R-L分数阶积分微分方程的小波解法及其力学应用研究

项目名称： R-L分数阶积分微分方程的小波解法及其力学应用研究

项目编号： No.11261041

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 韩惠丽

作者单位： 宁夏大学

项目金额： 45万元

中文摘要： 分数阶微积分建模方法和理论在诸多领域有着若干应用，而分数阶微积分方程的解析求解又存在很大的困难，所以发展分数阶微积分方程的数值解法是一个迫切需要解决的问题；另外分数阶拉普拉斯算子方程与Fredholm积分方程有着内在的联系和相似性.所以本项目结合函数逼近论、积分方程论和小波分析等理论，重点对R-L分数阶积分方程和分数阶拉普拉斯算子方程的数值解法从理论上展开研究，并开展在力学问题中的某些应用研究.本研究拟在三个工作点上展开：（1）依据传统的奇异积分方程数值方法和小波分析理论分别研究R-L分数阶积分方程和分数阶积分微分方程数值解的收敛性、精度和稳定性等问题；（2）依据Fredholm积分方程理论及其数值方法研究分数阶拉普拉斯算子方程的数值解法；（3）将此项目的前期研究工作应用于某些力学问题中，特别是分数阶黏弹性材料的应用研究. 本项目拟建立的理论推广了现有结果，并可以应用于工程领域.

中文关键词： 分数阶微积分；小波；收敛性；误差分析；材料力学

英文摘要： Modeling methods and theories of fractional calculus have a number of applications in many fields. But there are great difficulties to get the analytic solution of fractional calculus equation. The development of the numerical solution of fractional calculus equation has an urgent need. And Laplacian equation of fractional order with the Fredholm integral equation is intrinsically linked and similar. Combined with function approximation theory, integral equation theory and wavelet analysis theory, the study focuses on the theories of the numerical solutions of Riemann-Liouville fractional integral equation and on the theories of the numerical solution of fractional Laplacian equation, and attempt to carry out applied research work in mechanical problems. This study intends to commence in four working points. (1) Based on the numerical method of traditional singular integral equation and wavelet analysis theory, the convergence, accuracy and stability of numerical solution of Riemann-liouville fractional integral equation and of frcational integro-differential euqations are studied. (2) Based on the theory and the numerical solution of Fredholm integral equation, the numerical methods of fractional Laplacian equation are studied. (3) The preliminary research work will be applied to some mechanical problems,espe

英文关键词： Fractional order calculus；Wavelet；Convergence；Error analysis；Mechanics of materials