We prove the existence and uniqueness of global strong solution to the free boundary problem in one dimensional compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when the viscosity and heat conductivity depend on temperature in power law of Chapman-Enskog and the data is in the neighborhood of some background solution at initial time. We also study the large time behavior of the solution and obtain its decay property.

Fangzhu, which has been lost for thousands of years, is an ancient device for water collection from air, its mechanism is unknown yet. Here we elucidate its possible surface-geometric and related physical properties by the oldest the Yin-Yang contradiction. In view of modern nanotechnology, we reveal that Fangzhu’s water-harvesting ability is obtained through a hydrophilic-hydrophobic hierarchy of the surface, mimicking spider web’s water collection, lotus or desert beetle’s water intake. The convex-concave hierarchy of Fangzhu’s textured surface enables it to have low wettability(high geometric potential) to attract water molecules from air through the nano-scale convex surface and transfer the attracted water along the concave surface to the collector. A mathematical model is established to reveal three main factors affecting its effectiveness, i.e., the air velocity, the surface temperature and surface structure. The lost technology can play an extremely important role in modern architecture, ocean engineering, transportation and others to catch water from air for everyday use.

The interior Dirichlet boundary value problem for the diffusion equation in non-homogeneous media is reduced to a system of Boundary-Domain Integral Equations (BDIEs) employing the parametrix obtained in different from . We further extend the results obtained in for the mixed problem in a smooth domain with L²(Ω) right hand side to Lipschitz domains and PDE right-hand in the Sobolev space H−1(Ω), where neither the classical nor the canonical co-normal derivatives are well defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

Nanofluid as a special thermal transporting medium has recently received unprecedented attention due to its improved heat transfer performance compared to conventional fluids. Numerous researches have been conducted on the natural convection characteristics of different nanofluids in various configurations of cavities due to the important applications of natural convection in environmental, petrochemical, medical, aviation and space technology, industrial and many more areas. The emergence of a magnetic field as a tool for the manipulation of convective flow and heat transfer behaviours of nanofluids in non-square enclosures has been extensively reviewed. The influence of several variables such as controlling parameters, heat distribution methods, thermal and concentration boundary conditions, magnetic field types, numerical methods, correlation types, nanofluid types, heaters types, numbers and length, and slip conditions, etc., on the magnetohydrodynamic (MHD) natural convection flow and heat transfer behaviours of nanofluid in non-square cavities has been given great attention and brought to the spotlight for discussion. The concepts of bioconvection, micro-polar nanofluid, bio-nanofluid (green nanofluid), ionic nanofluid, and hybrid nanofluid have also been discussed for the first time in relation to natural convection. Special cases of MHD natural convection in non-square cavities involving hybrid nanofluids and micro-polar nanofluids are also presented herein. The application of several numerical methods (which is the major approach studied so far) to investigate the hydromagnetic behaviours of nanofluids in non-square cavities is the focus of this work.

We apply the Bielecki metric on the space C([a, b]), to analyze the different types of stabilities of non-linear fractional integral equation corresponding to fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional non-linear Volterra and Fredholm integral equations, given by Ulam, Hyer and Rassias. We extend the respective stability results to the fractional integral equations where the domain of integration is an unbounded interval. We provide numerical examples which asserts our stability results.

We establish nonexistence of nontrivial solutions (including sign-changing ones) for some partial differential inequalities of elliptic and parabolic type containing nonlinear terms that depend on the positive and negative part of the sought function in different ways. Systems of elliptic inequalities with similar structure are also considered. The proofs are based on the test function method.

In this paper we investigate the size of a bi-dimensional fragmentation process. A rectangle of dimensions x and y is considered, it is split into four sub-rectangles with some probability that depends on x and y, we iterate until the stop of the process. The total number of the all the obtained rectangles at the end of the process satisfies some equality in distribution which is resolved , using some tools on integral equations, via the contraction method.

The turbulent characteristics of heat transfer and flow have been determined by applying the Van Driest model of the eddy diffusivity for water and ethylene glycol-based nanofluids. The properties of CuO, Al2O3 and SiO2 nanofluids in two base liquids viz., water and EG-water mixture with the ratio of 60:40 are considered for various concentrations and bulk temperature ranges. Based on the observations, it is concluded that numerical outcomes are validated with experimental measurements for heat transfer properties. It is monitored that SiO2 reaches a higher temperature gradient in comparison to CuO at a similar temperature and concentration in EG-water with the mixture of 60: 40. The gradients are greater for the EGW mixture compared to water-based nanofluids. However, the water-based nanofluids have higher heat transfer coefficients compared to EG-water nanofluid at identical flow velocities.

In the present paper we provide the conditions guaranteeing the generalization of so-called {\it diauxic} behaviour of solutions of ODEs. This behaviour was described by Monod in 1949 in the context of bacterial growth. Then a similar behaviour was observed and described referring to dynamic of CDK1 protein during cell cycle. The diauxic behaviour is described in terms of inflection points.

In this paper, we investigate a fractional p-Laplacian system with different critical Sobolev-Hardy exponents. By variational methods and the Mountain-Pass lemma, positive minimizers of the related best Sobolev constants and the existence of positive solutions of the system are found.

