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1

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.

2

Parameter identification for stochastic burgers' flows via parabolic rescaling

Leonenko N. N, Woyczynski W. A.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

1--55

EN

The paper presents a systematic study of classical statistical inference problems (parameter estimation and hypothesis testing) for random fields arising as solutions of the one-dimensional nonlinear diffusion equation with random initial data (the Burgers' turbulence problem). This nonlinear, hydrodynamic-type partial differential equation is an ubiquitous model in physics and engineering. This work can be seen as part of a larger program of developing statistical inference tools for complex stochastic flows governed by nontrivial, physically constrained dynamics.

3

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.