This work addresses the study of the *L*^{p}-boundedness and compactness of abstract linear and nonlinear fractional integro-differential equations. The analysis is performed for the whole range of values of *p*, ie,
. In addition, theoretical results are complemented with illustrating particular cases of systems modeled by fractional evolution equations as heat conduction problems and problems arising in the theory of viscoelastic materials.

With the rapid growth of the amount of information stored on networks such as the internet, it is more difficult for information seekers to retrieve relevant information. This paper illustrates the design and improvement of a near neighborhood approach of information retrieval system to facilitate domain specific search. In exacting, a novel model depending on the notion of neighborhood system designed to rank documents according the searchers specific granularity requirements. The initial experiments confirm that our approach outperforms a classical vector-based information retrieval system. Our research work opens the door to the design and development of the next generation of internet search engines to alleviate the problem of information overload using more topological concepts.

]]>Fragmentation-coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper, we consider the discrete decay-fragmentation equation and prove the existence and uniqueness of physically meaningful solutions to this equation using the theory of semigroups of operators. In particular, we find conditions under which the solution semigroup is analytic, compact, and has the asynchronous exponential growth property. The theoretical analysis is illustrated by a number of numerical simulations.

]]>In this paper, we investigate the well-posedness and stability of mild solutions for a class of neutral impulsive stochastic integro-differential equations in a real separable Hilbert space. By the inequality technique combined with theory of resolvent operator, some sufficient conditions are established for the concerned problems. The obtained conclusions are completely new, which generalize and improve some existing results. An example is given to illustrate the effectiveness of our results.

]]>Lie group classification for 2 Burgers-type systems is obtained. Systems contain 2 arbitrary elements that depend on the 2 dependent variables. Equivalence transformations for the systems are derived. Examples of nonclassical reductions are given. A Hopf-Cole–type mapping that linearizes a nonlinear system is presented.

]]>The purpose of this paper is the presentation of a new extragradient algorithm in 2-uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.

]]>Bidirectional associative memory models are 2-layer heteroassociative networks. In this work, we prove the existence and the global exponential stability of the unique weighted pseudo–almost periodic solution of bidirectional associative memory neural networks with mixed time-varying delays and leakage time-varying delays on time-space scales. Some sufficient conditions are given for the existence, the convergence, and the global exponential stability of the weighted pseudo–almost periodic solution by using fixed-point theorem and differential inequality techniques. The results of this paper complement the previous outcomes. An example is given to show the effectiveness of the derived results via computer simulations.

]]>The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h-adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h-adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time-dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.

]]>Lie group classification for a diffusion-type system that has applications in plasma physics is derived. The classification depends on the values of 5 parameters that appear in the system. Similarity reductions are presented. Certain initial value problems are reduced to problems with the governing equations being ordinary differential equations. Examples of potential symmetries are also presented.

]]>In this paper we study the asymptotic behaviour of the solutions in linear models of population dynamics by means of the basic reproduction number *R*_{0}. Our aim is to give a practical approach to the computation of the basic reproduction number in continuous-time population models structured by age and/or space. The procedure is different depending on whether the density of newborns per time unit and the density of population belong to the same functional space or not. Three infinite-dimensional examples are illustrated: a transport model for a cell population, a model of spatial diffusion of individuals in a habitat, and a model of migration of individuals between age-structured local populations. For each model, we have highlighted the possible advantages of computing *R*_{0} instead of the Malthusian parameter.

We describe a mixture thin film as a membrane endowed with multiple out-of-tangent-plane vectors at each point, with vector sequence defined to within a permutation to account for the mixing of the mixture components. Such a description is motivated by a proposal for an atomistic-to-continuum derivation of a representation of multiatomic layer thin films, a view not requiring the introduction of phenomenological parameters. Differences between that proposal and the model discussed here are the definition of the values of the layer-descriptor map to within permutations and the explicit introduction of a bending-like term in the energy. The out-of-tangent-plane vectors satisfy a condition forbidding them to fall within the tangent plane after deformation. We consider a generic weakly surface-polyconvex membrane energy and a quadratic bending term involving the out-of-tangent-plane multiple vectors, a term which is also quasiconvex. Under appropriate energy growth assumptions and Dirichlet-type boundary conditions, we prove existence of ground states, i.e., equilibrium configurations described by the solutions to balance equations. An obvious corollary is the existence of equilibrium configurations of single out-of-tangent-plane vector Cosserat surfaces, a natural scheme for plates of simple materials.

