A hybrid collocation method based on combining the third kind Chebyshev polynomials and block-pulse functions for solving higher-order initial value problems

The purpose of this paper is to propose a new collocation method for solving high order linear and nonlinear differential equations on a very large interval as well as solving the differential equation problem on a small interval. The new collocation method is based on a hybrid method combining the third kind Chebyshev polynomials and Block-Pulse functions. In the proposed method, the large interval of the problems is divided into small sub-intervals and in each sub-interval, collocation method turns the differential equation into a set of algebraic equations. Solving such system makes an approximate solution of the differential equation on each sub-interval. The error of approximate solution has upper-bound of O(m(-r)/root N). It means that, the errors decrease as and increase. The proposed method is more accurate than the previous methods. Numerical examples show the capability and efficiency of the presented method compared to existing methods.