This paper is concerned with the global dynamics of a Holling-Tanner predator-prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.

]]>The study of collision-induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà-Ascoli and Banach contraction mapping principle.

]]>In this paper, a collocation spectral numerical algorithm is presented for solving nonlinear systems of fractional partial differential equations subject to different types of conditions. A proposed error analysis investigates the convergence of the mentioned algorithm. Some numerical examples confirm the efficiency and accuracy of the method.

]]>This paper is devotedto prove the existence of one or multiple solutions for a family of nonlinear fourth-order boundary value problems.

We use several fixed point theorems, previously developed by the author and her coauthor, to prove the existence of solutions of some simply supported beam problems. To finish the work, a particular case is studied, and the existence of multiple solutions is proved for 2 different particular nonlinear functions.

An exact knowledge of the mechanical and optical properties of crystals allows not only for theoretical advances, but it is also a useful tool to asses crystal quality in the technological processes of growth and production of advanced crystals like, eg, scintillators. In this paper, we study the elasto-optic behavior of tetragonal crystals to evaluate the photoelastic constants, associated to various states of stress, in terms of the components of the piezo-optic tensor. Moreover, we arrive at a generalization, for tetragonal crystals, of the Brewster law for optically isotropic materials.

]]>The initial-boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately.

]]>We consider a chemotaxis consumption system with singular sensitivity
, *v*_{t}=*ε*Δ*v*−*u**v* in a bounded domain
with *χ*,*α*,*ε*>0. The global existence of classical solutions is obtained with *n*=1. Moreover, for any global classical solution (*u*,*v*) to the case of *n*,*α*≥1, it is shown that *v* converges to 0 in the *L*^{∞}-norm as *t**∞* with the decay rate established whenever *ε*∈(*ε*_{0},1) with
.

During the 2013-2015 Ebola virus disease outbreak, admission into a health facility depended on the availability of hospital beds and health personnel. The limited number of such important logistics contributed to the escalation of the epidemic. We use a compartmental model to study the dynamics of Ebola virus disease when there is a limited number of beds for patients. We use a non-linear hospitalisation rate and formulate the rate at which the time-dependent number of available beds evolves. The model shows a backward bifurcation. Simulation results show that bed supply in Ebola treatment units contribute to the reduction of the number of individuals infected by Ebola virus. The model fitting results suggest that a timely supply of sufficient beds to Ebola treatment units limits the spread of the disease. Despite the fact that bed supplies to Ebola treatment units are not in themselves a control measure, they contribute to the reduction of the disease spread, by keeping the infectious in one place, during their infectious period.These results have important implications to the management and control of the disease.

]]>In this paper, we propose an improved human T-cell leukemia virus type 1 infection model with mitotic division of actively infected cells and delayed cytotoxic T lymphocyte immune response. By constructing suitable Lyapunov functional and using LaSalle invariance principle, we investigate the global stability of the infection-free equilibrium of the system. Our results show that the time delay can change stability behavior of the infection equilibrium and lead to the existence of Hopf bifurcations. Finally, numerical simulations are conducted to illustrate the applications of the main results.

]]>Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. The AFD merged with the greedy algorithm idea, and in particular, motivated the so-called pre-orthogonal greedy algorithm (pre-OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD-type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper constructs one novel method to perform the 1-D AFD algorithm. We make use of the FFT algorithm to reduce the algorithm complexity, from the original
to
, where *N* denotes the number of the discretization points on the unit circle and *M* denotes the number of points in [0,1). This greatly enhances the applicability of AFD. Experiments are performed to show the high efficiency of the proposed algorithm.

The purpose of the present paper is to define the GBS (Generalized Boolean Sum) operators associated with the two-dimensional Bernstein-Durrmeyer operators introduced by Zhou 1992 and study its approximation properties. Furthermore, we show the convergence and comparison of convergence with the GBS of the Bernstein-Kantorovich operators proposed by Deshwal et al 2017 by numerical examples and illustrations.

