Abstract
A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.
Highlights
The reproducing kernel method of solving differential equation is one of the important topics of reproducing kernel numerical methods
The forms of reproducing kernels obtained by solving high-order differential equation are complicated and have no unified structural properties
The above proof is independent of the inner product of W2s[0, 1] and the concrete form of reproducing kernel
Summary
The reproducing kernel method of solving differential equation is one of the important topics of reproducing kernel numerical methods. Zhang [6] and Wu and Lin [7] specially introduced the reproducing kernel methods of solving linear differential equation based on reproducing kernel theory. Shi et al [8] and Du and Zhang [9] obtained the analytical solutions and the corresponding approximate solutions of initial value problems of linear ordinary differential equations and error estimation was considered. Castro et al [11] introduced a new method to solve general initial value problems by means of reproducing kernel theory. These studies were generally concentrated on constructing the corresponding reproducing kernels for solving specific equations. Mathematical Problems in Engineering a unified reproducing kernel theory and method for solving arbitrary linear differential equation with arbitrary linear functional constraint
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