?）nwald-Letnikov公式近似一维Riesz导数,以及应用中心差分公式近似二维分数阶Laplace算子构造超慢扩散方程的全离散格式.同时,分别证明了相应数值格式的稳定性和收敛性,数值算例验证了数值格式的正确性.第四章研究了Caputo-Hadamard型时空分数阶扩散波方程的数值格式.采用了与第三章相同的空间离散方法,时间方向上分别利用H2N2和L2₁公式对Caputo-Hadamard导数进行离散,构造了超慢扩散波方程的全离散格式.证明了基于H2N2公式的数值格式的稳定性和收敛性,数值实验验证了基于H2N2和L2₁公式的数值稳定性和收敛性.最后一章总结全文主要结果,并给出未来拟研究的方向.更多还原

【Abstract】 In recent decades,fractional calculus has been widely applied in various fields such as natural sciences and engineering.Compared with classical calculus,fractional calculus ex-hibits integrable singularity and non-locality,enabling more accurate descriptions of complex physical processes involving materials and processes with memory or hereditary properties,as well as long-range interaction models.This paper primarily focuses on the numerical methods for time-space fractional diffu-sion equations and diffusion-wave equations.In the existing numerical studies on fractional diffusion and diffusion-wave equations,the time derivative is mainly in the Caputo sense,where the solution to the corresponding evolution equation has algebraic asymptotic behav-ior.In contrast,this study focuses on equations with the Caputo-Hadamard derivative as the time derivative,showcasing the logarithmic asymptotic behavior to more accurately describe complex processes like Lomnitz logarithmic creep law and super-slow diffusion dynamics.The innovative contributions of this paper can be summarized in the following two aspect-s:Firstly,based on numerical approximations to the Caputo-Hadamard derivative of orderα∈（0,1）,fully discrete schemes for time-space fractional diffusion equations with Riesz derivative in one space dimension and time-space fractional diffusion equations with fraction-al Laplacian in two space dimensions are constructed;Secondly,using numerical approxi-mations to the Caputo-Hadamard derivative of orderα∈（1,2）,fully discrete schemes for one-dimensional time-space fractional diffusion-wave equations and two-dimensional time-space fractional diffusion-wave equations are proposed.The specific research contents are as follows:Chapter 2 introduces several numerical approximation formulae for the Caputo-Hadamard derivative,including L1,L1-2,and L2-1_σformulae for the case of orderα∈（0,1）;and H2N2,L2₁formulae for the case of orderα∈（1,2）.Additionally,it presents the weighted and shifted Gr（?）nwald-Letnikov formula for one-dimensional Riesz derivative and the fraction-al centered difference formula for two-dimensional fractional Laplacian,along with the coef-ficient properties of these approximations.These contents lay a crucial foundation for estab-lishing and analyzing fully discrete schemes for time-space fractional diffusion and diffusion-wave equations in the subsequent chapters.Chapter 3 investigates the numerical methods for one-dimensional and two-dimensional time-space fractional diffusion equations with the temporal Caputo-Hadamard derivative.Ap-proximations to the Caputo-Hadamard derivative are based on L1,L1-2,and L2-1_σformulae,one-dimensional Riesz derivative is approximated using the weighted and shifted Gr（?）nwald-Letnikov formula,and the two-dimensional fractional Laplacian is evaluated by the fraction-al centered difference formula.Stability and convergence of the corresponding numerical schemes are discussed.Numerical examples are utilized to validate the correctness of the numerical algorithms.Chapter 4 investigates the numerical methods for one-dimensional and two-dimensional time-space fractional diffusion-wave equations with the temporal Caputo-Hadamard deriva-tive.Employing the same spatial discretization methods as those in Chapter 3,the discretiza-tion of the Caputo-Hadamard derivative is based on H2N2 and L2₁formulae.The stability and convergence of the numerical scheme based on the H2N2 formula are proven,and numerical experiments are conducted to verify the stability and convergence of the numerical algorithm based on H2N2 and L2₁formulae.The last chapter summarizes the main results of the entire paper and outlines the possible research directions in the future.更多还原