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1

A numerical approximation of 2D coupled burgers’ equation using modified cubic trigonometric B-spline differential quadrature method

Kapoor Mamta, Joshi Varun

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 3

79--102

EN

In the present paper, trigonometric B-spline DQM is applied to get the approximated solution of coupled 2D non-linear Burgers’ equation. This technique, named modified cubic trigonometric B-spline DQM, has been used to obtain accurate and effective numerical approximations of the above-mentioned partial differential equation. For checking the compatibility of results, different types of test examples are discussed. A comparison is done between 2L and L∞ error norms with the previous, present results and with the exact solution. The resultant set of ODEs has been solved by employing the SSP RK 43 method. It is observed that the obtained results are improved compared to the previous numerical results in the literature.

2

A hybrid solution approach to the Korteweg-de Vries and Burgers' equations

Shah Kunjan, Patel Himanshu

Mathematica Applicanda

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2021

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Vol. 49, no. 2

159--170

EN

The primary purpose of this paper is to analyze the application of a new integral transform together with a homotopy perturbation method to construct approximate solutions of the initial-value problem for Korteweg-de Vries and Burgers’ equations. The new integral transform homotopy perturbation method (NIHPTM) compared to other methods, offers the simple technique to handle such type partial differential equations. The 5th-order approximation results obtained in illustrative examples compared with the explicit solutions of the considered problems show the proposed approach’s efficiency and validity.

PL

Głównym celem tego artykułu jest analiza zastosowania nowej transformacji całkowej i metody homotopijnej perturbacji do konstrukcji przybliżonych rozwiązań zagadnienia początkowego dla równań Kortewega-de Vriesa i Burgersa. Nowa metoda homotopijnej perturbacji z transformacją całkową (NIHPTM) w porównaniu z innymi metodami oferuje prostą technikę do zastosowania w tego typu równaniach różniczkowych cząstkowych. Uzyskane aproksymacje piątego rzędu dla przykładów ilustracyjnych porównane z istniejącymi jawnymi rozwiązaniami rozważanych zagadnień pokazują skuteczność i trafność proponowanego podejścia.

3

Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination

Hadhoud Adel R., Alaal Faisal E. Abd, Abdelaziz Ayman A., Radwan Taha

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

436--451

EN

In this paper, we show how to approximate the solution to the generalized time-fractional Huxley-Burgers’ equation by a numerical method based on the cubic B-spline collocation method and the mean value theorem for integrals. We use the mean value theorem for integrals to replace the time-fractional derivative with a suitable approximation. The approximate solution is constructed by the cubic B-spline. The stability of the proposed method is discussed by applying the von Neumann technique. The proposed method is shown to be conditionally stable. Several numerical examples are introduced to show the efficiency and accuracy of the method.

4

On the evolution of solutions of Burgers equation on the positive quarter-plane

Hanaç Esen

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

237--248

EN

In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; vt + vvx - vxx = 0, x>0, t>0, v(x,0) = u+, x>0, v(0,t) = ub, t>0, where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

5

Reduced order controllers for Burgers' equation with a nonlinear observer

Atwell J. A., Borggaard J. T., King B. B.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1311-1330

EN

A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal decomposition (POD). However, the test problem was the two-dimensional heat equation, a problem in which the physics dominates the system in such a way that controller efficacy is difficult to generalize. Here, we additionally incorporate a nonlinear observer by including the nonlinear terms of the state equation in the differential equation for the compensator.

6

Second-order conditions for boundary control problems of the Burgers equation

Volkwein S.

Control and Cybernetics

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2001

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Vol. 30, no 3

249-278

EN

In this article control constrained optimal control problems for the Burgers equation are considered. First- and second-order optimality conditions are presented. Utilizing polyhedricity of the feasible set and the theory of Legendre-forms a second-order sufficient optimality condition is given that is very close to the second-order necessary optimality condition. For the numerical realization a prima-dual actrive set strategy is used.

7

Resonant interactions of nonlinear transverse waves in incompressible magnetoelasticity

Domański W.

Journal of Technical Physics

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1999

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Vol. 40, no 4

435-446

EN

The asymptotic expansion of high-frequency small-amplitude waves is applied to the equations of a nonlinear, elastic, incompressible solid interacting with a magnetic field. The single space dimension case is discussed. The evolution transport equations of weakly nonlinear geometric optics for the interacting nonlinear magnetoelastic waves are derived. In contrast to the zero background field, the nonzero prestrain initial data (around which the asymptotics is applied) imply that some of the resonant interacting coefficients are nonzero. All such coefficients are calculated explicitly and analysed for a certain nonzero constant state. Finally the equations for resonant triads are displayed.