Reproducing Kernel Methods For Solving Some Singular Differential Equations

Singular differential equations arise in a variety of applied mathematics and physicssuch as gas dynamics, nuclear physics, chemical reaction and studies of atomic structures.Therefore, the problems have attracted much attention and have been studied by manyscholars. There is considerable interest in numerical methods for singular differentialequations. In general, usual numerical methods fail to produce good approximations tothe solutions of the equations. Hence one has to go for new methods.This dissertation aims at finding reproducing kernel methods for solving singulardifferential equations. The methods are based on reproducing kernel theory. The mainadvantage of reproducing kernel methods is that many differential equations with com-plex boundary conditions such as period boundary condition, integral boundary condition,which are difficult to solve, can be solved easily in the reproducing kernel spaces. Thespecific treatment is as follows: First, construct reproducing kernel space satisfying com-plex boundary conditions and obtain its reproducing kernel; Next, solve differential equa-tions satisfying the boundary conditions by combining the good properties of reproducingkernel space and computational techniques. The main results obtained in this dissertationare summarized as follows:Firstly, reproducing kernel theory is studied. By defining a new inner product ina reproducing kernel Hilbert space, the new representation of reproducing kernel withthe form of polynomial is obtained, which is much simpler. This can reduce the size ofcomputation in solving problems. Also, this can improve the accuracy of computation bypreventing accumulating error of calculation.Secondly, reproducing kernel method is presented for solving linear singular two-point boundary value problems. The exact solution is given in the form of series and anapproximate solution is obtained by truncating the series. The error estimate is given.Based on this method, two iterative reproducing kernel methods are constructed for solv-ing nonlinear singular periodic boundary value problems. Also, the convergence of thesetwo methods is proved.Thirdly, for the insufficiencies of traditional variational iteration method, variational iteration-reproducing kernel method is proposed for solving nonlinear singular initialvalue problems. The method is the subtle combination of variational iteration methodand reproducing kernel method. The main advantage of this method is that it can over-come the restriction of the form of nonlinear term. Furthermore, the method can improvethe computation speed of conventional variational iteration method. In addition, homo-topy perturbation-reproducing kernel method is designed for solving nonlinear singulartwo-point boundary value problems by combining homotopy perturbation and reproduc-ing kernel methods. Homotopy perturbation method is based on the use of traditionalperturbation method and homotopy technique. The homotopy perturbation method canreduce a nonlinear singular two-point boundary value problem to a series of linear sin-gular two-point boundary value problems and generate a rapid convergent series solutionin most cases. Reproducing kernel method is also an analytical technique, which cansolve powerfully the reduced linear singular two-point boundary value problems. Ho-motopy perturbation-reproducing kernel method combines advantages of these two meth-ods. Therefore, nonlinear singular two-point boundary value problems can be solvedefficiently by using homotopy perturbation-reproducing kernel method.Fourthly, methods for solving singularly perturbed two-point boundary value prob-lems are studied. For second order linear singularly perturbed two-point boundary valueproblems, an effective method is presented. The method consists of the following steps:(1) The original problem is divided into two problems, a boundary layer region and anouter region problems using a terminal point; (2) The asymptotic inner condition at theterminal point is determined; (3) The boundary layer region and outer region problems aresolved respectively. For third-order linear singularly perturbed two-point boundary valueproblems and second order nonlinear singularly perturbed two-point boundary value prob-lems, based on asymptotic expansion and reproducing kernel methods, effective methodsare designed.Finally, reproducing kernel method is presented for solving a class of second or-der linear singular three-point boundary value problems. First, the problem is convertedinto an equivalent integral-differential equation, which is easy to solve by using reproduc-ing kernel method; Then the equivalent integral-differential equation is solved by usingreproducing kernel method. In the case of nonlinearity, three techniques are providedsuch as quasilinearization technique, iterative reproducing kernel method and homotopy perturbation-reproducing kernel method.Reproducing kernel theory has important applications in curve fitting, function es-timation, model description, probability and statistics. In this dissertation, based on re-producing kernel theory, a few analytical techniques are presented for solving singulardifferential equations such as singular two-point boundary value problems, second-orderand third-order singularly perturbed two-point boundary value problems, second-ordernonlinear singular periodic boundary value problems and singular three-point boundaryvalue problems. The solution obtained using the present methods has its own advantages.The main characteristic features of reproducing kernel methods are, in contrast with othernumerical methods like Runge-Kutta and linear multistep methods, that global approxi-mations can be established on the whole interval and the convergence is uniform. More-over, it also permits the study of the behaviour of derivatives of approximate solutions.