We consider the nonself-adjoint Sturm-Liouville operator with $q\in L_1[\mathrm{0,1}]$ and either periodic or antiperiodic boundary conditions. We obtain necessary and sufficient conditions for systems of root functions of these operators to be a Riesz basis in $L_2[\mathrm{0,1}]$ in terms of the Fourier coefficients of $q$.

The aim of this paper is to initiate the study of coincidence best proximity point problem in the setup of generalized metric spaces. Some results dealing with existence and uniqueness of a coincidence best proximity point of mappings satisfying certain contractive conditions in such spaces are obtained. An example is provided to support the result proved herein. Our results generalize, extend, and unify various results in the existing literature.

We show that the main result in the work by Mutlu et al. is not true. We explain point by point some of its main mistakes and we propose an alternative version to smooth away the defects of it.

An iterative learning control scheme is applied to a class of linear discrete-time switched systems with arbitrary switching rules. The application is based on the assumption that the switched system repetitively operates over a finite time interval. By taking advantage of the super vector approach, convergence is discussed when noise is free and robustness is analyzed when the controlled system is disturbed by bounded noise. The analytical results manifest that the iterative learning control algorithm is feasible and effective for the linear switched system. To support the theoretical analysis, numerical simulations are made.

A class of impulsive Cohen-Grossberg neural networks with time delay in the leakage term is investigated. By using the method of $\mathcal{M}$-matrix and the technique of delay differential inequality, the attracting and invariant sets of the networks are obtained. The results in this paper extend and improve the earlier publications. An example is presented to illustrate the effectiveness of our conclusion.

We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, and $g$-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaan $g$-circulant matrix.

We prove that the double inequality holds for all $a,b>0$ with $a\ne b$ if and only if $p\le 1/3$ and $q\ge \text{l}\text{o}\text{g}\mathrm{\hspace{0.17em}2}/(1+\text{l}\text{o}\text{g}\mathrm{\hspace{0.17em}2})=0.4093$…, where $X(a,b)$ and $M_r(a,b)$ are the Sándor and $r$th power means of $a$ and $b$, respectively.

A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local $L^p$ regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined for $x\in \left[0, 1\right]$ as $F\left(x\right)={\sum }_{j=0}^{\infty }{2}^{-\alpha j}\varphi \left(2^{j}x\right)$, where and $\varphi \left(x\right)=\text{dist}\left(x, \mathbb{z}\right)$. The Knopp function itself has everywhere the same p-exponent $\alpha$. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of the domain under the graph of F at each point $(x, F(x))$ and show that p-exponents, weak and strong accessibility exponents, change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.

We generalize the regularized sampling method introduced in 2005 by the author to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) to the case of vectorial SLP with parameter dependent boundary conditions. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed a general method capable of handling with ease very broad classes of SLPs, whether scalar or vectorial.

Boundary value problems arise while modeling various physical and engineering reality. In this communication we investigate windowed Fourier frames focusing two-point BVPs. We approximate BVPs using windowed Fourier frames. We present some numerical results to demonstrate the efficiency of such frame functions to approximate BVPs.

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graph $G$ and a suitable $G$-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

The main aim of this paper is to compute the generalized inverse $A_{T,S}^{(2)}$ over Banach spaces by using semi-iterative method and to present the error bounds of the semi-iterative method for approximating $A_{T,S}^{(2)}$.

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccine-age, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutative $C^{*}$-algebras and present a proof in the finite dimensional case.

Phase contrast tomography is a high sensitivity medical imaging technique. Several regularization methods have been used in the literature to obtain stable solutions for the phase retrieval or the phase contrast tomography problems. Yet, the functional framework and the convergence properties of the methods have not been studied in detail. In this work, the convergence properties of regularization approaches for phase retrieval and phase contrast tomography are investigated.

We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.

Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

We introduce the concept of L-fuzzy neighborhood systems using complete MV-algebras and present important links with the theory of L-fuzzy topological spaces. We investigate the relationships among the degrees of L-fuzzy r-adherent points (r-convergent, r-cluster, and r-limit, resp.) in an L-fuzzy topological spaces. Also, we investigate the concept of LF-continuous functions and their properties.

Optimal control problem with multipoint boundary conditions is considered. Sufficient conditions for the existence and uniqueness of the solution of boundary value problem for every fixed admissible control are obtained. First order increment formula for the functional is derived. Pontryagin’s maximum principle is proved by using the variations of admissible control.

