We analyze the onset of superconductivity, in a type II superconductor adjacent to a normal material, via a generalized Ginzburg–Landau energy functional, which models the effects of superconducting electron pairs diffusing into the normal part. We consider a superconductor and a normal material, each filling a half-space, in the presence of a constant magnetic field parallel to their interface. Among other results, we show that if the normal state conductivity of the superconductor is less than or equal to the conductivity of the normal material, then normal states are the only global minimizers down to the second critical field $H_{c_2}$. Hence, we analytically confirm experimental predictions that surface superconductivity may be suppressed by coating a superconductor with a normal metal.

In this paper, three iterative procedures (Landweber-Fridman, conjugate gradient and minimal error methods) for obtaining a stable solution to the Cauchy problem in slow viscous flows are presented and compared. A section is devoted to the numerical investigations of these algorithms. There, we use the boundary element method together with efficient stopping criteria for ceasing the iteration process in order to obtain stable solutions.

For the Cauchy problem for the nonlinear infiltration equation $$\left\{\begin{array}{@{}l@{\qquad}l} u_{t}=\frac{1}{m}(u^{m})_{xx},&x\in{\mathbb{R}}, t>0,m\geq{}1,\\[3pt] u|_{t=0}=u_{0}(x),&x\in{\mathbb{R}}, \end{array} \right.$$ we use its linear solution $u(x,t,1)$ to approach the nonlinear solution $u(x,t,m)$, and obtain the explicit estimate: $$\int_{0}^{T}\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|^{2}\,dx\,dt{} \leq{}(C^{\ast}(m-1))^{2},$$ where $C^{\ast}=O(T^{\gamma})$ and $\gamma=\frac{1+m-\alpha}{2(1+m)}$ for any $0<\alpha<1$.

We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in $L^2$ are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.

This paper is concerned with the positive solutions of the boundary-value problem \[ \left\{{\begin{array}{@{}l@{}}{\varepsilon u^{''} - \sigma (u) = -\gamma,}\\{u(0) = u(1) = 0,}\\ \end{array}}\right.\] where $\varepsilon$ is a small positive parameter and $\gamma$ is a positive constant. The nonlinear term $\sigma(u)$ behaves like a cubic; it vanishes only at $u=0$, where $\sigma'(0) > 0$ and $\sigma''(0) < 0$. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given $\gamma$, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of $\gamma$. Our approach is again based on the shooting method used previously by Ou & Wong (Stud. Appl. Math.112 (2004), 161-200).

Sufficient conditions are obtained for the existence of positive periodic solutions of a class of neutral delay differential equations of the form \begin{equation*} \left\{\begin{array}{@{}l@{}} {\normalsize N}^{\prime}{\normalsize (t)=N(t)F[t,N(t),N(t-\tau (t,N(t))),N}^{\prime}{\normalsize (t-\gamma (t)),P(t),P(t-\mu (t))]}\\ {\normalsize P}^{\prime}{\normalsize (t)=-}e(t)P{\normalsize (t)+k(t)N(t)+h(t)N(t-\sigma (t))}\end{array}\right. \end{equation*} by using the theory of topological degree. These results extend substantially the existing relevant existence results in the literature. As a demonstration, applying the obtained analysis results to a real complex neutral Lotka-Volterra population model, the existence criterion for positive periodic solutions is easily obtained and an example is used to give an impression of how restrictive these conditions are. Especially, this method is more suitable to state-dependent delay, and which have further applications in many fields.