]]>In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation:

where *ɛ* is a positive parameter, *a* and *b* are positive constants, and 3<*p*<5. Let
denotes the ground energy function associated with
,
, where
is regard as a parameter. Suppose that the potential *V*(*x*) decays to zero at infinity like |*x*|^{−α} with 0<*α*≤2, we prove the existence of positive solutions *u*_{ɛ} belonging to
for vanishing or unbounded *K*(*x*) when *ɛ* > 0 small. Furthermore, we show that the solution *u*_{ɛ} concentrates at the minimum points of
as *ɛ*0^{+}.

In this paper, we consider the 2D Boussinesq system with variable kinematic viscosity in the velocity equation and with weak damping effect to instead of the regularity effect for the thermal conductivity. Even if without thermal diffusion in the temperature equation, we establish the global well-posedness for the 2D Boussinesq system with general initial data.

]]>The main objectives of this article are to introduce stochastic parameterizing manifolds and to study the dynamical transitions of the two-dimensional stochastic Swift-Hohenberg equation. The study is based on the general framework developed by Chekroun, Liu and Wang. The detailed effect of the noise on the transition and on the stochastic low-dimensional parameterization of the system is obtained.

]]>In this paper, we investigate the generalized Stieltjes-Wigert and *q*-Laguerre polynomials. We derive the second- and the third-order nonlinear difference equations for the subleading coefficients of these polynomials and use them to find a few terms of the formal expansions in powers of *q*^{n/2}. We also show how the recurrence coefficients in the three-term recurrence relation for these polynomials can be computed efficiently by using the nonlinear difference equations for the subleading coefficient. Moreover, we obtain systems of difference equations with one of the equations being *q*-discrete Painlevé III or V equations and analyze them by a singularity confinement. We also discuss certain generalized weights.

In this paper, we consider a rotating Euler-Bernoulli beam. The beam is made of a viscoelastic material, and it is subject to undesirable vibrations. Under a suitable control torque applied at the motor, we prove the arbitrary stabilization of the system for a large class of relaxation functions by using the multiplier method and some ideas introduced by Tatar (J. Math. Phys. 52:013502, 2011).

]]>In this paper, we consider the uniqueness problems of finite-order meromorphic solutions to Painlevé equation. Our result says that such solutions *w* are uniquely determined by their poles and the zeros of *w*−*e*_{j} (counting multiplicities) for 2 finite complex numbers *e*_{1}≠*e*_{2}. As applications, we derive 2 uniqueness theorems about the Weierstrass *℘* function and Jacobi elliptic function *s**n*, respectively.

In this paper, we consider the Bresse-Cattaneo system with a frictional damping term and prove some optimal decay results for the *L*^{2}-norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number *δ* that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results

In this paper, we introduce (*p*,*q*)-Sturm-Liouville problems and prove that their solutions are orthogonal with respect to a (*p*,*q*)-integral space. We then present some illustrative examples for this kind of problems and obtain the (*p*,*q*)-hypergeometric representation of the polynomial solutions together with their 3-term recurrence relations. We also compute the norm square value of the polynomial solutions and obtain their limiting cases in the sequel.

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign-changing potential by the Nehari method.

]]>This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem for the nonlinear diffusion equation in an *unbounded* domain
(
), written as

which represents the porous media, the fast diffusion equations, etc, where *β* is a single-valued maximal monotone function on
, and *T*>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024-2050 and Adv Math Sci Appl 2017; 26:221-242) existence and uniqueness of solutions for were directly proved under a growth condition for *β* even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for *β* by confirming Cauchy's criterion for solutions of the following approximate problem _{ε} with approximate parameter *ε*>0:

The purpose of the study is to analyze the time-fractional reaction-diffusion equation with nonlocal boundary condition. The proposed model is used to predict the invasion of tumor and its growth. Further, we establish the existence and uniqueness of a weak solution of the proposed model using the Faedo-Galerkin method and compactness arguments.

