An analytical solution for two and three dimensional nonlinear Burgers' equation

Abstract

This paper derives analytical solutions for the two dimensional and the three dimensional Burgers' equation. The two-dimensional and three-dimensional Burgers' equation are defined in a square and a cubic space domain, respectively, and a particular set of boundary and initial conditions is considered. The analytical solution for the two dimensional Burgers' equation is given by the quotient of two infinite series which involve Bessel, exponential, and trigonometric functions. The analytical solution for the three dimensional Burgers' equation is given by the quotient of two infinite series which involve hypergeometric, exponential, trigonometric and power functions. For both cases, the solutions can describe shock wave phenomena for large Reynolds numbers (Re ≥ 100), which is useful for testing numerical methods.