]]>We propose a simple delay mathematical model for the dynamics of AIDS-related cancers with treatment of HIV and chemotherapy. The main goals are to study the effects of the delay and of treatment (HAART and chemotherapy) in cancer cells growth. The model was simulated for several biologically reasonable values of the delay, of HAART efficacies and of chemotherapeutic drugs decay rates. The results of the simulations reveal an epidemiologically well-defined model. Important inferences are drawn for designing future treatment protocols.

]]>Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass-like method for finding simultaneously *all* the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

In this paper, we study the following Schrödinger-Kirchhoff–type equation with critical or supercritical growth

where *a*>0, *b*>0, *λ*>0, and *p*≥6. Under some suitable conditions, we prove that the equation has a nontrivial solution for small *λ*>0 by variational method. Moreover, we regard *b* as a parameter and obtain a convergence property of the nontrivial solution as *b*↘0. Our main contribution is related to the fact that we are able to deal with the case *p*>6.

In this paper, we consider the following nonlinear Choquard equation driven by fractional Laplacian

where *V*(*x*) is a nonnegative continuous potential function, 0<*s*<1, *N*≥2, (*N*−4*s*)^{+}<*α*<*N*, and *λ* is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well *i**n**t*(*V*^{−1}(0)) as *λ* large enough.

A semilinear parabolic problem is considered in a thin 3-D star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter

The purpose is to study the asymptotic behavior of the solution *u*_{ε} as *ε*0, ie, when the star-shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors
and
in nonlinear perturbed Robin boundary conditions.

We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {*α*_{i}}and {*β*_{i}}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter *ε*0. Namely, in each case, we derive the limit problem (*ε*=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.

This paper deals with finite-time stabilization results of delayed Cohen-Grossberg BAM neural networks under suitable control schemes. We propose a state-feedback controller together with an adaptive-feedback controller to stabilize the system of delayed Cohen-Grossberg BAM neural networks. Stabilization conditions are derived by using Lyapunov function and some algebraic conditions. We also estimate the upper bound of settling time functional for the stabilization, which depends on the controller schemes and system parameters. Two illustrative examples and numerical simulations are given to validate the success of the derived theoretical results.

]]>A new defect-correction method based on the pressure projection for the stationary Navier-Stokes equations is proposed in this paper. A local stabilized technique based on the pressure projection is used in both defect step and correction step. The stability and convergence of this new method is analyzed detailedly. Finally, numerical examples confirm our theory analysis and validate high efficiency and good stability of this new method.

]]>In this paper, we consider stabilization of a 1-dimensional wave equation with variable coefficient where non-collocated boundary observation suffers from an arbitrary time delay. Since input and output are non-collocated with each other, it is more complex to design the observer system. After showing well-posedness of the open-loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed-loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non-collocated control.

]]>This paper is concerned with laminated beams modeled from the well-established Timoshenko system with time delays and boundary feedbacks. By using semigroup method, we prove the global well-posedness of solutions. Assuming the weights of the delay are small, we establish the exponential decay of energy to the system by using an appropriate Lyapunov functional.

]]>In this article, a finite element scheme based on the Newton method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank-Nicolson method is used for time discretization. Well-posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

]]>This work presents a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier-Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case.

]]>In this paper, the existence of antiperiodic solutions for fourth-order impulsive differential equation is obtained by variational approaches and results on the auxiliary system. It is interesting that there is no growth restraint on nonlinear terms and impulsive terms. Besides, any minimizing sequence is bounded in a closed convex set of a space composed of Lipschitzian functions with the appearance of antiperiodic boundary value conditions.

]]>In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram-type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results.

]]>In this paper, we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation

where *ε*>0 is a small parameter,
is the fractional Laplacian, *M* is a Kirchhoff function, *V* is a continuous positive potential, and *f* is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.

