Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials
Journal of Scientific Computing
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect
Journal of Computational Physics
Fourierization of the Legendre--Galerkin method and a new space--time spectral method
Applied Numerical Mathematics
Generalized Jacobi polynomials/functions and their applications
Applied Numerical Mathematics
A rational spectral method for the KdV equation on the half line
Journal of Computational and Applied Mathematics
Mathematics and Computers in Simulation
A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods
Mathematics and Computers in Simulation
Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations
Journal of Scientific Computing
Taylor's decomposition at several points for odd order ordinary DEs
Neural, Parallel & Scientific Computations
Journal of Computational Physics
A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations
Mathematical and Computer Modelling: An International Journal
Journal of Computational Physics
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A new dual-Petrov-Galerkin method is proposed, analyzed, and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the "dual" boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd-order equations.