We consider a PDE/ODE system for two pairs of competing species and study the spatial segregation limit as the interspecific competition rate tends to infinity. We show that the limiting problem is a one-phase Stefan problem for nonlinear diffusion equations.

In this paper, we investigate the wellposedness of the non-isentropic compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity proportional to a positive power of the temperature. This system describes the flow of a two-phase immiscible heat-conducting viscous compressible mixture. The phases are allowed to shrink or grow due to changes of density in the fluid and incorporates their transport with the current. We established the global existence and uniqueness of strong solutions for this system in 1-D, which means no phase separation, vacuum, shock wave, mass or heat or phase concentration will be developed in finite time, although the motion of the two-phase immiscible flow has large oscillations and the interaction between the hydrodynamic and phase-field effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov-Shelukhin’s result ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressible single-phase flow with constant heat conductivity to the non-isentropic compressible immiscible two-phase flow with degenerate and nonlinear heat conductivity.

We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem is of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data is bounded below away from zero in appropriate norms.

Melt superheated treatment can affect the solidification structure, but the direct evidence requires strict experiments. The molecular dynamics simulation method can break the limited experimental conditions, and provide advanced prediction for research. In this study, influences of superheated temperature (Ts) on the nucleation undercooling (ΔT) of metallic melts (Ti, Co, Mg, Al, Ni, Fe, Ag) were studied by the molecular dynamics simulation. The results show that the value of ΔT increases with the rise of Ts until the maximal ΔT approaches. In the curve of ΔT vs. Ts, there is an inflexion region where the nucleus cluster was broken. Above this inflexion region, the number of nucleus clusters decreases with the rise of Ts. Based on the simulated results, a model was proposed for describing the relation of ΔT and Ts, with which the maximal undercooling for metals can be predicted.

The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two- and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

The developed model of diffusion-limited and diffusionless solidification of eutectic alloy describes the relation “undercooling (ΔT)–velocity(V)–interlamellar spacing (λ)” for two cases. Namely, if the lamellar velocity V is smaller than the solute diffusion speed in bulk liquid VD, VVD, the solidification is mainly controlled by kinetic and thermal undercoolings. We show an influence of model parameters on the growth kinetics of eutectic solidification. Comparison of the model predictions with experimental data obtained on Fe-B samples processed in melt-glass fluxing is given.

Continuity, in particular sequential continuity, is an important subject of investigation not only in Topology, but also in some other branches of Mathematics. Connor and Grosse-Erdmann remodeled its definition for real functions by replacing {\sf lim} with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Then, this definition was extended to a topological group $X$ by replacing a linear functional $G$ with an arbitrary additive function defined on a subgroup of the group of all $X$-valued sequences. Also, some new theorems in generalized setting were given and some other theorems that had not been obtained for real functions were presented. In this study, we introduce neutrosophic $G$-continuity and investigate its properties in neutrosophic topological spaces.

Numerous experimental data on rapid solidification of eutectic systems exhibit the formation of metastable solid phases with the initial (nominal) chemical composition. This fact is explained by suppression of eutectic decomposition due to diffusionless (chemically partitionless) solidification beginning at a high but a finite growth velocity of crystals. A model considering the diffusionless growth is developed in the present work to analyze the atomic diffusion ahead of lamellar eutectic couples growing into supercooled liquid. A general solution of the model is presented from which two regimes are followed. The first presents diffusion-limited regime with the existence of eutectic decomposition if the solid-liquid interface velocity is smaller than characteristic diffusion speed in bulk liquid. The second shows suppression of eutectic decomposition under diffusionless transformation from liquid to one-phase solid if the solid-liquid interface velocity overcomes characteristic diffusion speed in bulk liquid.

The existence and uniqueness of the solution is proved for a nonlinear boundary value problem for ODE subject to an infinite condition \cite{1}, which describes the study of two-phase Stefan problems on the semi-infinite line [0,\infty). This result considerably extends the analysis of a recent work \cite{4}. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.

In this paper, authors introduce the generalized Bessel-Maitland transform whose kernel is the generalized Bessel-Maitland function. New identities are obtained for special cases of the generalized Bessel-Maitland function. Using these relations, several identities are obtained for generalized Bessel-Maitland integral transform. It is shown that some special cases of them are related with the Laplace transform and the Hankel transform. Also, some examples are given as representations of the outcomes presented here.

In this paper, we give a new method to show a monotonicity result for a function $f$ satisfying $(_{a}\nabla_{h}^{\nu}f)(t)\leq0$ ($(_{a}\nabla_{h,\ast}^{\nu}f)(t)\leq0$) with $\nu\in(0,1]$, which has never been solved in other papers. In addition, we give an example to illustrate one of our main results.

A two-dimensional tumor-immune model with the time delay of the adaptive immune response is considered in this paper. The model accounts for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid tumor with a necrotic kernel. The system has three equilibria. Both zero and maximum tumor volume equilibria are unstable, while the behavior of the positive equilibriums is closely related to the ratio of the immune killing rate to tumor volume growth rate. The positive equilibrium is more likely to be locally asymptotically stable when the ratio is smaller than a critical value, and unstable otherwise. We also derive conditions to guarantee the existence of Hopf bifurcation at the positive equilibrium. Applying the center manifold reduction and normal form method, we obtain explicit formulas to determine the properties of Hopf bifurcations. The global continuation of local Hopf bifurcation is studied based on the coincidence degree theory. The results reveal that long immune delay can lead to oscillation dynamics. We also carry out detailed numerical analysis for parameters to illustrate our qualitative analysis. Numerically, we find that a shorter immune response time leads to a longer patient survival time and the period and amplitude of a stable periodic solution increase with the immune response time. When CTLs recruitment rate and death rate vary, we observe how the ratio metioned above and the first bifurcation value change numerically, which yields further insights to the tumor-immune dynamics.

In this article, by using the notion of newly defined $q_{1}q_{2}$-derivatives and integrals some new Simpson’s type inequalities for co-ordinated convex functions are shown. The outcomes raised in this paper are extensions and generalizations of the comparable results in the literature on Simpson’s inequalities for co-ordinated convex functions.

In this paper, a fourth-order accurate conservative finite difference scheme for solving the coupled nonlinear Schr\”odinger (CNLS) equations is proposed. Conservation of the discrete energy and masses, priori estimates, existence and uniqueness of numerical solutions, convergence with second-order in time and fourth-order in space as well as stability of the present scheme are proved by discrete energy method. A convergent iterative method for the present scheme is developed. Numerical experiments are given to support the theoretical analysis.

A way to obtain approximate periodic solutions to nonlinear oscillators arising in a micro-electro-mechanical system (MEMS) is presented for the case of zero initial conditions and magnetostatic excitation. The frequency-amplitude relationship is derived by adopting He’s frequency formulation. The obtained analytical results are illustrated graphically. The proposed simple approach gives the fast insight into the dynamics of the singular oscillator and can be useful for design of MEMS devices.

The present work is concerned with the well-posedness and efficient numerical algorithm for a terminal value problem with a generalized Caputo fractional derivative. We investigate the existence and uniqueness of the solution of the terminal value problem, and consider the continuous dependence of the solutions on the given data. To illustrate our theoretical results, we present a one step algorithm for solving the considered problems. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the numerical method.

In this note, we show that the Hausdorff operator $H_{\Phi}$ is unbounded on a large family of Quasi-Banach spaces, unless $H_{\Phi}$ is a zero operator.

In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficients in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.