Abstract
An innovative scheme of collocation having quintic Hermite splines as base functions has been followed to solve Burgers’ equation. The scheme relies on approximation of Burgers’ equation directly in non-linear form without using Hopf–Cole transformation (Hopf in Commun Pure Appl Math 3:201–216, 1950; Cole in Q Appl Math 9:225–236, 1951). The significance of the numerical technique is demonstrated by comparing the numerical results to the exact solution and published results (Asaithambi in Appl Math Comput 216:2700–2708, 2010; Mittal and Jain in Appl Math Comput 218:7839–7855, 2012). Five problems with different initial conditions have been examined to validate the efficiency and accuracy of the scheme. Euclidean and supremum norms have been reckoned to scrutinize the stability of the numerical scheme. Results have been demonstrated in plane and surface plots to indicate the effectiveness of the scheme.
Highlights
Burgers’ equation is a partial differential equation that was originally proposed as a simplified model of turbulence as exhibited by the full-fledged Navier–Stokes equations
Let E = u(x, t) − uγ, where E defines the pointwise error, u(x, t) being the exact solution and uγ is the approximate solution obtained by quintic Hermite collocation method
5.3780 × 10−9 norms have been calculated for ε = 0.005, σ = 100 and t = 0.001. It has been distinguished from these tables that both Euclidean as well as supremum norms computed by Quintic Hermite collocation method (QHCM) are better than the norms estimated by modified cubic B-spline collocation method [25]
Summary
Burgers’ equation is a partial differential equation that was originally proposed as a simplified model of turbulence as exhibited by the full-fledged Navier–Stokes equations. A numerical solution of Burgers’ equation has been achieved by applying quintic Hermite collocation method directly, i.e., without transforming the non-linear term into linear one using Hopf–Cole transformation. The Burgers’ equation with ε as viscosity parameter, has been solved numerically using quintic Hermite collocation method. It has been elucidated in different forms using distinct initial conditions in terms of constant, and polynomial and trigonometric functions. Quintic Hermite polynomials of order 5, i.e., k = 2, have been taken as base function to approximate the trial function Details of graphical representation and structure of elements in QHCM using quintic Hermite interpolating polynomials are given in [3]
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