退化抛物型方程的一个初值反演问题

An Inverse Initial Value Problem for Degenerate Parabolic Equations

研究了一类重构退化抛物型方程初值的反问题.这类问题在应用科学的若干领域有着重要的应用.数值求解该问题的关键是构造相应正问题的高阶差分格式.然而,由于退化边界上的主项系数为零,目前广泛用于求解经典热传导方程的虚拟点法不能应用于该模型.该文提出了一种构造二阶精度差分格式的新方法,并证明了该方法的稳定性和收敛性.为了加快收敛速度,采用共轭梯度法求逆问题的数值解,并对算法的效率和精度进行了数值验证.

This paper investigates an inverse problem of reconstructing the initial value in a degenerate parabolic equation.Problems of this type have important applications in several fields of applied science.The key to numerically solve such problem is to construct highorder difference schemes for corresponding forward problem.However,the dumping point method which is widely-used for numerically solving classical heat conduction equations cannot be applied to degenerate parabolic equations,because the principal coefficients are zero on degenerate boundaries.In this paper,a new but quite simple technique is proposed to construct a difference scheme of second order accuracy,and the stability and convergence of the scheme are proved.In order to accelerate the convergence rate,the conjugate gradient method is adopted to obtain numerical solutions of the inverse problem.Numerical verification on the efficiency and accuracy of the proposed algorithm is also performed.