Numerical solution of backward heat conduction problems by a high order lattice-free finite difference method

We construct a high order finite difference method in which quadrature points do not need to have a lattice Structure. In order to develop our method we show two tools using Fourier transform and Taylor expansion, respectively. On the other hand, the backward heat conduction problem is a typical example of ill-posed problems in the sense that the solution is unstable for errors of data. Our aim is creation of a meshless method which can be applied to the ill-posed problem. From numerical experiments we confirmed that our method is effective in solving two-dimensional backward heat conduction equations subject to the Dirichlet boundary condition.