Method for constructing analytical solutions to the Dym initial value problem

摘要

We obtain approximate solutions to the Dym equation, and associated initial value problem, for general initial data by way of an optimal homotopy analysis method. By varying the auxiliary linear operator, thereby varying the time evolution of approximate solutions, we perform two independent homotopy analyses for the Dym equation. We obtain two separate approximate solutions, each dependent on the respective convergence control parameter. A method to control the residual error of each of these approximate solutions, through the use of the embedded convergence control parameter, is discussed. Treating each approximate solution individually, we select the value of the convergence control parameter which minimizes a sum of squared residual error. That is, we use the so-called optimal homotopy analysis method in order to pick the value of the convergence control parameter that leads to the most rapid convergence of the approximate solutions. Performing this error analysis with each approximate solution, we directly compare the deviation of each type of solution to determine which time evolution linear operator in the homotopy provides more accurate approximations. We illustrate such a comparison in a few explicit examples of initial data, to determine which solution offers a better representation of the way solutions to the Dym equation evolve in time. With this, we are able to determine the effect of a choice of linear operator on how accurately resulting solutions can approximate true solutions to the Dym equation. Since we have optimally selected the convergence control parameter in each case, our solutions are optimal in the sense that they minimize a global measure of the residual error over the domain (subject to the choice of auxiliary linear operator used).