| Time delay is a common phenomenon in real life.In order to describe these natural phenomena more accurately,various delay partial differential equations(DPDEs)have been proposed.Due to the presence of the delay term,the solution of the DPDEs is not only related to the current state of the system,but also to the historical state of the system.As a result,the exact solutions of DPDEs are often difficult to be expressed explicitly.Therefore,many scholars turn to construct efficient and stable numerical algorithms to obtain high-precision numerical solutions of such equations,so as to achieve numerical simulation of natural phenomena described by such equations.Meanwhile,one-parameter methods are a class of effective methods to solve differential equations.A large number of theoretical and numerical results show that the one-parameter methods have satisfactory accuracy and stability under appropriate parameter selection.In view of these,we first briefly review the research background of DPDEs,introduce some research results of one-parameter methods applied to DPDEs,outline research motivation and the research content of this paper.Then,we will extend the one-parameter methods to solve several classes of important DPDEs and obtain the appropriate parameter selection by analyzing the numerical properties of the method such as convergence and stability.For the initial-boundary value problems(IBVPs)of the semi-linear reaction-diffusion equations(RDEs)with time-variable delay(TVD),we construct a class of fully discrete one-parameter methods by using linear interpolation to approximate the term with TVD.Using the spectral radius of the matrix and mathematical induction,we prove the convergence and stability of the fully discrete single-parameter method,and verify the theoretical results and validity of the method by numerical experiments.For the IBVPs of two dimensional wave equations with discrete and distributed TVDs,we first construct a class of basic one-parameter methods.In order to improve the compu-tational efficiency of the methods,we transform the methods into a class of one-parameter alternate direction implicit(ADI)methods by combining the ADI technique.Then,by prov-ing some useful inequalities,we analyze the convergence and stability of the one-parameter ADI methods.In the numerical experiment part,we verify the theoretical results of the one-parameter ADI methods,and illustrate the computational advantages of the one-parameter ADI methods by comparing with the basic one-parameter methods.For the IBVPs of two-dimensional nonlinear Sobolev equations with TVD,we con-struct two kinds of one-parameter orthogonal spline collocation methods.Under appropri-ate conditions,we derive the error estimates of these two kinds of methods in the sense of L~2-and H~1-norm.The numerical results show that the two methods have the same conver-gence accuracy and similar computational efficiency,and the convergence order of the two methods is consistent with the theoretical results.For the Timoshenko beam system with delayed boundary feedback(DBF),we first establish the energy stability criterion for the external-force-free Timoshenkobeam system with DBF.Then a class of one-parameter spectral Galerkin method is constructed to solve the general Timoshenko beam system with DBF.For this kind of method,we first prove its unique solvability and energy stability,and then analyze its convergence by introduc-ing appropriate projection operator.Finally,the theoretical results and effectiveness of the methods are verified by numerical experiments.In order to solve the IBVPs of the viscoelastic plate equation with delay,we first intro-duce two variables to convert the original equation into a coupled system,and then construct a class of one-parameter spectral Galerkin methods to solve the resulting coupled equations.For this kind of method,the energy stability,unique solvability and convergence are ana-lyzed strictly,and the correctness of theoretical results of the method is verified by numerical experiments.In the end,we make a brief summary of the main content of the paper,expatiate some difficult problems related to this paper,and based on these difficult problems,the future research outlook. |
