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

Abstract

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B-spline as basis functions. In this technique, cubic B-splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank-Nicolson scheme is used for temporal domain and improvised cubic B-spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h 4 + Δt 2). Stability of the technique is examined using von-Neumann analysis. L 2 and L ∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time. KEYWORDS How to cite this article: Shallu, Kukreja VK. An improvised collocation algorithm with specific end conditions for solving modified Burgers equation.

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

“…Bratsos et al [30] employed the explicit finite difference scheme to numerically solve the equation.Numerical solution of nonlinear modified Burgers' equation EEJP. 4(2023) is obtained using an improvised collocation technique with cubic B-spline as basis functions in [31]. 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 Cubic B-Spline Collocation Method for Barrier Options under the CEV Model

“…Shallu and V.K. Kukreja [11] obtained a numerical solution to the nonlinear modified Burgers equation using an improvised collocation technique with cubic B-spline as basis functions. In this method, the cubic B-spline is forced to satisfy interpolant conditions with some specific end conditions.…”

Decatic B‐spline collocation scheme for approximate solution of Burgers' equation

“…Equation (1) is nonlinear partial differential equation which is applicable in many fields of sciences such as in approximation theory involving wave propagation in viscous fluid [16], heat conduction and continuous stochastic process [8], gas dynamics [34], longitudinal elastic waves in an isotropic solid [42]. Various efforts are taken to obtain the approximate solution of Equation (1) using different numerical schemes such as meshfree quasi‐interpolation scheme [33], variational scheme [1], optimized weighted essentially non‐oscillatory third order scheme [5], quasi‐linear technique [11], weighted average differential quadrature scheme [29], sinc differential quadrature scheme [31], septic B‐spline scheme [45], cubic B‐spline approaches [14, 51], non‐polynomial spline approach [44], hybrid numerical approach [27], quadratic and cubic B‐splines finite element scheme [13, 46], finite difference method [20, 32], finite element scheme [24, 41], automatic differentiation [6], differential quadrature scheme [36, 37], efficient numerical scheme [39], linearized implicit scheme [40], B‐spline Galerkin methods [15], Haar wavelet quasilinearization scheme [26], multiquadratic quasi‐interpolation technique [10], exponential twice continuously differentiable B‐spline scheme [18], quartic B‐spline collocation method [30], implicit and fully implicit exponential finite difference schemes [23], quintic B‐spline collocation scheme [47], combinations of the wavelet and finite volume scheme [38], semi‐implicit finite difference approach [43], modified cubic B‐spline scheme [35], improvised collocation scheme with cubic B‐spline as basis functions [19], radial basis functions with meshfree algorithms [28], two meshfree approaches [48], cubic trigonometric B‐spline approach [50], meshless technique by applying Lie group integrator with multiquadric radial basis functions [49], MIEELDLD technique [4], and so on.…”