We consider a 2-D study of a plate with finite thickness and infinite extent. The upper plate surface is considered traction free and subjected to an axisymmetric heating. The lower surface is thermally insulated and layed on a rigid foundation. A cylindrical heat source affects the plate. This problem is relevant to the generalized thermoelasticity theory with one relaxation time. Laplace and Hankel transforms are considered. We use Inverse Hankel and Laplace transforms numerically. All related functions are showed graphically.

How to choose the optimization criterion of the objective function is an important issue for uncertain optimal control. The Hurwicz criterion is a flexible optimization criterion attempting to find the intermediate area between the extremes posed by the optimistic and pessimistic criteria. Based on uncertainty theory, in this paper, we establish a new uncertain optimal control model with jump by making use of Hurwicz criterion to optimize an uncertain objective function. By applying Bellman's principle of optimality, the principle of optimality for the proposed model is presented and then the equation of optimality is derived. Finally, an example is given to show the the effectiveness of the results obtained.

In this paper we investigate the global existence of small data solutions for the following structurally damped σ-evolution model with nonlinear memory term % \[u_{tt}+(-\Delta)^\sigma u+\mu(-\Delta)^{\frac{\sigma}{2}}u_t=\int_0^t (1+\tau)^{-\gamma}|u_t(\tau,\cdot)|^p\,d\tau,\] % with σ>0. In particular, for $\gamma\in ((n-\sigma)/n,1)$ we find the sharp critical exponent, under the assumption of small data in~$L^1$. Dropping the~$L^1$ smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration~$1-\gamma$ in the nonlinear memory term, and the assumption that initial data are small in~$L^m$, for some~$m>1$.

Abstract. In this work we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form ψ + φ where ψ and φ are analytic on the annulus {z ∈ C,1/2 < |z| < 1} .

Acquired drug resistance syndrom (ADR) is one of the most important features associated with tumor treatment and it is therefore a~topic of intensive studies. We present two simple mathematical models reflecting different mechanisms of ADR with some Darwinian effects included. These effects allow resistant cells to become sensitive again. Basing on this mathematical approach we conclude that for constant continuous treatment, if no Darwinian effects are present then once resistant cells appear, sensitive cells are eliminated after a long time, independently of the mechanism of acquiring the resistance. However, with Darwinian effects the situation is a little better as the sensitive cells are not completely eliminated but they are still outcompeted by the resistant ones. Moreover, if the therapy is stopped resistant cells become dominated by sensitive cells and the situation changes completely in comparison to the case without Darwinian effects. We discuss these mechanisms on the example of gliomas.

The set of hybrid numbers is a noncommutative number system unified and generalized the complex, dual and double(hyperbolic) numbers with the relation IH = − HI = ε + I. Two hybrid numbers P and Q are said to be similar if there exist a hybrid number X satisfying the equality X−1QX = P. And it is denoted by p ∼ q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations PX = XQ and QX − XP = C for P, Q, C ∈ 𝕂. KEYWORDS : Hybrid numbers, quaternions, coquaternions, dual and double numbers. MSC CLASSIFICATION : 15A66, 11R52, 15A18, 15A06, 16S50.

Programmed to retain active responsivity to environmental stimuli, diverse types of synthetic gels have been attracting interests regarding various applications, such as versatile elastic biodevices. In a different approach, when the gels are made of tissue-derived biopolymers, they can act as an artificial extracellular matrix (ECM) for use as soft implants in regenerative medicine. To explore the physical properties of hydrogels in terms of statistical thermodynamics, the mean-field Flory-Huggins-Rehner theory has long been used with various analytical and numerical modifications. Here we suggest a novel mathematical model on the volume phase transition of a biological hybrid gel that is sensitive to ambient temperature. To mimic acellular soft tissues, the ECM-like hydrogel is modeled as a network of biopolymer chains, such as type I collagen and gelatin, which are covalently crosslinked and swollen in aqueous solvents. Within the network, thermoresponsive synthetic polymer chains are doped by chemical conjugation. Based on the Flory-Huggins-Rehner framework, our model phenomenologically illustrates a well-characterized volume phase behavior of engineered tissue mimics as a function of temperature by formulating the ternary mixing free energy of the polymer-solvent system and by generalizing the elastic free energy term. With this formalism, the decoupling of the Flory-Huggins interaction parameter between the thermoresponsive polymer and ECM biopolymer enables deriving a simple analytical equation for the volume phase transition as a function of the structural and compositional parameters. We show that the doping ratio of the thermoresponsive polymers affects the phase transition temperature of the ECM-like gels.

We introduce a neural field equation with a neuron-dependent Heaviside-type activation function and spatial-dependent delay. The basic object of the study is represented by a Volterra Hammerstein integral equation involving a discontinuous nonlinearity with respect to the state variable that is both time- and space-dependent. We replace the discontinuous nonlinearity by its multi-valued convexification and obtain the corresponding Volterra Hammerstein integral inclusion. We investigate the solvability of this inclusion using the properties of upper semi-continuous multi-valued mappings with convex closed values. Based on these results, we study the solvability of an initial-prehistory problem for the former neural field equation with the Heaviside-type activation function. The application of multi-valued analysis techniques allowed us to avoid some restrictive assumptions standardly used in the investigations of the solutions to neural field equations involving Heaviside-type activation functions.