Let $f$ be a diffeomorphism on a $C^{\infty }$ closed surface. In this paper, we show that if $f$ has the $C^2$-stably limit shadowing property, then we have the following: (i) $f$ satisfies the Kupka-Smale condition; (ii) if $\mathrm{P(f)}$ is dense in the nonwandering set $\Omega$($f$) and if there is a dominated splitting on $P_s$($f$), then $f$ satisfies both Axiom $A$ and the strong transversality condition.

In uniform spaces $\left(X, \mathcal{D}\right)$ with symmetric structures determined by the $\mathcal{D}$-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the $\mathcal{J}$-families of generalized pseudodistances on $X$ are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by $\mathcal{J}$-families. Results are new also in locally convex and metric spaces. Examples are provided.

We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tiling $\mathfrak{I}$. We investigate the JPD-coloring of the r-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles at r = 3 or squares at r = 4 or regular hexagons at r = 6. Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package.

We prove some results concerning Arens regularity and amenability of the Banach algebra $M_{\phi }\left(A\right)$ of all $\phi$-multipliers on a given Banach algebra $A$. We also consider $\phi$-multipliers in the general topological module setting and investigate some of their properties. We discuss the $\phi$-strict and $\phi$-uniform topologies on $M_{\phi }\left(A\right)$. A characterization of $\phi$-multipliers on $L_1\left(G\right)$-module $L_p\left(G\right)$, where $G$ is a compact group, is given.

We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

This paper is devoted to studying the boundary behavior of self-affine sets. We prove that the boundary of an integral self-affine set has Lebesgue measure zero. In addition, we consider the variety of the boundary of a self-affine set when some other contractive maps are added. We show that the complexity of the boundary of the new self-affine set may be the same, more complex, or simpler; any one of the three cases is possible.

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper.

The existence and uniqueness of the boundary value problem for linear systems equations of the mixed hyperbolic-elliptic type in the multivariate domain with the changing time direction are studied. Applying methods of functional analysis, “$\epsilon$-regularizing” continuation by the parameter and by means of prior estimates, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev space.

This paper formulates a delay model characterizing the competition between bacteria and immune system. The center manifold reduction method and the normal form theory due to Faria and Magalhaes are used to compute the normal form of the model, and the stability of two nonhyperbolic equilibria is discussed. Sensitivity analysis suggests that the growth rate of bacteria is the most sensitive parameter of the threshold parameter $R_0$ and should be targeted in the controlling strategies.

Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach.

We consider the reducing subspaces of $M_{z^N}$ on $A_{\alpha }^2({\mathbb{D}}^k)$, where $k\ge 3$, $z^N=z_1^{N_1}\cdots z_k^{N_k}$, and $N_i\ne N_j$ for $i\ne j$. We prove that each reducing subspace of $M_{z^N}$ is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that $\alpha =0$ and $\alpha \in (-1,+\mathrm{\infty })\setminus \mathbb{Q}$, respectively. Finally, we give a complete description of minimal reducing subspaces of $M_{z^N}$ on $A_{\alpha }^2({\mathbb{D}}^3)$ with $\alpha >-1$.

The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.

By making use of the concept of fractional $q$-calculus, we firstly define $q$-extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of $q$-analogue of $p$-valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the $q$-extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.

We use a new method for constructing some differential equations which are equivalent to a given equation in the sense of having the same reflecting function. We completely solve the problem: when is a polynomial differential equation equivalent to a given polynomial differential equation? Many sufficient conditions have been established for one differential equation to be equivalent to a given differential equation. We apply the obtained results to study the boundary value problem of two equivalent differential equations.

An abstract Hammerstein equation is an equation of the form $u+KFu=0$. A new method is introduced to prove the existence of a solution of this equation where $K$ and $F$ are nonlinear accretive (monotone) operators. The method does not involve the complicated technique of factorizing a linear map via a Hilbert space and does not involve the use of deep variational techniques.

