A new semi-analytical polynomial method (SAPM) for solving differential equations in mechanical engineering problems

In this paper, a new polynomial-based method is presented to solve the ordinary and partial differential equations system with various types of boundary conditions. A simple polynomial formulation is defined for the functions of both types of ordinary and partial differential equations system. Then the system discretized to an algebraic equations system. Whether the algebraic system is linear or nonlinear, the Newton-Raphson numerical method is applied to solve the algebraic system. The accuracy and the rate of calculations and convergence are compared with the other analytical and numerical methods such as finite difference method (FD), generalized differential quadrature method (GDQM), extended Kantorovich method (EKM) and with some problems that the analytical solutions are available. Several applied engineering problems such as the permanent Poisson heat transfer, bending of circular and graphene plates are analyzed using the present method and the accuracy of this method is proved by comparison with the other references. Although simplicity in formulation, the efficiency and accuracy of this method is proved by comparing the various methods and problems with different types of boundary conditions