]]>We propose the reconstruction of the solenoidal part of a vector field supported in the unit ball in 3 dimensions by using cone beam data from a curve surrounding it, and this curve satisfies the Tuy's condition of order 3. We use the quaternionic inversion formula to decompose the solenoidal part of a vector field into 2 parts. To recover the first one, which is the main part of the solenoidal component, another definition of a cone beam transform containing both Doppler and transverse data will be introduced. The second part will be reconstructed by using information from the first part as in Katsevich and Schuster's work with less data.

]]>We prove the existence and uniqueness of solution to the nonhomogeneous degenerate elliptic PDE of second order with boundary data in weighted Orlicz-Slobodetskii space. Our goal is to consider the possibly general assumptions on the involved constraints: the class of weights, the boundary data, and the admitted coefficients. We also provide some estimates on the spectrum of our degenerate elliptic operator.

]]>The purpose of this paper is to introduce iterative algorithm which is a combination of hybrid viscosity approximation method and the hybrid steepest-descent method for solving proximal split feasibility problems and obtain the strong convergence of the sequences generated by the iterative scheme under certain weaker conditions in Hilbert spaces. Our results improve many recent results on the topic in the literature. Several numerical experiments are presented to illustrate the effectiveness of our proposed algorithm, and these numerical results show that our result is computationally easier and faster than previously known results on proximal split feasibility problem.

]]>We study the homogenization of a slow viscous two-phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid-fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two-phase fluid in the free domain, a coupled Darcy law connecting two-phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.

]]>In this paper, a model problem that can be used for mathematical modeling and investigation of arc phenomena in electrical contacts is considered. An analytical approach for the solution of a two-phase inverse spherical Stefan problem where along with unknown temperature functions heat flux function has to be determined is presented. The suggested solution method is obtained from a new form of integral error function and its properties that are represented in the form of series whose coefficients have to be determined. Using integral error function and collocation method, the solution of a test problem is obtained in exact form and approximately.

]]>This paper is dedicated to the study of a family of nonlinear Volterra equations coming from the theory of viscoelasticity. We analyze the existence of local mild solutions to the problem and their possible continuation to a maximal interval of existence. We also consider the problem of continuous dependence with respect to initial data.

]]>The Loewner partial differential equation provides a one-parametric family of conformal maps on the unit disk. The images describe a flow of an expanding simply-connected domain, called the Loewner flow, on the complex plane. In this paper, we present a numerical algorithm for solving the radial Loewner partial differential equation. The algorithm is applied to visualization of Loewner flows with the performance and precision. From the theoretical point of view, our algorithm is based on a recursive formula for determining coefficients of polynomial approximations. We prove that each coefficient converges to true values with reasonable regularity.

]]>In this paper, we introduce different kinds of growth orders for the set of entire solutions to the most general framework of higher-dimensional polynomial Cauchy-Riemann equations
, where
is the hypercomplex Cauchy-Riemann operator, *λ*_{i} are arbitrarily chosen nonzero complex constants, and *k*_{i} are arbitrarily chosen positive integers. The core ingredient is a projection formula that establishes a relation to the *k*_{i}-monogenic component functions, which are null-solutions to iterates of the Cauchy-Riemann operator that we studied in earlier works. Furthermore, we briefly outline the analogies of the Lindelöf-Pringsheim theorem in this context.

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection-free equilibrium and no-immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.

]]>This research was motivated by a real technological problem of vibrations of bodies hanging on chains or ropes in tubes or spaces limited by walls or other bodies. The studied system has two degrees of freedom. It is formed by two pendulums moving between two walls. Its movement is governed by a set of nonlinear ordinary differential equations. The results of the simulations shown that the system exhibits regular and chaotic movement. The simulations were performed for 3 excitation amplitudes and the range of the excitation frequencies between 1 and 30 rad s^{−1}. The subject of the investigations was the determination of the character of the pendulums' motions and identification of their collisions with the sided walls.

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier-Stokes-Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier-Stokes-Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero-electron-mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier-Stokes equations with large forces, a system of stationary compressible Navier-Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier-Stokes equation, linear compressible Navier-Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier-Stokes-Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero-electron-mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier-Stokes-Poisson equations.