A novel bioinspired control strategy design is proposed for generalized synchronization of nonlinear chaotic systems, combining the bioinspired stability theory, fuzzy modeling, and a novel, simple-form Lyapunov control function design of derived high efficient, heuristic and bioinspired controllers. Three main contributions are concluded: (1) apply the bioinspired stability theory to further analyze the stability of fuzzy error systems; the high performance of controllers has been shown in previous study by Li and Ge 2009, (2) a new Lyapunov control function based on bioinspired stability theory is designed to achieve synchronization without using traditional LMI method, which is a simple linear homogeneous function of states and the process of designing controller to synchronize two fuzzy chaotic systems becomes much simpler, and (3) three different situations of synchronization are proposed; classical master and slave Lorenz systems, slave Chen’s system, and Rossler’s system as functional system are illustrated to further show the effectiveness and feasibility of our novel strategy. The simulation results show that our novel control strategy can be applied to different and complicated control situations with high effectiveness.

Water hammer on transient flow of hydrogen-natural gas mixture in a horizontal pipeline is analysed to determine the relationship between pressure waves and different modes of closing and opening of valves. Four types of laws applicable to closing valve, namely, instantaneous, linear, concave, and convex laws, are considered. These closure laws describe the speed variation of the hydrogen-natural gas mixture as the valve is closing. The numerical solution is obtained using the reduced order modelling technique. The results show that changes in the pressure wave profile and amplitude depend on the type of closing laws, valve closure times, and the number of polygonal segments in the closing function. The pressure wave profile varies from square to triangular and trapezoidal shape depending on the type of closing laws, while the amplitude of pressure waves reduces as the closing time is reduced and the numbers of polygonal segments are increased. The instantaneous and convex closing laws give rise to minimum and maximum pressure, respectively.

We give accurate estimates of the constants $C_n(\mathcal{A}(I),x)$ appearing in direct inequalities of the form $|L_{n}f(x)-f(x)|\le C_n(\mathcal{A}(I),x){\omega }_2$ $\left(f;\sigma (x)/\sqrt{n}\right)$, $f\in \mathcal{A}(I)$, $x\in I$, and $n=\mathrm{1,2},\dots ,$ where $L_n$ is a positive linear operator reproducing linear functions and acting on real functions $f$ defined on the interval $I$, $\mathcal{A}(I)$ is a certain subset of such functions, ${\omega }_2(f;·)$ is the usual second modulus of $f$, and $\sigma (x)$ is an appropriate weight function. We show that the size of the constants $C_n(\mathcal{A}(I),x)$ mainly depends on the degree of smoothness of the functions in the set $\mathcal{A}(I)$ and on the distance from the point $x$ to the boundary of $I$. We give a closed form expression for the best constant when $\mathcal{A}(I)$ is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.

The main motivation of this paper is to introduce the notion of cubic linear space. This inspiration is received from the structure of cubic sets. The notions of R-intersection, R-union, P-intersection, and P-union of cubic linear spaces are defined and we provide some results on these. We further introduce the notion of internal cubic linear space and external cubic linear space and establish some results on them.

This paper studies the finite-time synchronization problem for a class of complex dynamical networks by means of periodically intermittent control. Based on some analysis techniques and finite-time stability theory, some novel and effective finite-time synchronization criteria are given in terms of a set of linear matrix inequalities. Particularly, the previous synchronization problem by using periodically intermittent control has been extended in this paper. Finally, numerical simulations are presented to verify the theoretical results.

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality, and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions, and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, operator monotone functions, Borel measures, and certain operator equations.

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

We propose a three-dimensional autonomous nonlinear system, called the general $T$ system, which has potential applications in secure communications and the electronic circuit. For the general $T$ system with delayed feedback, regarding the delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics. We determine conditions for the existence of the Hopf bifurcations and analyze their direction and stability. Also, the fractional order general $T$-system is proposed and analyzed. We provide some numerical simulations, where the chaos attractor and the dynamics of the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic control and synchronization on schemes for the new fractional order chaotic system are verified by numerical simulations.

Synapses are key elements in the information transmission in the nervous system. Among the different approaches to study them, the use of computational simulations is identified as the most promising technique. Simulations, however, do not provide generalized models of the underlying biochemical phenomena, but a set of observations, or time-series curves, displaying the behavior of the synapse in the scenario represented. Finding a general model of these curves, like a set of mathematical equations, could be an achievement in the study of synaptic behavior. In this paper, we propose an exploratory analysis in which selected curve models are proposed, and state-of-the-art metaheuristics are used and compared to fit the free coefficients of these curves to the data obtained from simulations. Experimental results demonstrate that several models can fit these data, though a deeper analysis from a biological perspective reveals that some are better suited for this purpose, as they represent more accurately the biological process. Based on the results of this analysis, we propose a set of mathematical equations and a methodology, adequate for modeling several aspects of biochemical synaptic behavior.

