[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 RungeKutta 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 Â 10 À10 and valid for the entire positive real axis is then developed by matching a Pade 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.

The Boussinesq groundwater equation is widely used in hydrology to predict streamflow from an unconfined aquifer and derive the aquifer's saturated hydraulic conductivity and drainable porosity, and to predict water table height in drainage engineering. In this work, we solve this equation in an unconfined horizontal aquifer for nonhomogeneous boundary conditions for the water table height. The solution is found in the form of a Taylor series that has a finite radius of convergence, which is different for each initial condition. We also present an expression for the flux boundary condition at the origin as a function of the depth of the adjoining stream that automatically satisfies the boundary condition at infinity, and thus eliminates the need for a trial-and-error approach for the solution, which is accurate to 10 27 . In order to obtain an approximation for the water table height in the region where the series solution diverges, first we computed a diagonal Pad e approximation from the series coefficients, which converges in a larger interval than the series, and then we matched it with a new asymptotic approximation for large values of the independent variable. We found that the proposed matched solution is better suited to cases where the water head at the origin is close to the initial water head in the aquifer.

A higher-order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two-layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [1] and accounts for both a higher-order approximation to pressure coupling between the two layers and the effects of rapidly varying bottom variation. Motivated by the work of Rosales and Papanicolaou [2], an averaging technique is applied to the system for weakly nonlinear long internal waves propagating over periodic bottom topography. It is shown that the system reduces to an effective Intermediate Long Wave (ILW) equation, in contrast to the Korteweg-de Vries (KdV) equation derived for the surface wave case.

In this work we consider the Boussinesq equation applied to a one‐dimensional homogeneous aquifer with constant boundary conditions. Such a problem is generally solved (both analytically and numerically) by transforming it from a boundary value problem into an equivalent initial value problem that has to be solved by means of a trial‐and‐error approach. We devise a method that eliminates the need for trial and error that is purely analytical, which can benefit both numerical and analytical solutions. Furthermore, we present a general series solution that solves this equation analytically with arbitrary precision. The series solution generalizes two particular cases that already exist in the literature. With the new proposed series, the need for any numerical integration of differential equations is removed, and fully analytical benchmark solutions are obtained.

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