]]>We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle-node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

]]>It is easy to write down entire solutions of the Helmholtz equation: Examples are plane waves and Herglotz wavefunctions. We are interested in the far-field behaviour of these solutions motivated by the following question: When is it legitimate to split the far field of such an entire solution into the sum of an incoming spherical wave and an outgoing spherical wave? We review the relevant literature (there are disjoint physical and mathematical threads), and then we answer the question for Herglotz wavefunctions, using a combination of the 2-dimensional method of stationary phase and some explicit examples.

]]>Population aging is rapidly increasing in developing countries; thus, covering medical needs for breast cancer diagnosis and treatment is a priority in Latin America. We describe an approach for integrating differential expression analysis, biological pathway enrichment, in silico transcription-binding sites and network topology, to find key genes that may be used as biomarkers or therapeutic targets. This approach is based on publicly available data from microarrays of the MCF-7 breast cancer cell line treated with estrogen. We found significant estrogen-responsive genes, which were used as nodes to construct networks based on protein-protein interactions reported in the literature. Then, we conducted a topology analysis of the networks, revealing the most-connected nodes, ie, those responsible for maintaining the network structure corresponding to genes with well-acknowledged functions in G1/S cell cycle transition, such as cyclin-dependent kinase 2 (CDK2), which has been proposed as a therapeutic target in classical biochemical studies. In addition, analyses of biological pathway enrichment and in silico transcription-binding sites support the biological meaning and importance of these key genes and help to decide the best target genes. Therefore, we postulate that the integrative bioinformatics approach shown here, unlike the classical bioinformatics approach that only includes differentially expressed genes and enriched biological pathways, can be applied as an approach for finding novel biomarkers and/or therapeutic target genes for nonresponsive treatments.

]]>In this paper, the mathematical modeling and trajectory planning of a 3D rotating manipulator composed of a rotating-prismatic joint and multiple rigid links is considered. Possible trajectories of the end effector of the manipulator—following a sequence of 3D target points under the action of 2 external driving torques and an axial force—are modeled using zenithal gnomic projections and polar piecewise interpolants expressed as polynomial Hermite-type functions. Because of the geometry of the manipulator, the time-dependent generalized coordinates are associated with the spherical coordinates named the radial distance related to the manipulator length, and the polar and azimuthal angles describing the left and right and, respectively, up and down motion of the manipulator. The polar trajectories (left and right motion) of the end effector are generated using a inverse geometric transformation applied to the polar piecewise interpolants that approximate the gnomic projective trajectory of the 3D via-points. The gnomic via-points—located on a projective plane situated on the northern hemisphere—are seen from the manipulator base location, which represents the center of rotation of the extensible manipulator. The related azimuthal trajectory (up and down motion) is generated by polar piecewise interpolants on the azimuthal angles. Smoothness of the polygonal trajectory is obtained through the use of piecewise interpolants with continuous derivatives, while the kinematics and dynamics implementation of the model is well suited to computer implementation (easy calculation of kinematics variables) and simulation. To verify the approach and validate the model, a numerical example—implemented in Matlab—is presented, and the results are discussed.

]]>A second-order decoupled algorithm for the nonstationary Stokes-Darcy system, which allows different time steps in two subregions, is proposed and analyzed in this paper. The algorithm, which is a combination of the second-order backward differentiation formula and second-order extrapolation method, uncouples the problem into two decoupled problems per time step. We prove the unconditional stability and long-time stability of the decoupled scheme with different time steps and derive error estimates of this decoupled algorithm using finite element spatial discretization. Numerical experiments are provided to illustrate the accuracy, effectiveness, and stability of the decoupled algorithm and show its advantages of increasing accuracy and efficiency.

]]>In this work, the modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity is presented. Expressing the solution of the problem as a double-layer potential of an unknown density, we form the associated boundary integral equation that describes the problem. Exploiting that the discrete spectrum of the irregular values of the associated integral equation is identified with the spectrum of eigenvalues of the corresponding interior homogeneous Neumann problem for the transverse part of the elastic displacement field, we introduce a modification of the fundamental solution of the elastic field. We establish the sufficient conditions that the coefficients of the modification must satisfy to overcome the problem of nonuniqueness for the thermoelastic problem.

