The Long Time Behavior Of A Spectral Collocation Method For Delay Ordinary Differential Equations And Partial Integro-differential Equations Studying

The long time behavior for pantograph delay differential equation and a partial integro-differential equation with a weakly singular kernel are discussed in this paper. Delay differential equations (DDEs), also called the functional differential equations arises in many applications, like biology, medicine, con-trol theory, climate models, population dynamics, electrical networks, and many others, where DDEs are used to constitute basic mathematical models for such real phenomena. The delay term in these equations leads to the mem-ory property for the problem. Smooth data in these equation lead to solutions that are globally smooth on J, this fact motivated the numerical solution of DDEs with vanishing delays by spectral methods. This paper propose firstly an efficient numerical method for delay differential equations with vanishing proportional delay, the algorithm is a mixture of the Legendre Gauss-Lobatto collocation method and domain decomposition, numerical results show that these methods achieve the spectral convergence:then we present an algorithm to evaluate the nodal values of unknown function, this algorithm is a mixture of the Legendre-Gauss collocation method and domain decomposition, numer-ical results show that these methods achieve the spectral convergence again: and we have proved the existence and uniqueness of the spectral collocation solution and proved the long time error estimate. We provide a rigorous er-ror analysis for the proposed method, which indicates that the solution has global convergence and spectral accuracy provided that the data in the given pantograph delay differential equation are sufficiently smooth.The partial integro-differential equation with a weakly singular kernel of-ten occurs in applications such as heat conduction in material with memory, compression of poro-viscoelastic media, population dynamics, nuclear reactor dynamics, etc., The numerical solution of this equation was studied extensively in the literature. A lot of them use FEM, Spline collocation methods, finite difference methods, the error bounds of discretization in time are valid only on finite time intervals and point-wise. In this paper, Legendre-Galerkin spec- tral methods is used firstly for the spatial discretization, the global stability and convergence properties of semi-discretization are derived. Then, spectral collocation methods based on Legendre-Gauss-Lobatto points is used for the spatial discretization. we obtain the global stability and error estimates again. Finally, we discuss the full discretization based on the spectral collocation methods in space and the backward difference quotient combined with the first order convolution quadrature rule in time, the global stability and convergence properties of complete discretization are derived and numerical experiments are reported. The result in this work seems to be the first successful spec-tral approach (with theoretical justification) for the partial integro-differential equation with a weakly singular kernel.