IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

In this paper, we construct asymptotic periodic solutions of some generalized Burgers equations using a perturbative approach. These large time asymptotics (constructed) are compared with relevant numerical solutions obtained by a finite difference scheme. © 2018, Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature.

Department of Mathematics

Applied Mathematics

Asymptotic periodic solutions of some generalized Burgers equations

This paper concerns the separable solutions of a generalized Burgers equation. Existence of separable solutions to the generalized Burgers equation is proved under certain conditions. A careful numerical study shows that these separable solutions of the generalized Burgers equation describe the large time asymptotic behavior of solutions of initial boundary value problems. © 2015 Wiley Periodicals, Inc., A Wiley Company.

Studies in Applied Mathematics

A study of separable solutions of a generalized burgers equation

In this paper, we discuss the large-time asymptotics for periodic solutions of a generalized Burgers equation with variable viscosity (GBEV). Large-time asymptotics for the solutions of the GBEV depend on the parameters present in the partial differential equation and also the period of the solution of the GBEV. Large-time asymptotic expansions of the solutions are obtained by improving the solution of the linearized GBEV for certain parametric regionsviaa perturbative approach. These constructed large-time asymptotics are compared with the corresponding numerical solutions and are found to be in good agreement for large time. For certain other parametric region, our numerical study suggests that the solution of the inviscid GBEV describes the large-time behavior of the periodic solutions of the GBEV. © 2011 by the Massachusetts Institute of Technology.

Large-time asymptotics for solutions of a generalized Burgers equation with variable viscosity

Large-time behaviour of the entropy solution of an initial-value problem (IVP) for the inviscid non-planar Burgers equation is studied. The initial profile is assumed to be non-negative, bounded and compactly supported. The large-time behaviour of the support function of this entropy solution is also presented. This is achieved via the construction of the entropy solution of the IVP for the inviscid non-planar Burgers equation subject to the top-hat initial condition using the method of characteristics, Rankine-Hugoniot jump condition, and a similarity solution. © 2010 Springer Science+Business Media B.V.

Journal of Engineering Mathematics

Large-time behaviour of solutions of the inviscid non-planar Burgers equation

In this article, we construct solutions of a nonhomogeneous Burgers equation subject to certain unbounded initial profiles. In an interesting study, Kloosterziel [1] represented the solution of an initial value problem (IVP) for the heat equation, with initial data in , as a series of the self-similar solutions of the heat equation. This approach quickly revealed the large time behavior for the solution of the IVP. Inspired by Kloosterziel [1]'s approach, we express the solution of the nonhomogeneous Burgers equation in terms of the self-similar solutions of a linear partial differential equation with variable coefficients. Finally, we also obtain the large time behavior of the solution of the nonhomogeneous Burgers equation. © 2010 by the Massachusetts Institute of Technology.

Solutions of a Nonhomogeneous Burgers Equation

In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel (Journal of Engineering Mathematics 24, 213-236 (1990)) represented a solution to an initial value problem (IVP) for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel's approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained. © 2010 Shanghai University and Springer-Verlag Berlin Heidelberg.

Applied Mathematics and Mechanics (English Edition)

Large time asymptotics for solutions to a nonhomogeneous Burgers equation

In this paper, we construct asymptotic N-wave solutions for the nonplanar Burgers equation as t → ∞ via a balancing argument. These constructed asymptotics are compared with the approximate solutions of the nonplanar Burgers equation obtained by an approach due to Parker (Acoust. Lett. 4 (1981)). We also present a computationally convenient form for the N-wave solution of the nonplanar Burgers equation modifying Sachdev et al.'s (Stud. Appl. Math. 103 (1999)) approach. The asymptotic N-wave solutions obtained by balancing argument and modification to Sachdev et al.'s approach are validated by a careful numerical study. © 2008 by the Massachusetts Institute of Technology.

Asymptotic N-wave solutions of the nonplanar Burgers equation

In this survey, we explain shooting arguments for proving existence of solutions to boundary value problems posed for nonlinear ordinary differential equations. All the examples (except one) involve shooting on infinite intervals.

Indian Journal of Pure and Applied Mathematics

A survey on shooting arguments for boundary value problems

In this paper, we construct large-time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study. © 2005 by the Massachusetts Institute of Technology.

Large-time asymptotics for periodic solutions of the modified burgers equation

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming certain asymptotic conditions at plus infinity or minus infinity, we find a wide variety of solutions-(positive) single hump, monotonic (bounded or unbounded) or solutions with a finite zero. The existence or non-existence of positive bounded solutions with exponential decay to zero at infinity for specific parameter ranges is proved. The analysis relies mainly on the shooting argument. © 2003 Elsevier Science Ltd. All rights reserved.

World University root node

Nonlinear Analysis: Real World Applications

Self-similar solutions of a generalized Burgers equation with nonlinear damping

In this paper, we construct large-time asymptotic solutions of some generalized Burgers equations with periodic initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study. We also show that our asymptotic results agree with the approximate solutions obtained by Parker in certain limits.

Large-time asymptotics for periodic solutions of some generalized Burgers equations

The self-similar solutions of the generalized Burgers equation were analyzed. The existence of bounded, positive solutions with different types of decay to zero at x = +∞ or x = -∞ was shown for different parametric ranges. It was concluded that bounded, positive solutions with exponential decay to zero at x = ∞ or -∞ exist only for the case 1/(j+1) ≤α <1/j.

Nonlinear Analysis, Theory, Methods and Applications

Analysis of the self-similar solutions of the nonplanar Burgers equation

In this article, evolution of N-waves under the nonplanar Burgers equation, which takes into account geometrical expansion or contraction, is treated analytically. An exact asymptotic solution, generalizing that for the planar Burgers equation, is given for the case of expansion. An approximate treatement, using a balancing argument, gives asymptotic analytic results for both expansion and contraction. The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3].

Analytic and Numerical Study of N‐Waves Governed by the Nonplanar Burgers Equation

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions -(positive) single hump, unbounded or those with a finite zero. The existence and non-existence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

Fulltext

Mathematical Problems in Engineering

Analysis of the Self-similar Solutions of a Generalized Burgers Equation with Nonlinear Damping

The modified Burgers equation ut + unux = δuxx/2, where n ≥ 2 is even, is treated analytically for N-wave initial conditions. An exact asymptotic solution is presented, extending the validity of the linear solution far back in time. © 2000 Elsevier Science Ltd. All rights reserved.