Numerical solution of third order linear differential equations using generalized one-shot operational matrices in orthogonal hybrid function domain

Abstract

The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of orthogonal sample-and-hold functions (SHF) and triangular functions (TF). The HF set is used to approximate a time function in a piecewise linear manner. For such approximation, the mean integral square error (MISE) is much less than block pulse function based approximation which always provides staircase solutions and it is found that HF based approximation is a strong contender of approximations based upon orthogonal polynomials like Legendre polynomials. The respective MISE’s and elapsed times for MISE computations for these three kinds of approximations are also studied in detail.The one-shot operational matrices for integration of different orders in HF domain are also derived. These matrices are employed for more accurate multiple integration. An example is treated to determine efficiency of these matrices. Finally, these matrices are employed for solving third order non-homogeneous differential equations followed by a numerical example. The results, obtained via MATLAB, are compared with the exact solution as well as the results obtained via 4th order Runge–Kutta method.