Several Numerical Differentiation Algorithms And Applications

The numerical differential problem is a typical ill-posed problem in the sense of Hardamard,in which small errors in the measurement process can cause huge er?rors in the numerical results.In this paper,two methods are presented to solve the numerical differential problem.One method is the regular solution of constructing the numerical differential problem based on mollification method.The basic idea is to mollify the inaccurate measurement data and to derive an approximate mol-lification.function.This mollification is obtained by convolution with mollification kernel function.Then the method is applied to the error estimation of Abel inte-gral equation.Another method is generalized Hermite spectral and pseudospectral method,a mollification method based on generalized Hermite function.The basic idea is by properly choosing the scale factor a in generalized Hermite function to reduce the number of projection space bases so that the calculation efficiency can be improved.These two methods are both analyzed theoretically in this paper and the error estimates of the regular and exact solutions are obtained.The corresponding numerical experiments show the effectiveness of the two methods.