]]>The notion of electromagnetic chirality, recently introduced in the Physics literature, is investigated in the framework of scattering of time-harmonic electromagnetic waves by bounded scatterers. This type of chirality is defined as a property of the farfield operator. The relation of this novel notion of chirality to that of geometric chirality of the scatterer is explored. It is shown for several examples of scattering problems that geometric achirality implies electromagnetic achirality. On the other hand, a chiral material law, as for example given by the Drude-Born-Fedorov model, yields an electromagnetically chiral scatterer. Electromagnetic chirality also allows the definition of a measure. Scatterers invisible to fields of one helicity turn out to be maximally chiral with respect to this measure. For a certain class of electromagnetically chiral scatterers, we provide numerical calculations of the measure of chirality through solutions of scattering problems computed by a boundary element method.

]]>The present article deals with existence and uniqueness results for a nonlinear evolution initial-boundary value problem, which originates in an age-structured cell population model introduced by Lebowitz and Rubinow (1974) describing the growth of a cell population. Cells of this population are distinguished by age *a* and cycle length *l*. In our framework, daughter and mother cells are related by a general reproduction rule that covers all known biological ones. In this paper, the cycle length *l* is allowed to be infinite. This hypothesis introduces some mathematical difficulties. We consider both local and nonlocal boundary conditions.

In this article, we study the increasing stability property for the determination of the potential in the Schrödinger equation from partial data. We shall assume that the inaccessible part of the boundary is flat, and homogeneous boundary condition is prescribed on this part. In contrast to earlier works, we are able to deal with the case when potentials have some Sobolev regularity and also need not be compactly supported inside the domain.

]]>Controversial results concerning the effectiveness of bed net in reducing dengue fever transmission make further research necessary in this direction. At this aim, we consider a mathematical model of dengue transmission where the use by individuals of insecticide-treated bed nets is taken into account, combined or not with insecticide spraying. Furthermore, as climatic factors play a key role in mosquito-borne diseases, we model the effect of seasonality through a periodic mosquito birth rate. We numerically investigate some specific scenarios according to different rainfall and mean temperature values. We set an optimal control problem to minimize the number of human infections and the cost of efforts placed into bed net adoption and maintenance and insecticide spraying. To assess the most appropriate strategy to eliminate dengue with minimum costs, we perform a comparative cost-effectiveness analysis, which also shows how the cost-benefit of intervention efforts is affected by changes in the amplitude of seasonal variation. One general result is that in any case the combination of bed net use and insecticide spraying produces the highest ratio of infections averted, whereas in terms of cost-benefit only spraying campaigns should be implemented in control programs for regions with no large seasonality.

]]>Bor has recently obtained a main theorem dealing with absolute weighted mean summability of Fourier series. In this paper, we generalized that theorem for summability method. Also, some new and known results are obtained dealing with some basic summability methods.

]]>The subject of statistical convergence has attracted a remarkably large number of researchers due mainly to the fact that it is more general than the well-established theory of the ordinary (classical) convergence. In the year 2013, Edely et al introduced and studied the notion of weighted statistical convergence. In our present investigation, we make use of the (presumably new) notion of the deferred weighted statistical convergence to present Korovkin-type approximation theorems associated with the periodic functions , and defined on a Banach space . In particular, we apply our concept of the deferred weighted statistical convergence with a view to proving a Korovkin-type approximation theorem for periodic functions and also to demonstrate that our result is a nontrivial extension of several known Korovkin-type approximation theorems which were given in earlier works. Moreover, we establish another result for the rate of the deferred weighted statistical convergence for the same set of functions. Finally, we consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

]]>This paper deals with the blow-up solution to the following semilinear pseudo-parabolic equation

in a bounded domain
, which was studied by Luo (Math Method Appl Sci 38(12):2636-2641, 2015) with the following assumptions on *p*:

and the lifespan for the initial energy *J*(*u*_{0})<0 is considered. This paper generalizes the above results on the following two aspects:

- a new blow-up condition is given, which holds for all p> 1;
- a new lifespan is given, which holds for all p> 1 and possible
*J*(*u*_{0})≥0.

Moreover, as a byproduct, we refine the lifespan when *J*(*u*_{0})<0.

