Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation

  • Authors:

  • Kristina Mallory;Robert A. Gorder

  • Affiliations:

  • Department of Mathematics, University of Central Florida, Orlando, USA 32816-1364;Department of Mathematics, University of Central Florida, Orlando, USA 32816-1364

  • Venue:
  • Numerical Algorithms
  • Year:
  • 2013

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Abstract

We apply the method of homotopy analysis to the Zakharov system with dissipation in order to obtain analytical solutions, treating the auxiliary linear operator as a time evolution operator. Evolving the approximate solutions in time, we construct approximate solutions which depend on the convergence control parameters. In the situation where solutions are strongly coupled, there will be multiple convergence control parameters. In such cases, we will pick the convergence control parameters to minimize a sum of squared residual errors. We explain the error minimization process in detail, and then demonstrate the method explicitly on several examples of the Zakharov system held subject to specific initial data. With this, we are able to efficiently obtain approximate analytical solutions to the Zakharov system of minimal residual error using approximations with relatively few terms.