三类分数阶偏微分方程的有限元计算
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摘要
分数阶微积分作为整数阶(经典)微积分推广,在生物、物理、化学、工程等领域有着广泛的应用。特别地,在近几十年里,许多研究者指出分数阶微积分以及分数阶微分方程非常适合用来刻画具有记忆和遗传特性的材料和过程。由于应用的广泛性,使得分数阶微积分这一领域越来越受到人们的关注,与之相关的理论分析和数值计算等研究工作就显得尤为重要。
本文主要有两大部分。第一部分讨论的是函数的分数阶可积性和可微性问题。第二部分研究了三类分数阶偏微分方程的有限元计算问题。其中这三类方程分别从空间分数阶,时空分数阶,时空分数阶且时间方向为两项分数阶导数的角度研究了数值方法,给出了理论分析,并进行了数值模拟,数值结果与理论分析相吻合。
具体地说,第一章简要介绍了分数阶发展的概况和研究分数阶微分方程数值解法的实际意义。
第二章讨论函数的分数阶可积性和分数阶可微性。主要给出了函数关于Riemann-Liouville积分意义下的分数阶可积性定理,关于Riemann-Liouville导数和Caputo导数意义下的分数阶可微性定理。
第三章针对非线性空间分数阶Fokker-Planck方程,建立了有限元数值格式。时间方向采用差分格式,空间方向采用分数阶有限元格式,并对全局误差进行了理论分析。数值算例表明数值方法的可行性。
第四章考虑的是非线性时空分数阶亚扩散和超扩散方程。在空间方向上,我们利用分数阶有限元方法来逼近；时间方向上,分别利用分数阶欧拉向后差分格式和分数阶中心差分格式来逼近亚扩散和超扩散问题。同时研究了弱解的存在唯一性、半离散格式的稳定性、以及半离散和全离散格式的误差估计。最后所给出的数值例子验证了前面的理论分析。并在数值模拟中,我们观察到了有趣的分数阶扩散现象。
第五章为数值求解时空分数阶电报方程。在时间方向上我们同时使用分数阶欧拉向后差分格式和分数阶中心差分格式来逼近,在空间方向上使用分数阶有限元格式来逼近,建立了半离散格式和全离散格式,并给出了有限元理论分析。所给的数值例子验证了方法的可行性。
In this dissertation, the theory analysis for the fractional calculus and the finite element algorithms for three types of fractional partial differential equations are stud-ied. The first part focuses on the fractional integrability and differentiability of a given function, the second part deals with the finite element algorithms for the generalized nonlinear space fractional Fokker-Planck equation, the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion, and the time-space frac-tional telegraph eqaution.
In details, chapter II is devoted to discussing the fractional integrability and dif-ferentiability of the considered function, in the senses of Riemann-Liouville integral, Riemann-Liouville derivative and Caputo derivative, respectively. Important issues on these fractional integral and derivatives are also included.
Chapter III is to formulate a fully discrete scheme to numerically solve the gener-alized nonlinear space fractional Fokker-Planck equation, which can be used to describe the Levy flights. The error estimates for the fully discrete scheme are derived in details. The numerical examples are also included which agree with the theoretical analysis.
Chapter IV is to propose a new fractional finite element method for the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion. The semi-discrete and fully discrete numerical approximations are both analyzed. In spatial direction, we use the fractional finite element method, and in temporal direction, we use the fractional finite difference methods. For the subdiffusion problem, we use the fractional Euler backword difference method. For the superdiffusion problem, we use the fractional center difference method. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are also included to confirm the theoretical analysis. During our simulations, an interesting fractional diffusion phenomenon of particles is also observed.
Chapter V is to numerically study the time-space fractional telegraph equation, which has a multi-fractional order characteristically describing the random walks of the individual particles in the suspension flow, and the anomalous diffusion during the transmission of the voltage wave or the current wave. In temporal direction, we use the fractional difference method, including the fractional Euler backword difference scheme and the fractional center difference scheme. In spatial direction, we use the fractional finite element method. We derive the semi-discrete scheme and the fully discrete scheme separately for the considered equation. We prove the existence and uniqueness of the discrete solution and then derive the error estimates. The numerical results are inline with the theoretical results.