This paper deals with a stochastic system which models the population dynamics of a chemostat including species death rate. On the basis of the theory on Markov semigroup, we demonstrate that the probability densities of the distributions for the solutions are absolutely continuous. The densities will convergence in *L*^{1} to an invariant density or weakly convergence to a singular measure under appropriate conditions. We also give the sufficient criteria for extinction exponentially of the species. To be specific, when *D*_{1}>*D* and the strength of perturbation is relatively small, we derive a precise threshold for the species survival.

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long-term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.

]]>We address the question of mean biomass volumetric productivity optimization, which originates from the simplification of dynamics of microalgae in a batch bioreactor process with light incidence. In particular, the stability of the model is analyzed, some optimality necessary conditions for the nonsmooth optimization problem obtained through the inclusion of different photoperiods are studied, and the model is applied in the particular case of *Chlamydomonas reinhardtii* microalgae to validate our results.

Since Compton cameras were introduced in the use of single photon emission computed tomography, various types of conical Radon transforms, which integrate the emission distribution over circular cones, have been studied. Most of previous works did not address the attenuation factor, which may lead to significant degradation of image quality. In this paper, we consider the problem of recovering an unknown function from conical projections affected by a known constant attenuation coefficient called an attenuated conical Radon transform. In the case of a fixed opening angle and vertical central axis, new explicit inversion formula is derived. Two-dimensional numerical simulations were performed to demonstrate the efficiency of the suggested algorithm.

]]>Deciding participation level of a component to dubious information is essential, particularly all things considered are displaying issues. This paper will present a participation capacity of components that has a place with unverifiable information. The fundamental instrument is the similarity classes that came about because of the likeness connection of a data framework. We will likewise express a few properties and an examination between our work and the past one.

]]>In this paper, we study the existence of ground state solutions for a Kirchhoff-type problem in involving critical Sobolev exponent. With the help of Nehari manifold and the concentration-compactness principle, we prove that problem admits at least one ground state solution.

]]>In this work, we study coexistence states for a Lotka-Volterra symbiotic system with cross-diffusion under homogeneous Dirichlet boundary conditions. By using topological degree theory and bifurcation theory, we prove the existence and multiplicity of positive solutions under certain conditions on the parameters. Asymptotic behaviors of positive solutions are respectively studied as the cross-diffusion coefficient tends to infinity and the interaction rate tends to zero. Finally, we compare our results with those of the Lotka-Volterra predator and competition systems.

]]>The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in *L*_{p}-based Besov spaces on a bounded Lipschitz domain in
, with *p* in a neighborhood of 2. This system is obtained by adding the semilinear term |**u**|**u** to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well-posedness results for the Dirichlet and Neumann problems in *L*_{p}-based Besov spaces on bounded Lipschitz domains in
(*n*≥3) are also presented. Then, using integral potential operators, we show the well-posedness in *L*_{2}-based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in
(*n*≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some *L*_{p}-based Sobolev spaces. Next, we use the well-posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in *L*_{p}-based Besov spaces, with *p*∈(2−*ε*,2+*ε*) and some parameter *ε*>0.

For the detection of *C*^{2}-singularities, we present lower estimates for the error in Schoenberg variation-diminishing spline approximation with equidistant knots in terms of the classical second-order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second-order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.

This paper is focused on following time-harmonic Maxwell equation:

where
is a bounded Lipschitz domain,
is the exterior normal, and *ω* is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that *f* is asymptotically linear as
, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor
and permittivity tensor
, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

In this article, we present the equiform parameter and define the equiform-Bishop frame in Minkowski 3-space . Additionally, we investigate the equiform-Bishop formulas of the equiform spacelike case in Minkowski 3-space. Furthermore, some results of equiform spacelike normal curves according to the equiform-Bishop frame in are considered.

]]>This paper is concerned with the two-species chemotaxis-competition system

where Ω is a bounded domain in
with smooth boundary *∂*Ω, *n*≥2; *χ*_{i} and *μ*_{i} are constants satisfying some conditions. The above system was studied in the cases that *a*_{1},*a*_{2}∈(0,1) and *a*_{1}>1>*a*_{2}, and it was proved that global existence and asymptotic stability hold when
are small. However, the conditions in the above 2 cases strongly depend on *a*_{1},*a*_{2}, and have not been obtained in the case that *a*_{1},*a*_{2}≥1. Moreover, convergence rates in the cases that *a*_{1},*a*_{2}∈(0,1) and *a*_{1}>1>*a*_{2} have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all *a*_{1},*a*_{2}>0 which covers the case that *a*_{1},*a*_{2}≥1, and lead to convergence rates for solutions of the above system in the cases that *a*_{1},*a*_{2}∈(0,1) and *a*_{1}≥1>*a*_{2}.

