A table of solutions of the one-dimensional Burgers equation

Abstract

Abstract.The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.Introduction and survey of literature. The quasilinear parabolic equation now known as the "one-dimensional Burgers equation,"first appeared in a paper by Bateman [4], who derived two of the essentially steady solutions (1.3 and 1.5 of our Table). It is a special case of some mathematical models of turbulence introduced about thirty years ago by J. M. Burgers [10], [11]. The distinctive feature of (1) is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed thereby. Moreover, the transformation u = -(2v/6)(dd/dx)relates u{x, t) and d(x, t) so that if 6 is a solution of the linear diffusion equationthen u is a solution of the quasilinear Burgers equation (1). Conversely, if u is a solution of (1) then 9 from (2) is a solution of (3), apart from an arbitrary time-dependent multiplicative factor which is irrelevant in (2).In connection with the Burgers equation, transformation (2) appears first in a technical report by Lagerstrom, Cole, and Trilling [38, especially Appendix B], and was published by Cole [21]. At about the same time it was discovered independently by Hopf [30] and also-in the context of the similarity solution u = t~l/2S(z), z = (4vt)~1/2x-by Burgers [14, p. 250]. The similarity form of the Burgers equation-the quasilinear ordinary differential equation for S(z)-is a Riccati equation [51], and can thus be regarded as a basis for motivating transformation (2) inasmuch as (2) is a standard means of linearizing the Riccati equation. More general hydrodynamical applications of this transformation have been discussed by Ames [1, chapter 2], Chu [20], and Shvets and Meleshko [55].

Numerical Simulation of the Perturbed KdVB Equation

“…For this equation (7) there is a lot of exact solutions [5]. In our case, we use as the testing scheme the stationary analytical solution: As in the case of convection 4.2 for the initial distribution, we take a unit jump: fig.…”

A New Efficient Method for Solving Two-Dimensional Burgers' Equation

“…Let us consider system of Burgers' equations (1), with the following initial conditions [14]: To solve system (1) by LTNHPM, following the same procedure discussed in Section 2 and Example 1, we obtain the iterative relations (9); in this example we take initial approximations (15). The accuracy of LTNHPM for the system of two-dimensional Burgers' equation agrees good with the exact solution, and absolute errors are very small for the present choice of x, y, and t. These results are listed in Tables 1, 2 …”

“…They also illustrated independently that the Burgers equation can be solved exactly for an arbitrary initial condition. Benton and Platzman [9] have surveyed the analytical solutions of the one-dimensional Burgers equation. It can be considered as a simplified form of the Navier-Stokes equation [10] due to the form of the nonlinear convection term and the occurrence of the viscosity term.…”

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

“…Benton and Platzman [60] provide an exact solution for the above Burgers equation (18), with initial condition given by…”