Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

Abstract

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h4 + k2). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L2 and L∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

“…The stability analysis of proposed improvised collocation technique is proved using von‐Neumann method which is similar as used by [33]. Put y as a local constant l 1 = max y to linearize the nonlinear term and then apply Crank–Nicolson scheme to discretize the temporal domain in Equation (1.1): Separating the ( n + 1) th and n th time level terms: Substituting the values of y , y x , and y xx using improvised cubic B‐splines: …”

“…Comparison of L 2 and L ∞ error norms with μ = 0.01, 0.005, and 0.001 is represented in Tables 1–3 respectively for t ∈ [1, 10]. Results are compared with various existing works such as septic B‐spline [7], quintic B‐spline [14, 16] sextic B‐spline collocation methods [19], also with fourth order finite difference scheme [21], Petrov–Galerkin method [22], quintic B‐spline differential quadrature method [26], Haar wavelet collocation method [32], and quintic trigonometric B‐spline collocation method [33]. The improvised technique is giving better or almost similar results and such an accuracy is obtained with very less number of collocation points and Δ t = 0.1.…”

“…Nagaveni [32] solved the equation by Haar wavelet collocation method and computed L ∞ and L 2 error norms along with rate of convergence. Nair and Awasthi [33] applied Crank–Nicolson scheme for temporal domain and quintic trigonometric B‐spline for spatial domain as well as used quasilinearization process to tackle with nonlinear terms.…”

Numerical Simulation and Analysis of the Modified Burgers' Equation in Dusty Plasmas

“…The authors in [32] provided an orthogonal collocation technique with septic Hermite splines as basis function to obtain the numerical solution of non-linear modified Burgers' equation. A numerical method based on quintic trigonometric B-splines for solving modified Burgers' equation (MBE) is presented in [33]. The arrangement of the manuscript is as follows.…”