Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

Kundu, Sudeep; Pani, Amiya K.

doi:10.1007/s10915-019-01039-5

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…The 1‐D GBBMB equation is solved numerically by two high‐order implicit difference schemes in [19]. Kundu and Pani [20] proved global stabilization results for the BBMB equation under nonlinear Neumann boundary control laws. Readers can see [21–23] for further information about GBBM equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Dehghan

Shafieeabyaneh

Abbaszadeh

2020

Numerical Methods Partial

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This article is devoted to solving numerically the nonlinear generalized Benjamin-Bona-Mahony-Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi-discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss-Legendre-Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the oneand two-dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

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Section: Introductionmentioning

confidence: 99%

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Dehghan

Shafieeabyaneh

Abbaszadeh

2020

Numerical Methods Partial

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“…Subsequently in [19], optimal error estimates in the context of finite element method for the state variable and superconvergence result for the feedback control laws are derived. For related analysis on Benjamin Bona Mahony Burgers' (BBM-Burgers') type equations, we refer to [20]. Concerning extensive literature for one dimensional Burgers' and BBM-Burgers' problem, see the references in [19,20].…”

Section: Introductionmentioning

confidence: 99%

“…For related analysis on Benjamin Bona Mahony Burgers' (BBM-Burgers') type equations, we refer to [20]. Concerning extensive literature for one dimensional Burgers' and BBM-Burgers' problem, see the references in [19,20].…”

Section: Introductionmentioning

confidence: 99%

Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

Kundu

Pani

2020

J Sci Comput

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In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $$C^0$$ C 0 -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ L ∞ ( L 2 ) and in $$L^\infty (H^1)$$ L ∞ ( H 1 ) -norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

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“…Subsequently in [14], optimal error estimates in the context of finite element method for the state variable and superconvergence result for the feedback control laws are derived. For related analysis on BBM-Burgers' type equation, we refer to [15].…”

Section: Introductionmentioning

confidence: 99%

Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

Kundu¹,

Pani²

2018

Preprint

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In this article, global stabilization results for the two dimensional viscous Burgers' equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, using C 0 -conforming finite element method in spatial direction, global stabilization results for the semidiscrete solution are shown. Moreover, optimal error estimates in L ∞ (L 2 ) and in L ∞ (H 1 )norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

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Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

Cited by 4 publications

References 23 publications

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

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