A linearized compact difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂²u/∂x²+p(t)u+?(x,t),0<x<1,0<t?T$\mathrm{\partial }u/\mathrm{\partial }t={\mathrm{\partial }}^{2}u/\mathrm{\partial }{x}^2+p(t)u+?(x\text{,}t)\text{,}0<x<1\text{,}0<t?T$ satisfying u(x,0)=f(x),0<x<1$u(x\text{,}0)=f(x)\text{,}0<x<1$; u(0,t)=_g0(t)$u(0\text{,}t)=g_0(t)$, u(1,t)=_g1(t)$u(1\text{,}t)=g_1(t)$, u(x^∗,t)=E(t),0?t?T$u(x^{\ast }\text{,}t)=E(t)\text{,}0?t?T$ where ?(x,t),f(x),_g0(t),_g1(t)$?(x\text{,}t)\text{,}f(x)\text{,}{g}_0(t)\text{,}{g}_1(t)$ and E(t)$E(t)$ are known functions, u(x,t)$u(x\text{,}t)$ and p(t)$p(t)$ are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.     

Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H   we construct a group homomorphism BiGal(H)→BrPic(Rep(H))$\mathrm{BiGal}(H)\to \text{BrPic}(\mathrm{Rep}(H))$, from the group of equivalence classes of H  -biGalois objects to the group of equivalence classes of invertible exact Rep(H)$\mathrm{Rep}(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=_Tq$H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(_Tq)$\mathrm{Rep}(T_q)$-bimodule categories.     

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation _yt−_yxx+y_yx=u(t)$y_t-y_{xx}+y{y}_x=u(t)$ on the line segment [0,1]$[0,1]$. The right-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1)=0$y(t,1)=0$ and restrict ourselves to using only two controls (namely the interior one u(t)$u(t)$ and the boundary one y(t,0)$y(t,0)$). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole–Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.     

Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and _FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))$F_S(t,x,y)=\det (tI+x\Re (S)+y\Im (S))$ be a ternary form associated with S  . We investigate the number of singular points of the curve _FS(t,x,y)=0$F_S(t,x,y)=0$, and show that the number of singular points of _FS(t,x,y)=0$F_S(t,x,y)=0$ associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2$n(n-3)/2$. Furthermore, we verify the upper bound n(n−3)/2$n(n-3)/2$ is sharp for 4?n?7$4?n?7$.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.     

On an open problem concerning total domination critical graphs

A graph G   with no isolated vertex is total domination vertex critical if for any vertex v$v$ of G   that is not adjacent to a vertex of degree one, the total domination number of G-v$G-v$ is less than the total domination number of G  . We call these graphs _γt${\gamma }_t$-critical. If such a graph G has total domination number k, we call it k  -_γt${\gamma }_t$-critical. We verify an open problem of k  -_γt${\gamma }_t$-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(_γt(G)-1)+1$n=\Delta (G)({\gamma }_t(G)-1)+1$.