An exact series and improved numerical and approximate solutions for the Boussinesq equation



[1] We derive a series solution for the nonlinear Boussinesq equation in terms of the similarity variable of the Boltzmann transformation in a semi-infinite domain. The first few coefficients of the series have been known for a long time, having been obtained by a truncated inversion of the series solution of the Blasius equation, but no direct recurrence relation was known for the complete series representing the solution of the Boussinesq equation. The series turns out to have a finite radius of convergence, which we estimate with a numerical complex-plane integration method that identifies the singularities of the solution when the equation is extended to the complex plane. The homogeneous condition at the origin produces a singularity which complicates numerical solutions with Runge-Kutta methods. We present two variable transformations that circumvent the problem and that are best suited to the complex-variable and the real-variable versions of the equation, respectively. Using those tools, an approximate solution accurate to 1.75 × 10−10 and valid for the entire positive real axis is then developed by matching a Padé approximant of the exact series and an asymptotic solution (to overcome the restriction imposed by the finite radius of convergence of the series), along the same lines of the expression proposed by Hogarth and Parlange (1999). The accuracies of all of the existing and the newly proposed solutions are obtained.