The Research On Series Solutions And Spectral Collocation Methods To Emden-fowler Equation And Nonlinear Volterra Integral Equation

The singular differential and integral equations are important branchs of mathematics,as well as important tools in scientific research.They have been applied widely in some fields such as fluid mechanics,electromagnetic field theory,elasticity,and other disciplines.The typical feature of the solutions of these equations is that the derivatives of the solutions are usually singular at the origin,which deteriorates the calculation accuracy of standard numerical methods.Hence the design of efficiency numerical algorithms faces great difficulties for these singular problems.In this paper,we study the generalized series expansions of the solutions at the origin for the initial value problem of the singular Emden-Fowler differential equation and nonlinear Volterra integral equation.Based on these expansions,we design spectral collocation methods with ”infinite order convergence” and estimate the blow-up time for blow-up problems.This paper is divided into five chapters.In the first chapter,we briefly discuss the development of the Emden-Fowler equation and nonlinear Volterra integral equation and give the research objectives and main contents of this paper.In the second chapter,we give the preliminary knowledge in this paper,including the blow-up phenomenon of nonlinear equation,Padé approximation,and the definitions and main properties of some common used orthogonal polynomials(Chebyshev polynomials and Legendre polynomials).In the third chapter,we give a smoothing transformation for the initial value problem of the Emden-Fowler equation,and then the equation is further transformed into an equivalent Volterra integral equation of the second kind.The finite-term Taylor’s series expansion of the solution to the equation about the origin is obtained by Picard iteration,which shows that the solution of the transformed equation is sufficiently smooth at the origin.The series solution and its Padéapproximation are usually only accurate near the origin.Hence,we design a high-precision Chebyshev collocation method to obtain the approximate solution on a finite interval.We analyze the convergence of the scheme with respect to the maximum norm and then obtain the error estimation.Numerical examples illustrate that the Chebyshev collocation method does get high accuracy results for solving the initial value problem of the transformed Emden-Fowler equation.In the fourth chapter,we further design a singularity-separation Legendre collocation method for solving the initial value problem of the generalized Emden-Fowler equation.First,the EmdenFowler equation is transformed into an equivalent Volterra integral equation of the second kind.The psi-series expansion for the solution about the origin is obtained by Picard iteration,which has high approximation accuracy near the origin.By cutting off a small interval containing the origin,the singular equation is transformed into a regular one.Second,the Legendre collocation methods in integral form and differential form are constructed to obtain the approximate solutions of the equation on the remaining interval.The convergence of the scheme in integral form is analyzed and the error with maximum norm is estimated.The solution of a nonlinear equation may blow up in finite time.We also study the blow-up problem in this chapter.As an application of the series expansion for the solution about the origin,we successfully predict the blow-up time by performing Padé approximation for the series solution.For some special types of equations,we discuss the asymptotic property of the solution at the blow-up point and improve the calculation accuracy of the Legendre collocation methods using this expansion.By comparing the results of the Legendre collocation methods in integral form and differential form to solve the blow up problem,we discuss the advantages and disadvantages of these methods.Numerical examples show that the proposed methods have high accuracy to solve these kinds of singular problems.The chapter 5 is devoted to predicting the blow-up time for convolution Volterra-Hammerstein integral equations by using the series expansion for the solution about the origin.First,the finiteterm series expansion for the solution about the origin is obtained via Picard iteration and symbolic operation,which has high accuracy near the origin.Second,the Padé approximation of the series expansion is implemented to extend the convergence range of the series.The approximate blow-up time of the solution is estimated by calculating the minimal positive root of the denominator of the rational function.Third,an integral transform is performed to the obtained series expansion to convert the fractional order singularity to a pole of order one at the blow-up point,so the predicted accuracy of the blow-up time is improved remarkably.Numerical examples illustrate that the proposed method is very efficient for estimating the blow-up time of Volterra integral equations and some kinds of differential equations.Finally,a brief summary is given for the paper,and the further research topics are also considered.