]]>We evaluate a 1-loop, 2-point, massless Feynman integral *I*_{D,m}(*p*,*q*) relevant for perturbative field theoretic calculations in strongly anisotropic *d*=*D*+*m* dimensional spaces given by the direct sum
. Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions *D* and *m*. We obtain series expansions of *I*_{D,m}(*p*,*q*) in terms of powers of the variable *X*:=4*p*^{2}/*q*^{4}, where *p*=|** p**|,

The study of contractions of Lie algebras is profusely extended in the last decades. In this paper we study the graded contractions of some lower-dimensional filiform Lie algebras which have not been studied earlier. Particularly, we deal with graded contractions of model filiform Lie algebras of dimension less than or equal to 6 and with the ones of a nonmodel 6-dimensional filiform Lie algebra.

]]>In this paper, we obtain a Cauchy-type integral representation to solve certain Beltrami equations and apply it for research of the Riemann boundary value problem for that equations on nonrectifiable contours. While more or less studied for piecewise smooth contours, for the case of nonrectifiable curve, this is a pioneer result. There is only one relatively simple type of Beltrami equation that is solved under such condition (see below). Our next goal is to achieve solutions for more types of the Beltrami equations on nonrectifiable curves.

]]>In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic-quintic Ginzburg-Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz-continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique.

]]>The presence of a steady-state distribution is an important issue in the modelization of cell populations. In this paper, we analyse, from a numerical point of view, the approach to the stable size distribution for a size-structured balance model with an asymmetric division rate. To this end, we introduce a second-order numerical method on the basis of the integration along the characteristic curves over the natural grid. We validate the interest of the scheme by means of a detailed analysis of convergence.

]]>Let *L* be the *n*-th order linear differential operator *L**y*=*ϕ*_{0}*y*^{(n)}+*ϕ*_{1}*y*^{(n−1)}+⋯+*ϕ*_{n}*y* with variable coefficients. A representation is given for *n* linearly independent solutions of *L**y*=*λ**r**y* as power series in *λ*, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for *n*=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for *λ*=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of *n*-th order initial value problems and spectral problems.

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal *L*^{2}-norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal *L*^{2} error estimates of approximate solutions. Secondly, two-level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two-level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

This paper is devoted to the Cauchy problem for the nonlinear Schrodinger equation with time-dependent fractional damping term. We prove the local existence result, and we study the global existence and blow-up solutions.

]]>An inverse problem of determining a time-dependent source term from the total energy measurement of the system (the over-specified condition) for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional-order time derivative and Sturm-Liouville operator by fractional-order Sturm-Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.

]]>We consider a laminar boundary-layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3-dimensional boundary-layer equations that contains 2 physical parameters: pressure gradient (*β*) and shear-to-strain-rate ratio parameter (*α*). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller-box numerical method for the full nonlinear system. The results show that the flow field is divided into near-field region (mainly dominated by viscous forces) and far-field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3-dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary-layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing *α* when the wedge is moving in the *x*-direction, while the case is different when it is moving in the *y*-direction. Further, both analysis show that 3-dimensional boundary-layer solutions exist in the range −1<*α*<*∞*. These are some interesting results linked to an important class of boundary-layer flows.

This paper concerns itself with the development of an a priori error analysis of an eddy current problem when applying the well-known hybridizable discontinuous Galerkin (HDG) method. Up to the authors' knowledge, this kind of theoretical result has not been proved for this kind of problems. We consider nontrivial domains and heterogeneous media which contain conductor and insulating materials. When dealing with these domains, it is necessary to impose the divergence-free condition explicitly in the insulator, what is done by means of a suitable Lagrange multiplier in that material. In the end, we deduce an equivalent HDG formulation that includes as unknowns the tangential and normal trace of a vector field. This represents a reduction in the degrees of freedom when compares with the standard DG methods. For this scheme, we conduct a consistency and local conservative analysis as well as its unique solvability. After that, we introduce suitable projection operators that help us to deduce the expected a priori error estimate, which provides estimated rates of convergence when additional regularity on the exact solution is assumed.

