Two parameter singular perturbation problems for sine-Gordon type equations

In the real Sobolev space $H_0^1(\Omega)$ we consider the Cauchy-Dirichlet problem for sine-Gordon type equation with strongly elliptic operators and two small parameters. Using some {\it a priori} estimates of solutions to the perturbed problem and a relationship between solutions in the linear case, we establish convergence estimates for the difference of solutions to the perturbed and corresponding unperturbed problems. We obtain that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$

Quadratic Singular Perturbation Problems With Nonmonotone Transition Layer Properties

In this article, we consider the quadratic singular perturbation problems with Nonmonotone Transition Layer Properties. Under certain conditions, solutions are shown to exhibit nonmonotone transition layer behavior at turning point t=0. The formal approximation of problems is constructed using composite expansions, and then approximation solutions of left and right sides at t=0 are joined by joint method which exhibits spike layer behavior and boundary layer behavior respectively. As a result, an approximate solution is formed which exhibits nonmonotone transition layer behavior. In addition, the existence and asymptotic behavior of solutions are proved by the theory of differential inequalities.

Perturbation of Wavelet Frames of Quaternionic- Valued Functions

Let L2(R,H) denote the space of all square integrable quaternionic-valued functions. In this article, let Φ∈L2(R,H). We consider the perturbation problems of wavelet frame {Φm,n,a0,b0,m,n∈Z} about translation parameter b0 and dilation parameter a0. In particular, we also research the stability of irregular wavelet frame {SmΦ(Smx−nb),m,n∈Z} for perturbation problems of sampling.

Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.