Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.