To make effective use of model-based control system design techniques, one needs a good model which captures system’s dynamic properties in the range of interest. Here an analytical model of pneumatic muscle actuator with two pneumatic artificial muscles driving a rotational joint is developed. Use of analytical model makes it possible to retain the physical interpretation of the model and the model is validated using open-loop responses. Since it was considered important to design a robust controller based on this model, the effect of changed moment of inertia (as a representation of uncertain parameter) was taken into account and compared with nominal case. To improve the accuracy of the model, these effects are treated as a disturbance modeled using the recurrent (Elman) neural network. Recurrent neural network was preferred over feedforward type due to its better long-term prediction capabilities well suited for simulation use of the model. The results confirm that this method improves the model performance (tested for five of the measured variables: joint angle, muscle pressures, and muscle forces) while retaining its physical interpretation.

We present an analysis of corrected quadrature rules based on the method of undetermined coefficients and its associated degree of accuracy. The correcting terms use weighted values of the first derivative of the function at the endpoint of the subinterval in such a way that the composite rules contain only two new values. Using Taylor’s expansions and Peano’s kernels we obtain best truncation error bounds which depend on the regularity of the function and the weight parameter. We can minimize the bounds with respect to the parameter, and we can find the best parameter value to increase the order of the error bounds or, equivalently, the degree of accuracy of the rule.

The paper is mainly used to provide the equivalence of two algorithms of independent component analysis (ICA) based on the information bottleneck (IB). In the viewpoint of information theory, we attempt to explain the two classical algorithms of ICA by information bottleneck. Furthermore, via the numerical experiments with the synthetic data, sonic data, and image, ICA is proved to be an edificatory way to solve BSS successfully relying on the information theory. Finally, two realistic numerical experiments are conducted via FastICA in order to illustrate the efficiency and practicality of the algorithm as well as the drawbacks in the process of the recovery images the mixing images.

This paper focuses on a hybrid multistep and its twin one-leg methods and implementing them on implicit mixed differential algebraic equations. The orders of convergence for the above methods are discussed and numerical tests are solved.

We use the topological degree method to study the existence of solutions for nonlocal elliptic equations (systems) with a strong singular nonlinearity.

We present a deterministic model with isolation and lost to follow-up for the transmission dynamics of three strains of Mycobacterium tuberculosis (TB), namely, the drug sensitive, multi-drug-resistant (MDR), and extensively-drug-resistant (XDR) TB strains. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the model has locally asymptotically stable (LAS) disease-free equilibrium when the associated reproduction number is less than unity. Furthermore, the model undergoes in the presence of disease reinfection the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection. The result of the global sensitivity analysis indicates that the dominant parameters are the disease progression rate, the recovery rate, the infectivity parameter, the isolation rate, the rate of lost to follow-up, and fraction of fast progression rates. Our results also show that increase in isolation rate leads to a decrease in the total number of individuals who are lost to follow-up.

This paper is concerned with the asymptotic stability of delay differential-algebraic equations. Two stability criteria described by evaluating a corresponding harmonic analytical function on the boundary of a certain region are presented. Stability regions are also presented so as to show the method geometrically. Our results are not reported.

The closed water bodies, such as reservoirs and lakes, can be polluted by an inflow of pollutants in the upstream as well as a stratification caused by seasonal natural phenomena. The vertical circulation particularly plays an important role in reducing environmental pollutants. The factors of the vertical circulation are the temperature, wind, thermal diffusivity, sunlight, and so on. The wind is the most significant factor among all possible factors causing the vertical circulation. Thus, it is necessary to describe the validation and application of a three-dimensional numerical model of wind-driven circulation in a thermally stratified flow. In this study, the numerical model is conducted in three steps to calculate the velocity components from the momentum equations in x- and y-directions, the elevations from the free surface equation, and the temperature from the scalar transport equation. The present model was applied to two tests for verification of the numerical accuracy. Numerical results are compared with analytical solutions of the sloshing free surface movement in a rectangular basin and the model is applied to the circulation for the wind-driven flow in a thermal stratification. Consequently, the developed model is validated by two verifications and phenomena of the internal flow.

