三次样条插值及其在第一类积分方程求解中的应用

Cubic Spline Interpolation and Its Application in Solving the First Kind of Integral Equation

从三次样条插值的定义出发,通过研究第一类积分方程中未知函数的三次样条函数逼近,给出了第一类积分方程的三次样条插值离散化.利用该离散化形式,将第一类积分方程转化成线性方程组的形式.由于第一类积分方程的求解通常是不适定的,进而引起线性方程组的病态性.最后,为克服线性方程组的病态性,通过引入未知函数的多重光滑化约束,得到第一类积分方程的稳定解.

Starting from the definition of cubic spline interpolation,the discretization of cubic spline interpolation for the first kind of integral equation is given by studying the cubic spline approximation of unknown functions in the first kind of integral equation.Using the discretization form,the integral equation of the first kind is transformed into the form of linear equations.Because the solution of the first kind of integral equation is usually ill-posed,the ill-posed linear equations are caused.Finally,in order to overcome the ill-health of the linear equations,the stable solution of the first kind of integral equation is obtained by introducing multiple smoothing constraints of unknown functions.