Problem Of Second Order Numerical Differentiation

Numerical differential problems are to resolve the approximate derivatives when functions' value on some of discrete points. Differentiation is the inverse problem of Integral, though it look simple, it is much more complex than Integral. The errors of the obtained derivatives could be arbitrarily large when errors occur in the function's values, which are inevitable in practical application. Numerical differential problems are representative ill-posed problems. To obtain the reasonable results, it is necessary to use some special methods in ill-posed problems' resolving.There are a lot of methods for solving the numerical differential problems, such as difference and general difference method, smoothing method, integral method, Tikhonov method and other methods based on the general regularization theory. In which, Groetsch proposed an integral method with the advantages as follows: firstly it can give uniform errors' estimation with easy calculation. Secondly it can construct similar integral operators to improve errors' precision when functions' smooth properties are strengthened. In some practical applications, not only first-order numerical differentiation but also second or higher order numerical differentials are need. Presently there are many research works on second order numerical differential problems based on difference or Tikhonov regularization methods. But there are little works on second or higher order numerical differential problems through integral method.A novel integral operator method enlightened by Groetsch's ideas is proposed. It can approximate second-order derivative of approximately specified functions And it can give the corresponding error estimation when applied into Numerical differential problems. Finally, some numerical experiments on calculations of second-order derivative with the put method are done, and the results show that proposed approach is simple, stable and it can be realized fast.