The Solution Of The Complex KdV Equations In The Reproducing Kernel Space

Complex KdV equations arise in a variety of applied subjects of Mathematica and Physics such as plasma,soliton and so on.Due to its extensive applications,more and more domestic and foreign scholars show a considerable interest to solve complex KdV equations.However,because of the problem of the nonlinear of the equations,previous numerical methods failed to produce good convergence to the exact solutions of the equations.Hence,it is a practical significance to find an efficient method satisfy that the solutions of complex KdV equtions have better convergence.Firstly,for the complex KdV equation,the paper establishes the corresponding reproducing kernel space and reproducing kernel function.Secondly,by separating the real and imaginary parts of complex KdV equation,complex KdV equation translated into real KdV equations.Using homotopy perturbation parameter,nonlinear problem of equations are solved.Then,the definite solution of differential equation is converted into equivalently into a system of operator equation.We defined a new reproducing kernel space constituted by the product of two one-dimensional reproducing kernel spaces.The operator equation can be solved by the reproducing kernel method which gives the exact solution and approximate solution of the complex KdV equation.According to the system's form,the new reproducing kernel space is defined,a complete orthogonal system is constructed and the iterative form is designed for the system of equations.Finally,iterative convergence is proved.