Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials

In this article an efficient modification of the homotopy perturbation method is presented by using Chebyshev polynomials. Special attention is given to prove the convergence of the method. Some examples are given to verify the convergence hypothesis, and illustrate the efficiency and simplicity of the method. We compared our numerical results against the conventional numerical method, fourth-order Runge–Kutta method (RK4). From the numerical results obtained from these two methods we found that the proposed technique and RK4 are in excellent conformance. From the presented examples, we found that the proposed method can be applied to a wide class of linear and non-linear ODEs.