A proposed Method for the Solution of One-Dimensional Gas Dynamic Equation with Legendre and Chebyshev polynomials

Abstract

We have considered a proposed method to efficiently handle the piecewise formulation in the generation of approximate solutions for gas dynamic equation via the Legendre and Chebychev basis functions. Two cases of the gas dynamic equation were considered for numerical illustrations aided by MAPLE 18 software. We did obtained some fascinating results. To be precise, applying the proposed method to the homogeneous gas dynamic via the Legendre polynomial, we attained a maximum error of order 10−4 as against that of VIM with a maximum error of order 10−3 at t = 0.0001. Also, the proposed method via Chebyshev polynomials at t = 0.0001 attained a maximum error of order 10−4 as against the VIM which has a maximum error of order 10−2. In like manner, applying the proposed method to the nonhomogeneous gas dynamic via the Legendre polynomials we attained a maximum error of order 10−4 at t = 0.0001 as against that of VIM, which does not show convergent. Also, proposed method via Chebyshev polynomials at t = 0.0001 attained a maximum error of order 10−4 as against the VIM which does converge at every grid points. Thus, it is obvious that the proposed method is a better convergent iterative scheme than the in as much as t decreases.