The motion of multiple bubbles under gravity in two dimensions is numerically studied through the lattice Boltzmann method for the Eotvos number ranging from 1 to 12. Two kinds of initial arrangement are taken into account: vertical and horizontal arrangement. In both cases the effects of Eotvos number on the bubble coalescence and rising velocity are investigated. For the vertical arrangement, it has been found that the coalescence pattern is similar. The first coalescence always takes place between the two uppermost bubbles. And the last coalescence always takes place between the coalesced bubble and the bottommost bubble. For four bubbles in a horizontal arrangement, the outermost bubbles travel into the wake of the middle bubbles in all cases, which allows the bubbles to coalesce. The coalescence pattern is more complex for the case of eight bubbles, which strongly depends on the Eotvos number.

We present a counterexample to the main result of the abovementioned paper showing that this result is false and cannot be improved in a simple way.

We obtain sufficient conditions for the univalence, starlikeness, and convexity of a new integral operator defined on the space of normalized analytic functions in the open unit disk. Some subordination results for the new integral operator are also given. Several corollaries follow as special cases.

We establish some inequalities of Simpson type involving Riemann-Liouville fractional integrals for mappings whose first derivatives are h-convex.

We investigate the nonlinear matrix equation $X^r-{\sum }_{i=\mathrm{1}}^m{A}_i^{\mathrm{\ast }}{X}^{-{\delta }_i}{A}_i=I$, where $r$ is a positive integer and ${\delta }_i\in (\mathrm{0,1}],\hspace{0.17em}\text{for}i=\mathrm{1,2},\dots ,m$. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operator $L$ with bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

$L_2$-metric is used to find the distance between triangular fuzzy numbers. The mean and variance of a fuzzy random variable are also determined by this concept. The hazard rate is estimated and its relationship with mean residual life ordering of fuzzy random variable is investigated. Additionally, we have focused on deriving bivariate characterization of hazard rate ordering which explicitly involves pairwise interchange of two fuzzy random variables $X$ and $Y$.

The consistent tanh expansion (CTE) method has been succeeded to apply to the nonintegrable (3+1)-dimensional Kadomtsev-Petviashvili (KP) and (2+1)-dimensional Boussinesq equations. The interaction solution between one soliton and one resonant soliton solution for the (3+1)-dimensional KP equation is obtained with CTE method. The interaction solutions among one soliton and cnoidal waves for these two equations are also explicitly given. These interaction solutions are investigated in both analytical and graphical ways. It demonstrates that the interactions between one soliton and cnoidal waves are elastic with phase shifts.

The present survey contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Results on the stability of differential problems for second order equations and of difference schemes for approximate solution of the second order problems are presented.

We give refinements and generalizations of the Dresher and Bellman inequalities for positive linear functionals. We also give reverse of the new obtained generalized version of these inequalities. Finally, we apply our results on time scales integrals to obtain refinements and generalizations of time scales Dresher’s and Bellman’s inequalities.

This paper shows that the epidemic model, previously proposed under ordinary differential equation theory, can be generalized to fractional order on a consistent framework of biological behavior. The domain set for the model in which all variables are restricted is established. Moreover, the existence and stability of equilibrium points are studied. We present the proof that endemic equilibrium point when reproduction number $R_0>1$ is locally asymptotically stable. This result is achieved using the linearization theorem for fractional differential equations. The global asymptotic stability of disease-free point, when , is also proven by comparison theory for fractional differential equations. The numeric simulations for different scenarios are carried out and data obtained are in good agreement with theoretical results, showing important insight about the use of the fractional coupled differential equations set to model babesiosis disease and tick populations.

Making use of the operator ${\mathcal{L}}_{\upsilon }$ for functions of the form $f\left(z\right)=1/z+{\sum }_{k=1}^{\mathrm{\infty }}{a}_k{z}^{k-1}$, which are analytic in the punctured unit disc ${\mathbb{U}}^{\ast }=\{z:z\in \mathbb{C}$ and , we introduce two subclasses of meromorphic functions and investigate convolution properties, coefficient estimates, and containment properties for these subclasses.

This paper investigates the fault detection problem for discrete-time nonlinear impulsive switched systems. Attention is focused on designing the fault detection filters to guarantee the robust performance and the detection performance. Based on these performances, sufficient conditions for the existence of filters are given in the framework of linear matrix inequality; furthermore, the filter gains are characterized by a convex optimization problem. The presented technique is validated by an example. Simulation results indicate that the proposed method can effectively detect the faults.