]]>In this work we consider the higher order fractional differential equation with derivative defined in the sense of Katugampola. We present some equivalent integral form of the considered boundary value problem and using properties of an appropriate Green function and prove fractional counterpart of the Lyapunov inequality.

]]>In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann-Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi-Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.

]]>The purpose of this work is to investigate the problem of solutions to the time-fractional Navier-Stokes equations with Caputo derivative operators. We obtain the existence and uniqueness of the solutions to each approximate equation, as well as the convergence of the approximate solutions. Furthermore, we present some convergence results for the Faedo-Galerkin approximations of the given problems.

]]>This paper focuses on the construction of periodic solutions of nonlinear beam equations on the *d*-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the nonresonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.

This paper deals with the energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source term under null Dirichlet boundary condition. By constructing a new family of potential wells, together with logarithmic Sobolev inequality and perturbation energy technique, we establish sufficient conditions to guarantee the solution exists globally or occurs infinite blow-up and derive the polynomial or exponential energy decay estimates under some appropriate conditions.

]]>Topology is the most important branch of modern mathematics, which plays an important role in applications. In this paper, we use the concept of the topology, based on the concept of multiset to solve an important problems in life (DNA and RNA mutation) to detect diseases and help biologists in the treatment of diseases. Also, we introduce a new theory that explains if there is an existence of a mutation or not, and we have a set of metric functions through which we examine the congruence and similarity and dissimilarity between “types,” which may be a strings of bits, vectors, DNA or RNA sequences, …, etc. Finally, we will introduce a theory that which can be used to know the existence and place of mutation.

]]>A general nonlinear model of illiquid markets with feedback effects is considered. This equation with 2 free functional parameters contains as partial cases the classical Black-Scholes equation, Schönbucher-Wilmott equation, and Sircar-Papanicolaou equation of option pricing. We obtain here the complete group classification of the equation, and for every parameters specification we obtain the principal Lie algebra and its optimal system of 1-dimensional subalgerbras. For every such subalgebra we calculate the invariant submodel and invariant solution, when it is possible. Thus, the series of invariant submodels and invariant solutions are derived for the considered nonlinear model.

]]>In this article, we introduce the local fractional integral iterative method and the local fractional new iterative method for solving the local fractional differential equations. Also, we perform a comparison between the results obtained by these 2 local fractional methods with the results obtained by some other local fractional methods. The obtained results illustrate the significant features of the 2 methods that are both very effective and straightforward for solving the differential equations with local fractional derivative compared with the other local fractional methods.

]]>This work is based on using wavelet for calculating one-dimensional nonlinear Volterra-Hammerstein and mixed Volterra-Fredholm-Hammerstein integral equation of the second kind in a complex plane. So far, as we know, no study has yet been attempted for solving this integral equation in the complex plane. The main specificity of this method is to avoid solving any linear or algebra system for approximated of integral equations. In Section 2, we introduce the integral operator for RH wavelet and use it in our numerical methods. In Section 3, we show that our problems have a unique solution. Furthermore, we give an upper bound for the error analysis. Finally, we make some example in Section 4 and solve them.

]]>We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector-valued space of sequences for equations that can be modeled in the form

where *X* is a Banach space,
*A* is a closed linear operator with domain *D*(*A*) defined on *X*, and *G* is a nonlinear function. The operator Δ^{γ} denotes the fractional difference operator of order *γ*>0 in the sense of Grünwald-Letnikov. Our class of models includes the discrete time Klein-Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that *f* is small and that *G* is a nonlinear term.

The good Boussinesq equation is endowed with symplectic conservation law and energy conservation law. In this paper, some new highly efficient structure-preserving methods for the good Boussinesq equation are proposed by improving the standard finite difference method (FDM). The new methods only use and calculate values at the odd (or even) nodes to reduce the computational cost. We call this kind of methods odd-even method (OEM). Numerical results show that the OEM and the standard FDM have nearly the same numerical errors under the same mesh partition. However, the OEM is much more efficient than the standard FDM, such as the consumed CPU time and occupied memory.

