Boundary Value Methods For Several Classes Of Ordinary Differential Equations And Reaction–Diffusion Equations

Differential equations play a significant role in modeling various real phenomena arising in physics,chemistry reaction,control engineering,biological process and other scientific fields.Due to the diversity of real phenomena,there exist many kinds of differential equations to describe them.The exact solutions of these problems can not be analytically expressed in general.Moreover,as the complexity of the problems,it is also difficult to analyze the properties of their theoretical solutions.Therefore,some efficient numerical methods should be utilized to solve these problems.In this thesis,we deal with several classes of ordinary differential equations and reaction–diffusion equations,develop some efficient numerical methods,and investigate the theoretical properties of the numerical solutions.In Chapter 1,we first introduce some applications of several classes of differential equations,review the relevant research of these equations and boundary value methods,and then narrate the main work in this thesis.In Chapter 2,we focus on the numerical analysis of first-order singular initial value problems.Block boundary value methods(BBVMs)are firstly extended to solve such problems.It is proved under some suitable conditions that the extended methods are uniquely solvable,stable and convergent.Numerical examples verify the stability,efficiency and accuracy of the methods.Moreover,a comparison between the proposed methods and the IEM-based iterated defect correction method is presented.The numerical results show that the extended BBVMs are comparable in the computational accuracy and efficiency.In Chapter 3,we concern about the numerical computation and analysis of secondorder delay initial value problems.The generalized St¨ormer–Cowell methods are extended to solve this type of problems.The condition of unique solvability of the methods is given.Then the convergence and global stability of the methods are proved.Several numerical experiments are performed to illustrate the effectiveness and accuracy of the methods.Moreover,by using a transformation,the original problems can be converted into first-order delay initial value problems.The resulting problems are approximated by the extended trapezoidal rules(ETRs).Comparing these two methods,we find that the generalized St¨ormer–Cowell methods have some advantages over ETRs in the computational time.In Chapter 4,we study a class of compact boundary value methods(CBVMs)for solving semi-linear reaction–diffusion equations(SLREs).The presented CBVMs are constructed by combining the fourth-order compact difference method(CDM)with the p-order boundary value methods,where the former is for the spatial discretization and the latter for the temporal discretization.It is proved that CBVMs are locally stable,uniquely solvable and have fourth-order accuracy in space and p-order accuracy in time.The computational effectiveness and accuracy of CBVMs are further testified by applying the methods to the Fisher equation.Besides these researches,we also extend CBVMs to solve the twocomponent coupled system of SLREs.The numerical experiment shows that the extended CBVMs are effective and can arrive at the high-precision.In Chapter 5,we propose a class of efficient numerical methods for solving semi-linear convection–reaction–diffusion equations(SLCEs).In order to solve these problems,they are firstly converted into the equivalent reaction–diffusion equations by an exponential transformation.Then the resulting problems are discretized by the compact block boundary value methods(CBBVMs),which are constructed by combining the fourth-order CDE with the p-order BBVMs.Moreover,CBBVMs can also be extended to solve the two-component coupled system of SLCEs by using a special exponential transformation and an iterative algorithm.Two numerical examples are carried out to verify the effectiveness and highprecision of CBBVMs for solving SLCEs and their corresponding converting problems,and the extended CBBVMs for solving two-component coupled SLCEs and their corresponding converting problems.Furthermore,second-order block ETRs and second-order backward differential formulae are used as the temporal approximation methods.The resulting numerical results show that the block ETRs have some superiority in temporal discretization.In the last chapter,a brief summarisation is presented and some future work is stated.