The homogenization of kinetic laminates in the framework of time-dependent linearized elasticity is studied from a variational point of view through the Γ-convergence of the associated energies. The characterization of the effective coefficients is achieved by means of a finite dimensional minimization problem.

]]>In this paper, we study the homogeneous Dirichlet problem for an elliptic equation whose simplest model is

where
, *N*≥3 is an open bounded set, *θ*∈]0,1[, and *f* belongs to a suitable Morrey space. We will show that the Morrey property of the datum is transmitted to the gradient of a solution.

We study the existence of ground state solutions for the following Schrödinger-Poisson equations:

where
is the sum of a periodic potential *V*_{p} and a localized potential *V*_{loc} and *f* satisfies the subcritical or critical growth. Although the Nehari-type monotonicity assumption on *f* is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of *V*_{loc}.

The issue of justifying the eddy current approximation of Maxwell's equations is reconsidered in the time-dependent setting. Convergence of the solution operators is shown in the sense of strong operator limits.

]]>No abstract is available for this article.

]]>The discrete fractional calculus is used to fractionalize difference equations. Simulations of the fractional logistic map unravel that the chaotic solution is conveniently obtained. Then a Riesz fractional difference is defined for fractional partial difference equations on discrete finite domains. A lattice fractional diffusion equation of random order is proposed to depict the complicated random dynamics and an explicit numerical formulae is derived directly from the Riesz difference. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study *P*-type, *P**I*^{α}-type, and *D*-type iterative learning control for fractional impulsive evolution equations in Banach spaces. We present triple convergence results for open-loop iterative learning schemes in the sense of *λ*-norm through rigorous analysis. The proposed iterative learning control schemes are effective to fractional hybrid infinite-dimensional distributed parameter systems. Finally, an example is given to illustrate our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we model the growths of populations by means of local fractional calculus. We formulate the local fractional rate equation and the local fractional logistic equation. The exact solutions of local fractional rate equation and local fractional logistic equation with the Mittag-Leffler function defined on Cantor sets are presented. The obtained results illustrate the accuracy and efficiency for modeling the complexity of linear and nonlinear population dynamics (PD). Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this article, we study the problem of a bounded network of two triple junctions in a planar domain with fixed angle conditions at the junctions and at the points at which the curves intersect with the boundary. We introduce the evolution problem of this type of networks, identify the steady states, and study their stability in terms of the geometry of the boundary. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this article, we study the geometric relation between two different types of initial conditions (IC) of a class of singular linear systems of fractional nabla difference equations whose coefficients are constant matrices. For these kinds of systems, we analyze how inconsistent and consistent IC are related to the column vector space of the finite and the infinite eigenvalues of the pencil of the system and analyze the geometric connection between these two different types of IC. Numerical examples are given to justify the results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We relate soft universals and efficiency in linguistics to mathematical model theory and recursion. Mathematical models are built from language – complexity theory enters because mathematical model theory is a new higher-order level part of human cognition within previous linguistic context. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, it is shown how to change the integration basis in some Gaussian (weighted) quadrature rules in order to obtain new quadrature models and improve classical results in the sequel. The main advantage of this approach is its simplicity, which can be implemented in any numerical integration package. Several remarkable numerical evidences are then given to show the advantage and efficiency of the proposed approach with respect to classical methods. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we analyze a modification of a compartmental model for Ebola disease previously introduced in the literature. For this initial model, the basic reproduction number is obtained. Moreover, a new dimensionless model is introduced. For this new model, we analyze the local stability and obtain the basic reproduction number. Finally, a fractional analogue of the dimensionless model is presented. Some numerical experiments are also included. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The inverse problem with unknown source function is studied for the nonlinear Oskolkov's system of partial differential equations that describes the dynamics of viscoelastic Kelvin–Voigt fluid. It is reduced to the problem for a first order semilinear ordinary differential equation with operator coefficients in Banach spaces. The main difficulty of the problem is a presence of the operator with a nontrivial kernel at the derivative. By the results on the unique solvability of the abstract problem, obtained by the authors before, the existence of a unique classical solution for the Oskolkov's system inverse problem is proved. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Entropy plays an important role in the simulation of traffic flow distribution. This paper studies the entropy condition for the Lighthill–Whitham–Richards model of the fractal traffic flows described by local fractional calculus. We also discuss the solutions of non-differentiability with graphs by using the local fractional variational iteration method. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We simulate a fractional feed-forward network. This network consists of three coupled identical ‘cells’ (aka, oscillators). We study the behaviour of the associated coupled cell system for variation of the order of the fractional derivative, 0 < *α* < 1. We consider the Caputo derivative, approximated by the Grünwald–Letnikov approach, using finite differences of fractional order. There is observed amplification of the small signals by exploiting the nonlinear response of each oscillator near its intrinsic Hopf bifurcation point for each value of *α*. The value of the Hopf bifurcation point varies with the order of the fractional derivative *α*. Copyright © 2016 John Wiley & Sons, Ltd.

