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Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation. (English) Zbl 1283.65090
Summary: 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.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q82 PDEs in connection with statistical mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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