本文主要有两大部分。第一部分讨论的是函数的分数阶可积性和可微性问题。第二部分研究了三类分数阶偏微分方程的有限元计算问题。其中这三类方程分别从空间分数阶,时空分数阶,时空分数阶且时间方向为两项分数阶导数的角度研究了数值方法,给出了理论分析,并进行了数值模拟,数值结果与理论分析相吻合。
具体地说,第一章简要介绍了分数阶发展的概况和研究分数阶微分方程数值解法的实际意义。
第二章讨论函数的分数阶可积性和分数阶可微性。主要给出了函数关于Riemann-Liouville积分意义下的分数阶可积性定理,关于Riemann-Liouville导数和Caputo导数意义下的分数阶可微性定理。
第三章针对非线性空间分数阶Fokker-Planck方程,建立了有限元数值格式。时间方向采用差分格式,空间方向采用分数阶有限元格式,并对全局误差进行了理论分析。数值算例表明数值方法的可行性。
第四章考虑的是非线性时空分数阶亚扩散和超扩散方程。在空间方向上,我们利用分数阶有限元方法来逼近；时间方向上,分别利用分数阶欧拉向后差分格式和分数阶中心差分格式来逼近亚扩散和超扩散问题。同时研究了弱解的存在唯一性、半离散格式的稳定性、以及半离散和全离散格式的误差估计。最后所给出的数值例子验证了前面的理论分析。并在数值模拟中,我们观察到了有趣的分数阶扩散现象。
第五章为数值求解时空分数阶电报方程。在时间方向上我们同时使用分数阶欧拉向后差分格式和分数阶中心差分格式来逼近,在空间方向上使用分数阶有限元格式来逼近,建立了半离散格式和全离散格式,并给出了有限元理论分析。所给的数值例子验证了方法的可行性。
In this dissertation, the theory analysis for the fractional calculus and the finite element algorithms for three types of fractional partial differential equations are stud-ied. The first part focuses on the fractional integrability and differentiability of a given function, the second part deals with the finite element algorithms for the generalized nonlinear space fractional Fokker-Planck equation, the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion, and the time-space frac-tional telegraph eqaution.
In details, chapter II is devoted to discussing the fractional integrability and dif-ferentiability of the considered function, in the senses of Riemann-Liouville integral, Riemann-Liouville derivative and Caputo derivative, respectively. Important issues on these fractional integral and derivatives are also included.
Chapter III is to formulate a fully discrete scheme to numerically solve the gener-alized nonlinear space fractional Fokker-Planck equation, which can be used to describe the Levy flights. The error estimates for the fully discrete scheme are derived in details. The numerical examples are also included which agree with the theoretical analysis.
Chapter IV is to propose a new fractional finite element method for the nonlinear time-space fractional differential equations with subdiffusion and superdiffusion. The semi-discrete and fully discrete numerical approximations are both analyzed. In spatial direction, we use the fractional finite element method, and in temporal direction, we use the fractional finite difference methods. For the subdiffusion problem, we use the fractional Euler backword difference method. For the superdiffusion problem, we use the fractional center difference method. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are also included to confirm the theoretical analysis. During our simulations, an interesting fractional diffusion phenomenon of particles is also observed.
Chapter V is to numerically study the time-space fractional telegraph equation, which has a multi-fractional order characteristically describing the random walks of the individual particles in the suspension flow, and the anomalous diffusion during the transmission of the voltage wave or the current wave. In temporal direction, we use the fractional difference method, including the fractional Euler backword difference scheme and the fractional center difference scheme. In spatial direction, we use the fractional finite element method. We derive the semi-discrete scheme and the fully discrete scheme separately for the considered equation. We prove the existence and uniqueness of the discrete solution and then derive the error estimates. The numerical results are inline with the theoretical results.
引文
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