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

TL;DR

This paper derives an exact, computationally efficient solution for Burgers' equation on bounded intervals using inverse Laplace transforms, offering a new analytical approach that improves upon traditional series or numerical methods.

Contribution

It introduces an exact solution method for Burgers' equation on bounded intervals via inverse Laplace transforms, enhancing computational efficiency and analytical clarity.

Findings

Solutions match classical exact solutions in tests.

Closed-form expressions offer an innovative alternative to series or numerical methods.

Inverse Laplace transform approach is more efficient and serves as a reference for other methods.

Abstract

Burgers' equation with fixed Dirichlet boundary conditions is considered on generic bounded intervals. By using the Hopf-Cole transformation and the exact operational solution recently established for linear reaction-diffusion equations with constant coefficients, an exact solution in the time domain is implicitly derived by means of inverse Laplace transforms. Analytic inverses, whenever they exist, can be obtained in closed form using Mellin transforms. However, highly efficient algorithms are available, and numerical inverses in the time domain are always possible, regardless of the complexity of the Laplace domain expressions. Two illustration tests show that the results coincide well with those of classical exact solutions. Compared to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent an innovative alternative and new perspectives can be envisaged. The exact solution via the inverse Laplace transform is shown to be more computationally efficient and thus provides a reference point for numerical and semi-analytical methods.