In the sixties, Loud obtained interesting results of continuation on periodic solutions in driven nonlinear oscillators with small parameter (Loud, 1964). In this paper Loud’s results are extended out for periodically driven Duffing equations with odd symmetry quantifying the continuation parameter for a periodic odd solution which is elliptic and emanates from the equilibrium of the nonperturbed problem.

A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.

Let $X$ be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space $X^{\ast }$. Let $T:X\supseteq D(T)\to 2^{X^{\ast }}$ be maximal monotone and $S:X\supseteq D(S)\to X^{\ast }$ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space $W\subset D(S)$, dense and continuously embedded in $X$. Assume, further, that there exists $d\ge 0$ such that $\langle v^{\ast }+Sx,x\rangle \ge -$d${‖x‖}^2$ for all $x\in D(T)\cap D(S)$ and $v^{\ast }\in Tx$. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type $T+S$. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator $L:X\supseteq D(L)\to X^{\ast }$ is given as a result of surjectivity of $L+S$, where $S$ is of type $(M)$ with respect to $L$. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in $X=L^p(0,T;W_0^{1,p}(\Omega ))$ of a nonlinear parabolic problem of the type $u_t-{\sum }_{i=1}^n(\partial /\partial x_i)a_i(x,t,u,\nabla u)=f(x,t)$, $(x,t)\in Q$; $u(x,t)=0$, $(x,t)\in \partial \Omega \times (0,T)$; $u(x,0)=0$, $x\in \Omega$, where $p>1$, $\Omega$ is a nonempty, bounded, and open subset of ${\mathbb{R}}^N$, $a_i:\Omega \times (0,T)\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ $(i=\mathrm{1,2},\dots ,n)$ satisfies certain growth conditions, and $f\in L^{p^{\prime }}(Q)$, $Q=\Omega \times (0,T)$, and $p^{\prime }$ is the conjugate exponent of $p$.

We study the oscillation behavior for some higher order integrodynamic equations on timescales. We establish some new sufficient conditions guaranteeing that all solutions of theses equations are oscillatory. Some numerical examples in the continuous case are given to validate the theoretical results.

This work is based on the application of Fourier and quasi-solution methods for solving the continuation inverse problem for 3D steady-state diffusion model inside a cylindrical layered medium. The diffusion coefficient is supposed to be a piecewise constant function, Cauchy data are given on the outer boundary of the cylinder, and we seek to recover the temperature at the inner boundary of the cylinder. Numerical experiments are investigated and show the capacity of proposed method only for smooth boundary condition. Under the suitable choice of regularization parameters we recover the distribution of temperature on the inner boundary with satisfactory quality for noisy data.

We show that the sublinearity hypothesis of some well-known existence results on multipoint Boundary Value Problems (in short BVPs) may allow the existence of infinitely many solutions by using Tietze extension theorem. This is a qualitative result which is of concern in Applied Analysis and can motivate more research on the conditions that ascertain the existence of multiple solutions to sublinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differential equations with multiple solutions.

We consider boundary value problems for scalar differential equation $x^{\mathrm{\prime \prime }}+\lambda f\left(x\right)=0$, $x(0)=0$, $x(1)=0$, where $f(x)$ is a seventh-degree polynomial and $\lambda$ is a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.

We investigate the unsteady peristaltic transport of a viscoelastic fluid with fractional Maxwell model through two coaxial vertical tubes. This analysis has been carried under low Reynolds number and long wavelength approximations. Analytical solution of the problem is obtained by using a fractional calculus approach. The effects of Grashof number, heat parameter, relaxation time, fractional time derivative parameter, amplitude ratio, and the radius ratio on the pressure gradient, pressure rise, and the friction forces on the inner and outer tubes are graphically presented and discussed.

This paper is devoted to both theoretical and numerical study of boundary value problems for higher-order nonlinear fractional integrodifferential equations. Existence and uniqueness results for the considered problem are provided and proved. The numerical method of solution for the problem is based on a conjugate collocation and spline approach combined with shooting method. Some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm.

The existence and uniqueness results for the Tricomi problem of Chaplygin’s hodograph equation are shown, in the case that the domain considered is close to the parabolic degenerate line, by adopting the energy integral methods and choosing judiciously suitable multipliers.

