Research On Spectral Methods For Delay Differential Equations And Integral Algebraic Equations

Spectral methods,finite element methods and finite difference methods are all effective numerical methods for solving linear and non-linear differential equations.Spectral methods,which discretize differential equations with respect to space variable(s),mainly consist of tentative functions(generally called primary functions or expansion functions)and test functions.Tentative functions of spectral methods are infinitely differentiable global functions(which are usually the characteristic functions in singular or non-singular Sturm-Liouville problems).According to different selections about test functions,spectral methods could be divided into three types,namely,the spectral Galerkin methods,spectral collocation methods(also called pseudo-spectral methods),and spectral Tau methods.The greatest advantage of spectral methods is that they possess so-called infinite order convergence.But the infinite order convergence will be obtained only if the solutions of original problems are sufficiently smooth.This situation causes spectral methods can not be flexibly used in complicated domains.In this dissertation,spectral collocation methods and spectral Tau methods are introduced to solve delay differential equations and integral algebraic equations,and the convergence of those methods is studied.In spectral collocation methods,Dirac-? functions which center on collocation points are chosen as test functions so that differential equations accurately hold at the collocation points.Collocation methods that choose the Legendre-Gauss type quadrature points as collocation points and Legendre polynomials as tentative functions are known as Legendre-Gauss collocation methods.In this dissertation,Legendre-Gauss collocation methods are used to solve non-linear neural delay differential equations and non-linear Volterra delay-integro-differential equations.The solutions of these two kinds of equations are not smooth enough in solving intervals.In solving intervals,the accurate solutions do not have ideal global smooth property,because accurate solutions own bad smooth properties at some points which are determined by delay functions ?(t)in equations.To deal with this problem,multiple-domain Legendre-Gauss collocation methods are presented in this dissertation.In the methods,solving intervals are decomposed sufficiently so that accurate solutions could be sufficiently smooth in each subinterval,then the numerical solutions can be found in every subinterval and the global numerical solutions can be found.The numerical solutions obtained by multiple-domain Legendre-Gauss collocation methods own spectral accuracy,that is,the numerical solutions own infinite order convergence as long as the accurate solutions are sufficiently smooth except at discontinuity points determined by ?(t).Different from the general spectral Tau methods,Lanczos and Ortiz presented an operational Lanczos Tau methods.In Lanczos Tau methods,there is no need of approximating integrals.Lanczos Tau methods directly convert a differential equation to a system of algebraic equations.In this dissertation,Lanczos Tau methods are introduced to solve pantograph linear Volterra delay-integro-differential equations.From comparison results in numerical examples,it is known that the superiority of Lanczos Tau methods is highly efficiency compared with high accuracy of Legendre collocation methods.Lanczos Tau methods need very less time than Legendre collocation methods under the same degree of accuracy.Meanwhile,the convergence analysis of Lanczos Tau methods is given in general situations and the key factors which determine speed of convergence are pointed out in this dissertation.These theoretical results are infrequent in existing studying results.If the corresponding points,which are obtained by mapping the equal grid points jh(j = 0,±1,±2,· · ·,h > 0)on solving interval,are selected as the collocation points and Sinc cardinal functions are selected as tentative functions in collocation methods,then the methods are called Sinc collocation methods.Sinc methods are another numerical methods with high accuracy,which do not need equations owning higher regularity.When dealing with complicated equations,it makes Sinc methods own many superiorities that Sinc cardinal functions as tentative functions could not only excellently approximate many problems such as singular and oscillation problems,but also own good stability.In this dissertation,Sinc collocation methods are used to solve pantograph linear Volterra delay-integro-differential equations and integral algebraic equations of index-1,which is the new attempt to apply Sinc collocation mthods.After error analysis,it is known that Sinc collocation methods are high accuracy numerical methods which can converge according to exponential order.