A Finite Element Method for Singularly Perturbed Reaction—diffusion Problems

A finite element method is proposed for the sing ularly perturbed reaction-diffusion problem.An optimal error bound is derived,independent of the perturbation parameter.     

Singularly Perturbed Generalized Boundary Value Problems for Semilinear Elliptic Equation of Fourth Order

The singularly perturbed generalized boundary value problems for semi-linear elliptic equations of fourth order are considered. Under suitable conditions the existence, uniqueness and asymptotic behavior of generalized solutions for the boundary value problems are studied.     

Nonmonotone Interior Layer Solutions for Singularly Perturbed Semilinear Boundary Value Problems with a Turning Point

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A Class of Nonlinear Singularly Perturbed Problems of Fourth Order

ConsiderthefollowingnonlinearboundaryvalueproblemwhereEisasmallpositiveparameter,f,Warecontinuouspartialderivativefunctionsregardingtheirvariablesincorrespondenceranges,andfisasecondordercontinuouspartialderivativeaboutu(3).MoJiaqiconsideredaseriesofsingularlyperturbednonlinearproblemsusingthe.differentialinequalitytheory['J-['J.Mo[9Jandothers[3Jstudiedthesingularlyperturbedprob-lemsoffourthorder.Assumethat(I)Thereducedproblemof(1)-(5)hasafourthordercontinuouspartialderivativesolutionU(x)o…     

A Class of Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Boundary Perturbation

A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.     

The Singularly Perturbed Nonlinear Nonlocal Initial Boundary Value Problems for Reaction Diffusion Equations

The singularly perturbed nonlinear nonlocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.     

A Sampling Method for Solving Inverse Scattering Problems with a Locally Perturbed Half Plane

In this paper,we will consider the scattering of time—harmonic electromagnetic waves by aninfinitely long cylinder with a perfectly conducting boundary.The problem is modelled by aDirichlet problem for the Helmholtz equation.Adopting Cartesion coordinates(Xl,X2,X3)in础we assume that the boundary is the XlX3一plane with the perturbation independent of z3and local with respect to X1.Furthermore,we assume that the elect:tic and magnetic fieldsare invariant with respect to X3 and we restrict …     

Spectral Collocation Method for Caputo Fractional Multi-point Value Problems北大核心CSCD

Spectral collocation method is investigated for the nonlinear Caputo fractional multi-point value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular Volterra–Fredholm integral equations obtained from the nonlinear Caputo fractional multi-point value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with Volterra–Fredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multi-point value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects. © 2022 Chinese Academy of Sciences. All rights reserved.     

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.     

A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation

A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.     

The Nonlinear Singularly Perturbed Initial Boundary Value Problems of Nonlocal Reaction Diffusion Systems

In this paper the singularly perturbed initial boundary value problems for the nonlocal reactiondiffusion system are considered.Using the iteration method and the comparison theorem,the existence and itsasymptotic behavior of the solution for the problem are studied.     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

On One Method of Solving Singularly Perturbed Systems of Tikhonov’s Type

To investigate Tikhonov systems arising in problems of control theory we apply the method of holomorphic regularization, which allows one to obtain solutions to singularly perturbed problems in the form of series converging in the usual sense in powers of a small parameter.     

A Homotopy Interior point Method for Nonlinear Complementarity Problems

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On S-Shaped Bifurcation Curves for a Class of Perturbed Semilinear Equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.     

The Interior Layer Problem for the Strongly Nonlinear Singularly Perturbed System

The nonlinear strongly singularly perturbed systemεyi" = fi(t, y,y'), a < t < b, i = 1, 2, …, n, yi(a,ε)-pily'i(a,ε) = Ai, yi(b,ε) pi2y'i(b,ε) = Biare considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of interior solution for the problems are studied.     

A Restarted Conjugate Gradient Method for Ill—posed Problems

This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems.The damped Morozov‘s discrepancy principle is used as a stopping rule,Numerical experiments are given to illustrate the efficiency of the method.     

A Smoothing Method for Zero–One Constrained Extremum Problems

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.