Fourier truncation method for high order numerical derivatives

The truncation method for identifying an unknown source in the Poisson equation

“…Eldén et al [8] used the truncation method to analyze and compute one-dimensional IHCP, Xiong et al [9] used it to consider the surface heat flux for the sideways heat equation, Fu et al [10] used it to solve the BHCP, Qian et al [11] used it to consider the numerical differentiation, Regiń ska and Regiń ski [12] applied the idea of truncation to a Cauchy problem for the Helmholtz equation. In [8][9][10][11][12], the ill-posedness of the problem were caused by the high frequency components, they all used the truncation method to eliminate all high frequencies, and called this truncation method Fourier method. In [13], the authors ever identified the unknown source on Poisson equation on half unbounded domain using Fourier Transform.…”

“…11), small errors in the components of large n can blow up and completely destroy the solution. A nature way to stabilize the problem is to eliminate all the components of large n from the solution and instead consider (1.11) only for n 6 N. Then we get a regularized solution e Àn ðg d ; X n ÞX n :ð2:1ÞNote (2.1), if the parameter N is large, f d,N (x) is close to the exact solution f(x).On the other hand, if the parameter N is fixed, f d,N (x) is bounded.…”

Fourier regularization for a backward heat equation

“…The essence of Fourier regularization method is just to eliminate all high frequencies from the solution, and instead consider (1.5) only for |ξ | < ξ max , where ξ max is an appropriate positive constant. Recently, Fourier regularization method has been effectively applied to solve the sideways heat equation [17,18], a more general sideways parabolic equation [19] and numerical differentives [20]. This regularization method is rather simple and convenient for dealing with some ill-posed problems.…”

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

“…The Fourier truncation regularization method is a very effective method for dealing with ill-posed problems. Many authors have used it to deal with different ill-posed problems, such as in [23][24][25][26][27][28][29][30][31][32]. In [33], the authors extended the Fourier method to the general filtering method and solved the semi-linear ill-posed problem in the general framework.…”