Study On Some Nonlinear Problems In The Reproducing Kernel Space

In the many fields of science and engineering, a lot of problems can be modeled by the nonlinear equation. Obtaining the solution (the numerical solution or the exact solution) of the nonlinear equation, will help to analyze and study the practical phenomenon described by the nonlinear equation. Thus, this thesis discusses the following problems with respect to the nonlinear equation.Firstly, the reproducing kernel space, proposed by the Professor Cui in 1986, is studied. The two-dimensional reproducing kernel space is constructed by employing the direct product space of one-dimensional reproducing kernel spaces, and its reproducing kernel function is obtained. Other multi-dimensional reproducing kernel spaces can be extended similarly. Consequently, the theory of space solving all kinds of equations is established.Secondly, in the reproducing kernel space W₂¹[a,b] , the exact solution of the nonlinear operator equation A₁(Ï…BÏ…) + A₂Ï… = f is given. Employing the special properties of the reproducing kernel function and the dimension raising method, the bounded and linear operator is defined in the reproducing kernel space W₁(Ω) . So, the nonlinear operator equation is equivalently changed into the linear operator equation Tu = f . In the case that the solution is unique, the exact solution of the nonlinear operator equation is obtained. And numerical examples show the effectiveness of this algorithm.Generalizing the above conclusions, this thesis discusses the nonlinear integro-differential equation AÏ…²+KÏ…+PÏ…+EÏ… = f in the two-dimensional reproducing kernel space W₂(Ω). The nonlinear equation is solved by transforming into the linear operator equation in the four-dimensional reproducing kernel space. When the solution for the linear equation is not unique, this thesis discusses the solution space of homogeneous equation and gives an orthonormal system, which yields all solutions of linear equation. From the relations of solutions for both equations, a numerical algorithm of solving nonlinear operator is obtained. To prove the algorithm valid, numerical experiences are proceeded.In addition, the Black Body Radiation Problem, arisen in practical problems, shows that the ill-posed problem is changed into well-posed problem in the reproducing kernel space, when the integral kernel function satisfies certain conditions for the first kind Fredholm integral equation. At the same time, a stably...