Application of Homotopy Perturbation Method with Chebyshev Polynomials to Nonlinear Problems

Nonlinear differential equations arise in a wide variety of problems such as fluid dynamics, quantum field theory, and plasma physics to describe the various phenomena. These problems, a limited number of them apart, do not have a precise analytical solution, so these nonlinear equations should be solved using approximate methods. The homotopy perturbation method (HPM), first proposed by He in 1998, was developed and improved by He [1 – 3]. Very recently, the new interpretation and new development of the homotopy perturbation method have been given and well addressed in [4 – 7]. Homotopy perturbation method [1 – 7] is a novel and effective method, and can solve various nonlinear equations. This method has successfully been applied to solve many types of linear and nonlinear problems, for example, nonlinear oscillators with discontinuities [8], nonlinear wave equations [9], limit cycle and bifurcations [10 – 13], nonlinear boundary value problems [14], asymptotology [15], Volterra’s integro-differential equation by El-Shahed [16], some fluid problems [17 – 18], Chen system and other systems of equations [19-20], singular problems [21], and many other problems [22 – 23] and the references therein. These applications also verified that the HPM offers certain advantages over other conventional numerical methods. Numerical methods use discretization which gives rise to rounding off errors causing loss