Almost nothing decisive has been said about collocation methods for solving SPDEs. Among the best of such SPDEs the Burgers equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. This paper discusses spectral collocation method to reduce stochastic Burgers equation to a system of stochastic ordinary differen...

Higher-order accurate finite-difference schemes for solving the unsteady Burgers’ equation which often arises in mathematical modeling used to solve problems in fluid dynamics are presented. The unsteady Burgers’ equation belongs to a few nonlinear partial differential equations which has an exact solution, and it allows one to compare the numerical solution with the exact one, and the properti...

A generalization of the Hopf–Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is presented. The existence of the travelling wave solution and the interaction of nonlocal perturbation are considered. The nonlocal generalizations of the one-dimen...

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equa...

In this paper, new classes of symmetries for partial differential equations (PDE) which can be written in a conserved form are introduced. These new symmetries called nonclassical potential symmetries, are neither potential symmetries nor nonclassical symmetries. Some of these symmetries are carried out for the Burgers equation ut + uux − uxx = 0. (1) by studying the nonclassical symmetries of ...

The Burgers equation with linear damping has been subjected to Lie's group theoretic method of infinitesimal transformation to derive its solutions. To the best of our knowledge, we are the first to obtain a new similarity variable for an equation of the Burgers type and therefore provide new solutions.

In this paper, we discuss a link of Itô’s stochastic differential equations to nonlinear partial differential equations of Burgers type. Under certain conditions, we derive a generalised Burgers equation from a stochastic differential equation. We also give some economic interpretation of our result as well as the relevant conditions. Mathematics Subject Classification (2000): 60H10, 35K58, 91G99.

The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent. The results agree with the one found recently by Polyakov, who used the operator product conjecture. If this conjecture is true, then our WKB asymptotics of the Wyld functional i...