Exponential robust consensus of stochastic multiagent systems is studied. Coupling structures of multiagent systems are Markov jump switching; that is, multiagent systems contain Markov jump parameters. Sufficient conditions of almost surely exponential robust consensus are derived by utilizing the stochastic method and the approach of the matrix inequality. Finally, two simulations are shown to demonstrate the validity of the achieved theoretical results.

Herpes simplex virus type 2 (HSV-2) is the most prevalent sexually transmitted disease worldwide, despite the availability of highly effective antiviral treatments. In this paper, a basic mathematical model for the spread of HSV-2 incorporating all the relevant biological details and poor treatment adherence is proposed and analysed. Equilibrium states of the model are determined and their stability has been investigated. The basic model is then extended to incorporate a time dependent intervention strategy. The aim of the control is tied to reducing the rate at which HSV-2 patients in treatment quit therapy before completion. Practically, this control can be implemented through monitoring and counselling all HSV-2 patients in treatment. The Pontryagin’s maximum principle is used to characterize the optimal level of the control, and the resulting optimality system is solved numerically. Overall, the study demonstrates that though time dependent control will be effective on controlling new HSV-2 cases it may not be sustainable for certain time intervals.

The concept of “white noise,” initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of “noises” are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable “noises.” The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application.

Let $\Omega$ be a smoothly bounded pseudoconvex domain in ${\mathbb{C}}^n$ with one degenerate eigenvalue and assume that there is a smooth holomorphic curve $V$ whose order of contact with $b\Omega$ at $z_0\in b\Omega$ is larger than or equal to $\eta$. We show that the maximal gain in Hölder regularity for solutions of the $\overline{\partial }$-equation is at most $1/\eta$.

New classes of univalent harmonic functions are introduced. We give sufficient coefficient conditions for these classes. These coefficient conditions are shown to be also necessary if certain restrictions are imposed on the coefficients of these harmonic functions. By using extreme points theory we also obtain coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. Radii of convexity and starlikeness of the classes are also considered.

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains of ${\mathbb{R}}^m$ ($m\ge \mathrm{3}$) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

The existence and uniqueness of a mild solution to nonlinear fuzzy differential equation constrained by initial value were proven. Initial value constraint was then replaced by delay function constraint and the existence of a solution to this type of problem was also proven. Furthermore, the existence of a solution to optimal control problem of the latter type of equation was proven.

The problem of positive solutions for nonlinear $q$-fractional difference eigenvalue problem with nonlocal boundary conditions is investigated. Based on the fixed point index theory in cones, sufficient existence of positive solutions conditions is derived for the problem.

This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.

The variability ordering for more and less variables of fuzzy random variables in terms of its distribution function is defined. A property of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) is derived as an application to the variability ordering of fuzzy random variables. The concept of generalized variability orderings of nonnegative fuzzy random variables representing lifetime of components is introduced. The domination is a generalized variability ordering. We proposed an integral inequality to the case of fuzzy random variables using ordering. The results included equivalent conditions which justify the generalized variability orderings.

We prove the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions. The first existence and uniqueness result is based on Banach’s contraction principle. Moreover, other existence results are also obtained by using the Krasnoselskii fixed point theorem. An example is given to illustrate the main results.

Sequence of $q$-Bleimann, Butzer, and Hahn operators which is based on a continuously differentiable function $\tau$ on ${\mathbb{R}}^{+}$, with $\tau \left(\mathrm{0}\right)=\mathrm{0}$, $\inf {\tau }^{\text{'}}\left(x\right)\ge \mathrm{1}$, has been considered. Uniform approximation by such a sequence has been studied and degree of approximation by the operators has been obtained. Moreover, shape preserving properties of the sequence of operators have been investigated.

We study a degenerate evolution system containing the $p$-curl system in a bounded domain with initial and boundary conditions for the magnetic field $\mathbf{H}$ under the influence of a system force $\mathbf{F}$. This is concerned with an approximation of Bean’s critical-state model for type II superconductors. We will show the existence, uniqueness, and regularity of solutions. Moreover we will get the properties of the limit solution as $p\to \mathrm{\infty }$.

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefficients. We use a wave operator, which provides a convenient way of controlling all initial and boundary conditions. The proposed choice of the auxiliary operator helps to find the approximate series solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method.

The authors discuss multiple solutions for the nth-order singular boundary value problems of nonlinear integrodifferential equations in Banach spaces by means of the fixed point theorem of cone expansion and compression. An example for infinite system of scalar third-order singular nonlinear integrodifferential equations is offered.

Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.