]]>For two-dimensional autonomous linear incommensurate fractional-order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh-Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two-dimensional fractional-order FitzHugh-Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer-order FitzHugh-Nagumo model.

]]>We study the Korteweg-de Vries equation
subject to boundary condition in nonrectangular domain
where
, with some assumptions on functions (*φ*_{i}(*t*))_{1≤i≤2} and the coefficients of equation. The right-hand side and its derivative with respect to *t* are in the Lebesgue space *L*^{2}(Ω). Our goal is to establish the existence, the uniqueness, and the regularity of the solution.

In this paper, we present a relation on graphs that induces new types of topological structures to the graph and then study some of the properties of this graph. Also, we investigate an algorithm to generate the topological structures from different graphs. Finally, some applications in medicine and geographs will be given, and we verify our results in the real life.

]]>In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.

]]>Finite-region stability (FRS), a generalization of finite-time stability, has been used to analyze the transient behavior of discrete two-dimensional (2-D) systems. In this paper, we consider the problem of FRS for discrete 2-D Roesser models via dynamic output feedback. First, a sufficient condition is given to design the dynamic output feedback controller with a state feedback-observer structure, which ensures the closed-loop system FRS. Then, this condition is reducible to a condition that is solvable by linear matrix inequalities. Finally, viable experimental results are demonstrated by an illustrative example.

]]>We analyze existence and asymptotic behavior of a system of semilinear diffusion-reaction equations that arises in the modeling of the mitochondrial swelling process. The model itself expands previous work in which the mitochondria were assumed to be stationary, whereas now their movement is modeled by linear diffusion. While in the previous model certain formal structural conditions were required for the rate functions describing the swelling process, we show that these are not required in the extended model. Numerical simulations are included to visualize the solutions of the new model and to compare them with the solutions of the previous model.

]]>Singularity analysis of a parallel manipulator (PM) is important in its trajectory planning and workspace design. This paper presents a new method based on geometric algebra (GA) for the singularity analysis of the 3/6-SPS Gough-Stewart PM, where S denotes a spherical joint and P a prismatic pair. The 6 line vectors associated with the SPS limb are expressed using GA. An analytic singular polynomial is then derived as the coefficient of the outer product of all 6 line vectors. This polynomial provides an overall description of the singularity of the 3/6-SPS Gough-Stewart PM. Position-singularity loci and orientation-singularity loci can be drawn based on this polynomial, the latter of which have seldom been addressed. It is also shown that the proposed GA-based method is geometrically intuitive and computationally efficient.

]]>In this paper, we consider low-order stabilized finite element methods for the unsteady Stokes/Navier-Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low-order element (*P*_{1}−*P*_{1} or *P*_{1}−*P*_{0}) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.

In this article, we establish some relationships between several types of partial differential equations and ordinary differential equations. One application of these relationships is that we can get the exact values of the blowup time and the blowup rate of the solution to a partial differential equation by solving an ordinary differential equation. Another application of these relationships is that we can give the estimates for the spatial integration (or mean value) of the solution to a partial differential equation. We also obtain the lower and upper bounds for the blowup time of the solution to a parabolic equation with weighted function and space-time integral in the nonlinear term.

]]>In this paper, we study the well posed-ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane. By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter *m*∈(0,1) and establish local and global existences of solution to the semilinear equation. Meanwhile, we derive the existence of solutions to semilinear generalized Euler-Poisson-Darboux equation with a negative parameter *α*∈(−1,0).

No abstract is available for this article.

]]>The time fractional Fokker-Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial-boundary value problems of the time fractional Fokker-Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.

]]>We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two-by-two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric-to-magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.

]]>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.

