The Computation Of Reproducing Kernel And Its Application In Numerical Solution Of The Equation

In this paper, we have discussed the method for constructing and computing the reproducing kernel in the reproducing kernel space W₂^m[a,b]. We also have discussed the best approximating operator and the best approximation of the linear functionals. Especially, we discuss the numerical solution of boundary-value problem of the differential equation in two special reproducing kernel spaces.In chapter three, we introduce the differential equation method for computing the reproducing kernel in the situation of defining specific inner product, and discuss the relationship between the reproducing kernel of W₂^m[a,b] possessing the specific inner product and the Green's function of some specific differential operator. And then, based on the method for constructing the reproducing kernel by using differential operator, we discuss the recursive property of this method. The reproducing kernels of two special Hilbert spaces are also given.In chapter four, we have discussed the approximating problem and the numerical solution problem of the differential equation in W₂^m[a,b]. We have demonstrated that the uniformity of spline interpolating operators and the best operators of interpolating approximation in W₂^m[a,b]. At the same time, we discuss the best approximation of the linear functionals. And then, we introduce the reproducing kernel method for solving differential equation. We discuss the numerical solution of boundary-value problem of the differential equation in W₂¹ (*) and W₂² (*), and give the numerical example. We also compare the reproducing kernel method with the traditional method. These exemplify that the reproducing kernel method for solving differential equation is not only feasible in theory but also in practice.