The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of ...The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.展开更多
How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave＇s propagation and scattering. Levin method is a classical quadrature method for ...How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave＇s propagation and scattering. Levin method is a classical quadrature method for this type of integrals. Unfortunately it is susceptible to the system of linear equations＇ ill-conditioned behavior. We bring forward a universal quadrature method in this paper, which adopts Chebyshev differential matrix to solve the ordinary differential equation （ODE）. This method can not only obtain the indefinite integral＇ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations in our method is ill-conditioned, TSVD method can be adopted to solve them properly and eventually obtain accurate integral results, thus making a breakthrough in Levin method＇s susceptivity to the system of linear equations＇ ill-conditioned behavior.展开更多
文摘The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.
文摘How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave＇s propagation and scattering. Levin method is a classical quadrature method for this type of integrals. Unfortunately it is susceptible to the system of linear equations＇ ill-conditioned behavior. We bring forward a universal quadrature method in this paper, which adopts Chebyshev differential matrix to solve the ordinary differential equation （ODE）. This method can not only obtain the indefinite integral＇ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations in our method is ill-conditioned, TSVD method can be adopted to solve them properly and eventually obtain accurate integral results, thus making a breakthrough in Levin method＇s susceptivity to the system of linear equations＇ ill-conditioned behavior.