At first, the existence of a unique solution for the Cauchy problem to nondegenerate fractional differential equation was proved. These results were used for research of the unique solvability for the initial Cauchy and Showalter–Sidorov problems to differential equations in Banach spaces with degenerate operator at fractional Caputo derivative in linear and nonlinear cases. Abstract results are applied to the research of an initial boundary value problem for time-fractional order Oskolkov system of equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We present a numerical method based on the fuzzy transform for solving second-order differential equations with boundary conditions. We demonstrate the effectiveness of the method by some examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We present a comparison between two different mathematical models used in the description of the Ebola virus propagation currently occurring in West Africa. In order to improve the prediction and the control of the propagation of the virus, numerical simulations and optimal control of the two models for Ebola are investigated. In particular, we study when the two models generate similar results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We introduce the reproducing kernel method to approximate solutions of difference equations. Reproducing kernel functions for difference equations are obtained. Examples that illustrate the accuracy and power of the method are given. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study properties of the localized solutions to the sine-Gordon equation excited on the attractive impurity by a moving kink. The cases of one-dimensional and two-dimensional spatially extended impurities are considered. For the case of one-dimensional impurity the possibility of excitation of the first even and odd high-amplitude impurity modes by the moving kink is demonstrated. By linearizing the sine-Gordon equation the dispersion relations for the small-amplitude localized impurity modes were obtained. The numerically obtained dispersion relations in the case of low oscillation amplitudes are in a good agreement with the results of analytical calculations. For the case of two-dimensional impurity we show the possibility of excitation of the nonlinear high-amplitude waves of new type called here a breathing pulson and a breathing 2D soliton. We suggest analytical expressions to model these nonlinear excitations. The breathing pulson and breathing 2D soliton are long-lived and can be of both symmetric and asymmetric type depending on the impurity type. The range of the impurity parameters where the breathing pulson and breathing 2D soliton can be excited was determined. The dependencies of the oscillation frequency and the amplitude of the excited impurity modes on the impurity parameters are reported. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An important question of electroencephalography and magnetoencephalography is associated with the possibility to identify the number of simultaneously activated areas in the brain. In the present paper, employing a homogeneous spherical conductor, serving as a geometrical model of the brain, we provide a criterion that determines whether the measured surface potential is caused by a single or a double localized neuronal excitations. We present the necessary and sufficient conditions, which decided whether the collected data originates from a single or from a set of two or more dipoles. Furthermore, we investigate the impossibility of deciding when we cannot be sure for the existence of one or two excitation centres. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this research article, we present theoretical techniques that can be used to investigate and comprehend the convergence behavior patterns of single parent evolution strategies. In the process, we determine instances of divergence or prove log-linear convergence and estimate the related speed, for a single parent evolution strategies class. The tools and results provided herein entertain a wide range of evolution strategies variants of interest and can be readily adapted for global convergence studies of a multitude of continuous optimization methods, evolving a single solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.

]]>