]]>We present a systematic mathematical analysis of the qualitative steady-state response to rate perturbations in large classes of reaction networks. This includes multimolecular reactions and allows for catalysis, enzymatic reactions, multiple reaction products, nonmonotone rate functions, and non-closed autonomous systems. Our *structural sensitivity analysis* is based on the stoichiometry of the reaction network, only. It does not require numerical data on reaction rates. Instead, we impose mild and generic nondegeneracy conditions of algebraic type. From the structural data, only, we derive which steady-state concentrations are sensitive to, and hence influenced by, changes of any particular reaction rate—and which are not. We also establish transitivity properties for influences involving rate perturbations. This allows us to derive an *influence graph* which globally summarizes the influence pattern of any given network. The influence graph allows the computational, but meaningful, automatic identification of functional subunits in general networks, which hierarchically influence each other. We illustrate our results for several variants of the glycolytic citric acid cycle. Biological applications include enzyme knockout experiments and metabolic control.

Under the displacement and stress satisfying Riemann boundary value condition, the decoupled quasistatic linear thermoelasticity system is discussed on bounded simply connected domain. The quasistatic equilibrium equation is solved by using Riemann boundary value problem theory. Also decoupled temperature equation is studied by applying the contractive mapping principle. Finally, existence and analyticity of the solution are proved.

]]>We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, and frequency decomposition, we obtain the global existence and the optimal time-decay estimate of solutions.

]]>In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô-Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's-Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.

]]>In this paper, we are concerned with optimal decay rates for higher-order spatial derivatives of classical solution to the compressible Navier-Stokes-Maxwell equations in three-dimensional whole space. If the initial perturbation is small in -norm, we apply the Fourier splitting method to establish optimal decay rates for the second-order spatial derivatives of a solution. As a by-product, the rate of classical solution converging to the constant equilibrium state in -norm is .

]]>The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro-differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro-differential equation and examines their convergence, stability, and accuracy.

]]>The initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely dependence on time. These results ensure, for instance, the well-posedness of a class of vehicular traffic models with time-dependent speed limits. A traffic management problem is then shown to admit an optimal solution.

]]>This paper is devoted to the analysis of a linearized theta-Galerkin finite element method for the time-dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on *θ*-time stepping scheme with
including the standard Crank-Nicolson (
) and the shifted Crank-Nicolson (
, where *δ* is the time-step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in *L*^{2}-norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank-Nicolson, the shifted Crank-Nicolson, and the general case where
with *k*=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy-Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy-Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing.

]]>In this paper, the Cauchy problem for the 3D diffusion approximation model in radiation hydrodynamics is considered. By using the embedding theorem and interpolation technique, we establish the global well-posedness of strong solutions in *H*^{2}.

A Hamilton-Poisson system is an approach for the motion of a spacecraft around an asteroid or for the motion of an underwater vehicle. We construct a coordinate chart on the symplectic leaf which contains a specific generic equilibrium point and we establish stability conditions for this equilibrium point.

]]>To protect fishery populations on the verge of extinction and sustain the biodiversity of the marine ecosystem, marine protected areas (MPA) are established to provide a refuge for fishery resource. However, the influence of current harvesting policies on the MPA is still unclear, and precise information of the biological parameters has yet to be conducted. In this paper, we consider a bioeconomic Gompertz population model with interval-value biological parameters in a 2-patch environment: a free fishing zone (open-access) and a protected zone (MPA) where fishing is strictly prohibited. First, the existence of the equilibrium is proved, and by virtue of Bendixson-dulac Theorem, the global stability of the nontrivial steady state is obtained. Then, the optimal harvesting policy is established by using Pontryagin's maximum principle. Finally, the results are illustrated with the help of some numerical examples. Our results show that the current harvesting policy is advantageous to the protection efficiency of an MPA on local fish populations.

]]>In this paper we solve an initial-boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first-order pde; the stochastic version yields a second-order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first-order case, the analysis does not readily extend to the second-order case. We develop a method for solving the second-order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.

]]>This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state-space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time-varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.

]]>In this paper, a new generalized 5×5 matrix spectral problem of Ablowitz-Kaup-Newell-Segur type associated with the enlarged matrix Lie superalgebra is proposed, and its corresponding super soliton hierarchy is established. The super variational identities are used to furnish super Hamiltonian structures for the resulting super soliton hierarchy.

