A Research Of Numerical Differentiation Based On Two-variable Spline

Numerical differentiation is to approximate the derivative of the function at a certain point by using a linear combination of the values of the function at some discrete points[1].It can be widely used in many fields of science and engineering to solve many practical problems,so it is of great significance to obtain a stable numerical differential formula for the study of differential problems.Since last century,many scholars proposed many methods for constructing stable numerical differential schemes.In 2015,D.Barrera et al.proposed a method for constructing numerical differential formats based on the idea of differential quadrature:first,constructed an interpolation operator(?)(f)based on B spline,and the approximate form can be obtained by taking the first derivative of the operator.Based on the construction method of D.B ararrera et al.,this paper explore how to con-struct numerical differential formulas in binary cases.The main work of this paper includes two aspects:first,an interpolation operator based on binary product splines is constructed by using the relevant theory obtained in[17],by taking the first partial derivative of the op-erator,the approximate form of the corresponding first partial derivative of the function f can be obtained;the second aspect is the main research content,that is,the construction of numerical differential format based on Box spline.Box spline N(2,2,2)is the main research object,to construct an interpolation operator that can accurately approximate polynomials of degree 3 and below,the differential forms(?)(2,2,2)(1,0)(f)and(?)(2,2,2)(0,1)(f)are obtained,and to any 3 times and the following accurate approximation of the second order partial derivative polynomial differential format(?)(2,2,2)(1,1)(f),(?)(2,2,2)(2,0)(f),(?)(2,2,2)(0,2)(f).By numerical experiments,it approximates the first and second partial derivatives of the given function,and the results show that the derivative formula based on Box spline has good approximation property.