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基于精细积分方法,结合差分格式,提出了含有分数阶导数微分方程的数值求解方法,并对含有分数导数的一阶和二阶微分方程进行求解.所论方法首先引入差分格式,将含有分数阶导数的一阶和二阶微分方程变为一阶的常微分方程,然后再用精细积分方法逐步积分进行求解.文中不仅给出了详细的理论推导,而且还给出了相关的数值算例.数值算例对不同的分数阶微分方程进行了讨论,探讨了不同的分数阶和时间步长对计算结果的影响,并将计算结果与文献中相关算法的计算结果进行了比较.数值结果表明了所提的求解策略在求解分数阶导数模型时的可行性,以及较高的计算精度和较好的稳定性.
Abstract：Relatively new numerical techniques for solving differential equations of fractional order are presented based on high precision direct integration and difference method.The method can be used for obtaining approximate solution for different types of fractional differential equations.Using difference method,the first-order and the second-order differential equations of fractional derivative can become the first-order ordinary differential equation,and the high precision direct integration can be used for the solution.In the article,the detailed theory inferential reasoning is presented,moreover the numerical experiments is also given.Satisfactory numerical validation is presented including a preliminary investigation of time step effect and the value of fractional derivative order for different types of fractional differential equations.A numerical comparison is also conducted between the presented method and the calculation methods in the related literature.The numerical results demonstrates the feasibility of the method for solving different types of fractional differential equations with high computation precision and good stability.
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基本信息:
中图分类号:O241.82
引用信息:
[1]薛齐文,魏伟.含分数阶导数微分方程的数值求解[J].大连交通大学学报,2009,30(05):88-92.
基金信息:
国家自然科学基金资助项目(10802015);; 辽宁省重点实验室基金资助项目(2008S036);; 工业装备结构分析重点实验室开放基金资助项目(GZ0811)
2009-10-15
2009-10-15