Some analytic studies on generalized burgers equations

Abstract

Thesis Summary: Analysis of Generalized Burgers Equations This thesis analyzes a class of nonlinear parabolic equations known as generalized Burgers equations. The study combines analytical techniques with numerical solutions of both partial differential equations (PDEs) and ordinary differential equations (ODEs) derived via self-similar transformations. The numerical results support and confirm the analytical findings. The thesis is divided into two main parts:

Chapters 2 and 3: Focus on N-wave initial conditions, both discontinuous and continuous. Chapters 4 to 6: Address single hump initial conditions, including oscillatory solutions for specific parameter sets.

Chapter 1: Introduction

Discusses background and special solutions of the plane Burgers equation (N-waves and single hump). Reviews previous work on generalized Burgers equations. Introduces methods for studying ODEs: shooting method and upper and lower solutions method.

Chapter 2: Nonplanar Burgers Equation

Studies N-wave evolution under nonplanar Burgers equation (includes geometric expansion/contraction). Derives exact asymptotic solution for expansion. Uses balancing arguments for approximate asymptotics in both expanding and contracting geometries. Supported by accurate numerical solutions.

Chapter 3: Modified Burgers Equation

Equation: ut+unux=?uxxu_t + u^n u_x = \delta u_{xx}ut?+unux?=?uxx?, with even n>2n > 2n>2, small ?>0\delta > 0?>0. Analyzes N-wave initial conditions with equal and unequal lobes. Uses nonlinearization of old-age solutions and singular perturbation methods (Lee-Bapty & Crighton, Harris).

Chapter 4: Self-Similar Solutions of Modified Burgers Equation

Equation: ut+uaux=?uxxu_t + u^a u_x = \delta u_{xx}ut?+uaux?=?uxx? Transformation: u=?1/2at?1/2f(?),?=x/(2?t)1/2u = \delta^{1/2} a t^{-1/2} f(\eta), \eta = x/(2\delta t)^{1/2}u=?1/2at?1/2f(?),?=x/(2?t)1/2 Analyzes:

Initial value problem Connection problem

Proves existence and behavior of solutions depending on parameter ?\gamma?.

Chapter 5: Generalized Burgers Equation with Second Derivative

Equation: utt+uauxx+12t=?2uttu_{tt} + u^a u_{xx} + \frac{1}{2t} = -2u_{tt}utt?+uauxx?+2t1?=?2utt? Transformation: u=?1/2at?1/2f(?),?=x/(2?t)1/2u = \delta^{1/2} a t^{-1/2} f(\eta), \eta = x/(2\delta t)^{1/2}u=?1/2at?1/2f(?),?=x/(2?t)1/2 Analyzes:

Initial value problem Connection problem

Proves existence and decay behavior of solutions based on parameters aaa and jjj.

Chapter 6: Generalized Burgers Equation with Nonlinear Damping

Equation: ut+u(a?1)/2ux+Aua=?uxxu_t + u^{(a-1)/2} u_x + A u^a = -u_{xx}ut?+u(a?1)/2ux?+Aua=?uxx? Transformation and analysis similar to previous chapters. Studies:

Initial value problem Connection problem

Proves existence and decay types of solutions depending on aaa and AAA.

Publications and Work in Progress

With P. L. Sachdev: Numerical study of Euler–Painlevé transcendents. With P. L. Sachdev and K. T. Joseph: Analytic and numerical study of N-waves governed by nonplanar Burgers equation (Studies in Applied Mathematics). With P. L. Sachdev: N-wave solution of the modified Burgers equation (submitted to European Journal of Applied Mathematics). With P. L. Sachdev and Mythily Ramaswamy: Analysis of self-similar solutions of generalized Burgers equation with nonlinear damping (in preparation). With P. L. Sachdev and Mythily Ramaswamy: Analysis of self-similar solutions of generalized Burgers equation (in preparation).