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Differential equation is an important branch of modern mathematics. Delay differential equations(DDE) is one type of differential equation. The derivative of unknown function is determined by the function in the previous time. There are a wide range of applications in modern science. The delay dif-ferential equations is better than ordinary differential equations to characterize a more realistic in mechanics, physics, ecology, biology and other applied techniques. For x∈Rn, the general form of delay differential equations is where is the track of solution at the last time. In this equation,f is a functional operator from R×Rn×C1 to Rn.Spectral Method is a numerical method for solving differential equations. Which is expanding solution approximately into a finite series expansion of a smooth function(generally orthogonal polynomials). Computational costs of the spectral methods are small, especially for more than two-dimensional problem. The difference method needs to set a considerable number of grid points, but spectral methods generally do not need to take a lot of items to obtain high accuracy solution.In this article we consider the following vanishing delay differential equa- tion where a(x),b(x)are smooth functions on[0,T].The vanishing delay functionθ(x)satisfies the following three terms:(V1)θ(0)=0,θ(x)is strictly increasing on[0,T].(V2)0<9(x)≤(?)x,x∈(0,T]for some (?)∈(0,1).(V3)θ∈Cd([0,T])for some d≥1.We will analyze(1)on the standard intervalⅠ:=[一1,1].And we de-scribe the spectral method onⅠ.So we use transformation Then problem(1)becomes WhereWe use spectral method based on Legendre basal to solve equations(2). Notate{tk}kN=0 be the set of the(N+1)-point Legendre Gauss,Legendre Gauss-Radau or Legendre Gauss-Lobatto on[一1,1],PN-is the space of polynomials with degrees not exceeding N.Y∈PN is the form of which Fj(t) is the standard Lagrange interpolation polynomial in the points {tk}k=0N as we mentioned above. We can get the numerical scheme for solving (2) Where tk= vk,k= 0,1…N,We introduce some useful lemmas, and finally prove the convergence of the method, the theorem is as followsTheorem 1.1 Consider the problem (2) and its Spectral Method (4). Assume the functions a(t), b(t) are sufficiently smooth functions, when N is sufficiently large, then we have Y∈PN is the approximate solution by using numerical scheme (3)-(4). The vanishing delay functionθ(t) satisfies the nature of (V1)-(V3), C is positive constant that doesn’t depend on NWe list two numerical examples in Chapter 2. The results confirm a very good text of the conclusions of the previous. We further extend the method to multiple delays in Chapter 3. Numerical examples verify the spectral method in case of multiple delays are still available.
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