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A Legendre–Petrov–Galerkin and Chebyshev collocation method for third-order differential equations. (English) Zbl 0986.65095
Authors’ abstract: A Legendre-Petrov-Galerkin method for a third-order differential equation is developed. By choosing appropriate base functions, the method can be implemented efficiently. Also, this new approach enables us to derive an optimal rate of convergence in \(L^2\)-norm. The method is applied to some nonlinear problems such as the Korteweg-de Vries equation with a Chebyshev collocation treatment for the nonlinear term. It is a Legendre-Petrov-Galerkin and Chebyshev collocation method. Numerical experiments are given to confirm the theoretical result.

MSC:
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
35Q53 KdV equations (Korteweg-de Vries equations)
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