小波分析在R-L分数阶微分方程数值解中的应用

Theories and modeling methods of fractional calculus are widely applied in many fields. But there are great difficulties to get the analytic solution of fractional differential equation. The development of the numerical solution of fractional differential equation has an urgent need. Combined with wavelet analysis theory, function approximation theory and singular integral equation theory, the project focuses on the theories of the numerical solutions of Riemann-Liouville fractional differential equation and Riesz space fractional diffusion equation. The problems under consideration are: (1)Based on the wavelet analysis theory and numerical method of singular integral equation, the convergence, accuracy and stability of numerical solution of Riemann-Liouville fractional differential equation are studied. (2)Based on the numerical analysis theory and the numerical solution of integral equation, the numerical methods of Riesz space fractional diffusion equation are studied. (3)The new fractional wavelets are constructed and will be applied to solve fractional differential equations numerically.This project intends to establish the theories and methods of promotion of the existing results and these theories have important application prospects.

分数阶微积分的理论和建模方法在诸多领域有广泛的应用, 但要得到分数阶微分方程的解析解则存在很大困难, 因此研究分数阶微分方程的数值解法是一个迫切需要解决的问题. 本项目依据小波分析、函数逼近论和奇异积分方程等理论, 重点对R-L分数阶微分方程和Riesz空间分数阶扩散方程的数值解法从理论上展开研究. 具体研究内容包括: (1)依据小波分析理论和奇异积分方程数值方法研究R-L分数阶微分方程数值解的收敛性、精度和稳定性等问题; (2)依据数值分析理论和积分方程数值方法研究Riesz空间分数阶扩散方程的数值解法; (3)构造新的分数阶小波, 并用于分数阶微分方程数值解的研究. 本项目拟建立的理论和方法将推广现有结果, 具有重要的应用前景.

分数阶微积分的理论和建模方法在诸多领域有广泛的应用, 但要得到分数阶微分方程的解析解则存在很大困难, 因此研究分数阶微分方程的数值解法是一个迫切需要解决的问题. 本项目依据小波分析、函数逼近论和奇异积分方程等理论,主要研究了1、具有弱奇异核的第二类Volterra积分方程和具有弱奇异核的分数阶积分微分方程值解问题, 以及算法的收敛性; 2、分数阶偏微分方程数值解的收敛性; 3、构造了Euler小波, 并将其用于求解分数阶非线性Volterra积分微分方程, 得到了误差估计. 本项目拟建立的理论和方法将推广现有结果, 具有重要的应用前景.