a new reproducing kernel method for solving the second order partial differential equation

In this study, a reproducing kernel hilbert space method with the chebyshev function is proposed for approximating solutions of a second-order linear partial differential equation under nonhomogeneous initial conditions. based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be erected in the reproducing kernel spaces spanned by the shifted chebyshev polynomials. the exact solution is given by reproducing kernel functions in a series expansion form, the approximation solution is expressed by an n-term summation of reproducing kernel functions. this approximation converges to the exact solution of the partial differential equation when a sufficient number of terms are included. convergence analysis of the proposed technique is theoretically investigated. this approach is successfully used for solving partial differential equations with nonhomogeneous boundary conditions.