Trigonometric and hyperbolic type solutions to a generalized Drinfel'd-Sokolov equation
Abbreviated Journal Title
Appl. Math. Comput.
Generalized Drinfel'd-Sokolov equations; Exact solution; Analytical; solution; Nonlinear partial differential equation; HOMOTOPY ANALYSIS METHOD; NONLINEAR DIFFERENTIAL-EQUATIONS; TANH METHOD; W-ALGEBRAS; PERIODIC-SOLUTIONS; WILSON EQUATION; WAVE-EQUATIONS; REDUCTION; OPERATORS; KDV; Mathematics, Applied
A class of trigonometric and hyperbolic type solutions to the generalized Drinfel'd-Sokolov (GDS) equations u(t) + alpha(1)uu(x) + beta(1)u(xxx) + gamma(v(delta))(x) - 0 and v(t) + alpha(2)uv(x) + beta(2)v(xxx) - 0 is obtained for the case in which alpha(2) = 0, for various values of the other model parameters. The method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter alpha(2), in effect extending the exact solutions. We do not assume traveling wave solution forms for the analytical solutions; that is, we solve the generalized Drinfel'd-Sokolov equations as PDEs without resorting to transforming the system to ODEs. An error analysis of the obtained approximate local analytical solutions is provided. Then, we outline a general framework by which one many construct solutions in either sine/cosine or sinh/cosh basis. We provide the general perturbation expansion via homotopy analysis, and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors. Travelling solutions with time-dependent amplitude are then discussed. (C) 2010 Elsevier Inc. All rights reserved.
Applied Mathematics and Computation
"Trigonometric and hyperbolic type solutions to a generalized Drinfel'd-Sokolov equation" (2010). Faculty Bibliography 2010s. 852.