]]>We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by

with the initial condition *u*|_{(t=0)}=*u*_{0}, *u*_{t}|_{(t=0)}=*u*_{1}, and *θ*|_{(t=0)}=*θ*_{0} in Ω and the boundary condition *u*=*∂*_{ν}*u*=*θ*=0 on Γ, where *u*=*u*(*x*,*t*) denotes a vertical displacement at time *t* at the point *x*=(*x*_{1},⋯,*x*_{n})∈Ω, while *θ*=*θ*(*x*,*t*) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half-space case. We prove the existence of
-bounded solution operators of the corresponding resolvent problem. Then, the generation of *C*_{0} analytic semigroup and the maximal *L*_{p}-*L*_{q}-regularity of time-dependent problem are derived.

In this study, modelling and identification of prestress state in functionally graded plate within the framework of the Timoshenko theory are discussed. With the help of variational principles, statements of boundary problems for stationary vibration of inhomogeneous prestressed plates have been derived taking into account various factors of prestress state. The comparative analysis of classical and nonclassical models has been conducted. The effect of the prestress state factors on the solution characteristics has been estimated. New approaches to solving the inverse problems on a reconstruction of inhomogeneous prestress functions in a functionally graded plate have been proposed on the basis of derivation of reciprocity relations and iterative regularization. The results of numerical reconstruction experiments are presented; practical recommendations on a selection of frequency range for the purpose of getting the highest reconstruction accuracy are given.

]]>In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher-order moduli of smoothness and of best approximation quantity are obtained.

]]>In this paper, we consider a Kudryashov-Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1-dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.

]]>We are concerned with a family of dissipative active scalar equation on
. By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution *θ* of satisfying . In this paper, we consider some countable sum-closed frequency sets (see Remark ). We show that the property of the solution is rather different from Chae et al and obtain that
with some initial data *θ*_{0} for all *t*≥0,
and 0≤*α*≤*ω*, where *ω* is a fixed constant. Furthermore, arranging the elements of a countable sum-closed frequency set *F*_{δ} as in Remark , we have for any 0≤*α*≤*ω* that
belongs to
, where *F*_{δ} is defined in or .

The inverse scattering transform for the derivative nonlinear Schrödinger-type equation is studied via the Riemann-Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann-Hilbert problem is established for the derivative nonlinear Schrödinger-type equation. In the inverse scattering process, *N*-soliton solutions of the derivative nonlinear Schrödinger-type equation are obtained by solving Riemann-Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.

In this paper, we are concerned with a rumor propagation model with L vy noise. We first prove that there exists a positive global solution. Then, the asymptotic behaviors around the rumor-free equilibrium and rumor-epidemic equilibrium are obtained. Lastly, simulations verify our results.

]]>In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite-time blow-up result for a large class of solutions with positive initial energy.

]]>This paper is devoted to the study of the blow-up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions:

Here,
is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow-up or non–blow-up criterion under some appropriate assumptions on the functions *f*,*g*,*ρ*,*k*, and *u*_{0}. Moreover, we dedicate an upper bound and a lower bound for the blow-up time when blowup occurs.

A partial inverse problem for an integro-differential Sturm-Liouville operator on a star-shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.

]]>In this paper, we study global existence and asymptotic stability of solutions for the initial value problem of the three-dimensional (3-D) generalized incompressible micropolar system in Fourier-Besov spaces. Besides, we also establish some regularizing rate estimates of the higher-order spatial derivatives of solutions, which particularly imply the spatial analyticity and the temporal decay of global solutions.

]]>An efficient algorithm for the computation of a *C*^{2} interpolating clothoid spline is herein presented. The spline is obtained following an optimisation process, subject to continuity constraints. Among the 9 various targets/problems considered, there are boundary conditions, minimum length path, minimum jerk, and minimum curvature (energy). Some of these problems are solved with just a couple of Newton iterations, whereas the more complex minimisations are solved with few iterations of a nonlinear solver. The solvers are warmly started with a suitable initial guess, which is extensively discussed, making the algorithm fast. Applications of the algorithm are shown relating to fonts, path planning for human walkers, and as a tool for the time-optimal lap on a Formula 1 circuit track.