时滞分数阶微分积分方程高性能算法研究与可计算建模

Delay Volterra integro-differential equations arise as mathematical model in many areas of physics and biology, such as fluid mechanics, memory materials in petroleum mining, nuclear reactors, etc. The importance of research attracts attention of many scholars. Efficient numerical methods which can substantially reduce associated computational work are forming. Now we plan to investigate fractional Volterra integral and differential equations with delay. We use the spectral method to analyze their convergence. Firstly we use the spectral collocation method to investigate the convergence of Volterra integral equation in one dimension with pantograph delay, in the condition that the kernel and solution of the equation are sufficient smooth. We use some transformation to change the equation into a new equation defined on [-1,1]. Taking the test function, making discretization scheme by using quadrature formular to approximate the integral terms, we get the conclusion of convergence and provide a rigorous error analysis using spectral method. The desire conclusion is the errors of approximated solution decay exponentially in appointed norm. After that, we give some numerical examples to confirm our conclusion. Then we pay attention to the cases of nonlinear delay and normal delay. In the end, we plan to research delay Volterra integral equation that contains a weakly singular kernel and try to extend the conclusion to the case of nonlinear equations and multivariate dimension.

Volterra型微分积分方程出现在许多物理及生物领域数学模型中, 如流体力学、记忆性材料热传导问题、核反应堆问题, 其重要的研究意义使得该课题备受学者们关注, 呼吁着高效数值解法的产生. 本项目拟采用谱方法对时滞影响下的分数阶Volterra型微积分方程进行分析, 证明敛散性. 本项目拟先处理被积函数带比例时滞的一元分数阶Volterra型微积分方程, 在核和解充分光滑的条件下, 通过变换, 把原方程组化为定义在[-1,1]上, 再取检验函数, 并用相应的求积公式计算从而给出离散格式, 最后证明谱配置方法的收敛性并给出了严格的误差算法分析. 本项目研究期望获得的结论是: 在指定模意义下, 方程精确解与近似解的误差呈指数收敛, 并且给出数值例子进行验证. 紧接着将着手研究在非线性时滞乃至一般时滞影响下方程的敛散情况. 最后研究带弱奇异核以及方程为非线性情况, 并试图将结论延拓至多元情形.

Volterra型微分积分方程出现在许多物理及生物领域数学模型中,如流体力学、记忆性材料热传导问题、核反应堆问题,其重要的研究意义使得该课题备受学者们关注,呼吁着高效数值解法的产生.本项目采用谱方法对时滞影响下的分数阶Volterra型微积分方程进行分析,证明敛散性.首先处理被积函数带比例时滞的Volterra型积分方程,通过适当的变量变换,把原方程组化为定义在[-1,1]上,再取检验函数,并用相应的求积公式计算从而给出离散格式,最后证明谱配置方法的收敛性并给出了严格的误差算法分析，获得方程的真解与近似解的误差在和空间中呈现指数收敛的结论；接下来把研究的情形拓展至线性与非线性的情形，同样获得收敛结论，并且提供相应数值试样验证理论分析的正确性；然后再延拓至非线性分数阶的情况，不仅获得在指定模意义下, 方程精确解与近似解的误差呈指数收敛,还推导出精确导数与逼近导数误差关系；最后研究微积分Volterra方程收敛情形，期间也研究在一般时滞影响下方程的敛散情况，以及在核函数和解函数充分光滑的条件下研究具有弱奇性的时滞Volterra型方程的收敛情形。