A qualocation method for Burgers’ equation

Mantri, Pritam S.; Nataraj, Neela; Pani, Amiya K.

doi:10.1016/j.cam.2006.12.028

Cited by 6 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Error estimates for one dimensional Burgers equation are established and verified by a numerical experiment in [39]. The paper [37] obtains an optimal error estimate for one dimensional Burgers equation with the numerical investigation. Using a weak Galerkin finite element method, [19] establishes optimal order error estimates for one dimensional Burgers equation.…”

Section: Introductionmentioning

confidence: 88%

“…Due to the wide applications of Burgers equations, various numerical methods are proposed to solve it. The finite element method is used extensively to solve Burgers equations, and we refer to [17,19,22,37,39] and references therein. Error estimates for one dimensional Burgers equation are established and verified by a numerical experiment in [39].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Local stabilization of viscous Burgers equation with memory

Akram¹,

Mitra²

2022

EECT

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this article, we study the local stabilization of the viscous Burgers equation with memory around the steady state zero using localized interior controls. We first consider the linearized equation around zero which corresponds to a system coupled between a parabolic equation and an ODE. We show the feedback stabilization of the system with any exponential decay <inline-formula><tex-math id="M1">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \omega\in (0, \omega_0) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M3">\begin{document}$ \omega_0>0 $\end{document}</tex-math></inline-formula>, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, <inline-formula><tex-math id="M4">\begin{document}$ \omega_0 $\end{document}</tex-math></inline-formula> is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay <inline-formula><tex-math id="M5">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \omega>\omega_0 $\end{document}</tex-math></inline-formula>, using any <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-control. Finally, we obtain the local stabilization result for the nonlinear system by means of the feedback control stabilizing the linearized system using the Banach fixed point theorem.</p>

show abstract

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Local stabilization of viscous Burgers equation with memory

Akram¹,

Mitra²

2022

EECT

View full text Add to dashboard Cite

show abstract

“…Based on cubic B-spline quasi-interpolation, Zhu et al [121] proposed another solution. Mantri et al [122] presented a qualocation technique for (1). Whereas [123][124][125] …”

Section: Survey Of Different Techniquesmentioning

confidence: 99%

Contemporary review of techniques for the solution of nonlinear Burgers equation

Dhawan

Kapoor

Kumar

et al. 2012

Journal of Computational Science

View full text Add to dashboard Cite

“…In recent years, Burgers' equation has been motivated considerable research into numerical methods by many authors (Refs. ). In this article, we present a new method for solving the following one‐dimensional homogeneous variable‐coefficient Burgers' equation with initial and boundary conditions

true{\begin{matrix} u_{t} + b_{1} (x, t) u_{x x} + b_{2} (x, t) u + b_{3} (x, t) u_{x} + b_{4} (x, t) u u_{x} = f (x, t), 0 \leq x \leq 1, 0 \leq t \leq 1, \\ u (x, 0) = 0, \\ u (0, t) = 0, u (1, t) = 0, \end{matrix}

where

b_{1} (x, t), b_{2} (x, t), b_{3} (x, t), b_{4} (x, t)

and

f (x, t) \in W_{1} (D) .

…”

Section: Introductionmentioning

confidence: 97%

Numerical solution of one‐dimensional Burgers' equation

Fei

Han

et al. 2014

Numerical Methods Partial

View full text Add to dashboard Cite

In this article, an iterative method for the approximate solution of a class of Burgers' equation is obtained in reproducing kernel space W 2 (D). It is proved the approximation w n (x, t) converges uniformly to the exact solution u(x, t) for any initial function w 0 (x, t) ∈ W 2 (D) under trivial conditions, the derivatives of w n (x, t) are also convergent to the derivatives of u(x, t), and the approximate solution is the best approximation under the systemwhere b 1 (x, t), b 2 (x, t), b 3 (x, t), b 4 (x, t) and f (x, t) ∈ W 1 (D).

show abstract

A qualocation method for Burgers’ equation

Abstract: In this paper, a qualocation method for the one-dimensional Burgers' equation is proposed. A semidiscrete scheme along with optimal error estimates is discussed. Results of a numerical experiment performed support the theoretical results.

Cited by 6 publications

References 19 publications

Local stabilization of viscous Burgers equation with memory

Local stabilization of viscous Burgers equation with memory

Contemporary review of techniques for the solution of nonlinear Burgers equation

Numerical solution of one‐dimensional Burgers' equation

Contact Info

Product

Resources

About