Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682214220230201A new reproducing kernel method for solving the second order partial differential equation327339691310.22075/ijnaa.2022.24802.2832ENMohammadreza ForoutanDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran0000-0002-7373-617XSoheyla Morovvati DarabadDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, IranKamal FallahiDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, IranJournal Article20211007In 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.https://ijnaa.semnan.ac.ir/article_6913_f19d741b07a8bb3b33c61667f8d93be7.pdf