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

• Published in: Numerical algorithms Vol. 64; no. 4; pp. 633 - 657
• Main Authors: Mallory, Kristina, Van Gorder, Robert A.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2013; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Convergence; Dissipation; Error analysis; Errors; Evolution; Exact solutions; Homotopy theory; Linear operators; Mathematical analysis; Mathematical models; Numeric Computing; Numerical Analysis; Original Paper; Parameters; Theory of Computation; Zakharov system; Time evolution auxiliary operator; Nonlinear partial differential equation; Control of residual error
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-012-9683-6

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.