Abstract
In this paper we consider the inverse time problem for the axisymmetric heat equation which is a severely ill-posed problem. Using the modified quasi-boundary value (MQBV) method with two regularization parameters, one related to the error in measurement process and the other related to the regularity of solution, we regularize this problem and obtain the Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.
Highlights
Partial differential equations (PDEs) associated with various types of boundary conditions are a powerful and useful tool to model natural phenomena
For time-dependent phenomena, they are usually joined by a time condition, which can be considered as the data
The backward heat conduction problem (BHCP) is the time-inverse boundary value problem, i.e., given the information at a specific point of time, say t = T, the goal is to recover the corresponding structure at an earlier time t < T
Summary
Partial differential equations (PDEs) associated with various types of boundary conditions are a powerful and useful tool to model natural phenomena. Trong and Tuan in [ ] used the method of integral equation to regularize backward heat conduction problem and get some error estimates. Similar to the case of a constant coefficient, the axisymmetric BHCP is an ill-posed problem: a small perturbation in the final data may cause dramatically large errors in the solution. In [ ], with a prior condition on the solution, we use the spectral method with a regularizing filter function to approximate problem
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