Mathematical Sciences: Determination of Nonlinear Terms in Parabolic and Elliptic Partial Differential Equations and Overposed Data

本课题的目的是研究含有未知系数的因变量函数的椭圆型和抛物型偏微分方程的若干反问题。这些系数可以通过给出叠加的边界数据来恢复，它规定了Dirichlet型数据和位于边界区域的选定曲线。该方法将反问题转化为一个不动点问题，其中有一个从未知系数到叠加数据的映射。通过迭代方案求解不动点问题。作者已经能够在某些情况下证明该方案的收敛性。他们现在提出了一些需要调查的研究问题，以提高对该技术的认识。反问题出现在许多应用中，在这些应用中，人们已经观察到数据，问题是找到描述现象的方程的参数。连续介质力学、数学生物学和化学燃烧理论是这种逆方法的潜在应用领域。

The aim of this project is to investigate certain inverse problems for elliptic and parabolic partial differential equations that contain unknown coefficients which are functions of the dependent variables. These coefficients can be recovered by giving overposed boundary data, that it prescribed Dirichlet type data and a chosen curve lying on the boundary region. The method consists of converting the inverse problem into a fixed-point problem in which one has a mapping from the unknown coefficients to the overposed data. The fixed point problem is solved through an iteration scheme. The authors have been able to demonstrate the convergence of that scheme in some cases. They propose now a number of research questions to be investigated to improve the knowledge of the technique. The inverse problems arise in a number of applications where one has observed data and the problem is to find the parameters of the equation describing the phenomenon. Continuum mechanics, mathematical biology and chemical combustion theory are among potential applications of this inverse method.