2019/03/14 by Sten A. Reijers, Reijers, Sten A.
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Thermoelastic and Magnetoelastic Phenomena #math-ph #math.AP #math.MP #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.1903.05916
18 pages, 3 figures
openalex publication_date 2019/03/14 · arxiv created 2019/09/18 · arxiv updated 2019/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Burgers' equation is a one-dimensional momentum equation for a Newtonian fluid. The Cole-Hopf transformation solves the equation for a given initial and boundary condition. However, in most cases the resulting integral equation can only be solved numerically. In this work a new semi-analytic solving method is introduced for analytic and bounded series solutions of the Burgers' equation. It is demonstrated that a sequence transformation can split the non-linear Burgers' equation into a sequence of linear diffusion equations. Each consecutive sequence element can be solved recursively using the Green's function method. The general solution to the Burgers' equation can therefore be written as a recursive integral equation for any initial and boundary condition. For a complex exponential function as initial condition we derive a new analytic solution of the Burgers' equation in terms of the Bell polynomials. The new solution converges absolutely and uniformly and matches a numerical solution with arbitrary precision. The presented semi-analytic solving method can be generalized to a larger class of non-linear partial differential equations which we leave for future work.