Exact solution to a class of problems for the Burgers’ equation on bounded intervals

Abstract

In this study, we consider Burgers’ equation with fixed Dirichlet boundary conditions on generic bounded intervals. By employing the Hopf–Cole transformation and a recently established exact operational solution for linear reaction–diffusion equations, an exact solution in the time domain is derived through inverse Laplace transforms. In the event that analytic inverses do in fact exist, they can be obtained in closed form through the use of Mellin transforms. Nevertheless, highly efficient algorithms are available, and numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. Two illustrative tests demonstrate that the results align closely with those of classical exact solutions. In comparison to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent a novel alternative, offering new insights and perspectives. The exact solution via the inverse Laplace transform is shown to be more computationally efficient, providing a reference point for numerical and semi-analytical methods.

Keywords

• Burgers’ equation
• Hopf–Cole transformation
• Integral transforms
• Closed-form solution
• Laplace domain
• Numerical efficiency