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We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

Plasma generated by hypersonic aerocraft is one of the research subjects in computational electromagnetics. When the aerocraft flies at an ultra-high speed in the near space, plasma sheath will be generated in the surroundings and it will trigger varying changes in aerocraft radar electromagnetic scattering properties. Plasma sheath brings about tremendous impacts on the communication between the aerocraft and the outside world, and even ends with "blackout effects" to interrupt communication in severe cases. The interplay between the plasma sheath and the electromagnetic wave involves multiple disciplines covering aerodynamics, plasma sheath physics, and electromagnetic field theory. Therefore, it is challengeable to build a rigorous numerical modeling of the plasma sheath. In this paper, the latest technologies in the electromagnetic simulation of hypersonic aerocraft with plasma sheath are reviewed. Applied cases review and illustrate extensive approaches from approximate calculations to accurate calculations, including analytical methods, differential equation methods, integral equation methods and high-frequency approximation methods. The paper assesses the RCS of electromagnetic wave in analyzing the electromagnetic scattering problem in plasma. With the considerations the reality of electromagnetic simulation and future challenges, the paper talks about the efficiency, advantages and disadvantages of these approaches in simulating hypersonic aerocraft targets.

The paper considers the existence of one- or two-hump solutions of a singularly perturbed nonlinear Schrodinger (NLS) equation, which is the standard NLS equation with a third order perturbation. In particular, this equation appears in the field of nonlinear optics, where it is used to describe pulses in optical fibers near a zero dispersion wavelength. It has been shown formally and numerically that the perturbed NLS equation has one- or multi-hump solutions with small oscillations at infinity, called generalized one- or multi-hump solutions. The main purpose of the paper is to provide the first rigorous proof of the existence of generalized one- or two-hump solutions of the singularly perturbed NLS equation. The several invariant properties of the equation, i.e., the translational invariance, the gauge invariance and the reversibility property, are essential to obtain enough free constants to prove the existence. The ideas and methods presented here may be applicable to show existence of generalized 2 k -hump solutions of the equation.

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system: { − Δ u + V ( x ) u = f 1 ( x , v ) , x ∈ R 2 , − Δ v + V ( x ) v = f 2 ( x , u ) , x ∈ R 2 , where V ∈ C ( R 2 , ( 0 , ∞ ) ) and f 1 , f 2 : R 2 × R → R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity f i , we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain, { − Δ u = f 1 ( v ) , x ∈ Ω , − Δ v = f 2 ( u ) , x ∈ Ω , u = 0 , v = 0 , x ∈ ∂ Ω . Our results improve and extend the related results of de Figueiredo-do O-Ruf (2004) [20] ; (2011) [21] , of Lam-Lu (2014) [30] , and of de Figueiredo-do O-Zhang (2020) [22] .

Consider the advective Fisher-KPP equation u t = u x x − β u x + f ( u ) on the half line [ 0 , ∞ ) with Dirichlet boundary condition at x = 0 . In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0 ) and constructed a new type of entire solution U ( x , t ) , which, under the additional assumption f ″ ( u ) ≤ 0 , is concave and U ( ∞ , t ) = 1 for all t ∈ R . In this paper, we consider the equation with advection and without the additional assumption f ″ ( u ) ≤ 0 . In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ ( ∞ , t ) ≡ 1 and U ˜ ( ⋅ , t ) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞ ) property rather than the concavity is more essential for such entire solutions. In case β 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ ( ∞ , t ) increases from 0 to 1 as t increasing from −∞ to ∞.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

This paper describes a new efficient approach based on the concept of reduced basis for large deformation analysis. The domain problem is discretized using the meshfree particle radial point interpolation method (RPIM), which inherently possesses the Kronecker’s delta property. Meshless numerical integration is evaluated by the Cartesian transformation method (CTM), which enhances the performance of the RPIM. In addition, we also introduce a new approach to further improve the capability of the current CTM in evaluation of numerical integration for problems with complex geometries, i.e., by incorporation of the non-uniform rational B-splines function (NURBS) into the CTM. The emphasis of the paper is on the nonlinear nature of the large deformation problems, which are often solved by an iterative scheme. Conventional Newton–Raphson technique usually requires high cost due to the fact that several load steps are usually performed, and multiple iterations are needed in each load step. This low computational efficiency can be overcome, as proposed in this work, by using the so-called combined approximation, which approximates the full-size solution by a set of reduced bases. In other words, reduction of the problem size can be obtained, leading to reduction of the computational time, while accuracy is almost preserved.

We study the homogenization of the evolutionary compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain perforated with a large number of very tiny holes. We show that under suitable assumptions on the smallness and distribution of the holes the limit system remains the same in the unperforated domain. One of the main novelty in the paper consists in the treatment of the entropy inequality and thus the paper also improves the related result in the steady case from Lu and Pokorný (J Differ Equ 278:463–492, 2021).

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by$\mathcal{J}_{\nu , \lambda }^{\mu }$Jν,λμ, where$\mu >0$μ>0and$\lambda ,\nu \in \mathbb{C\ }$λ,ν∈Cget increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμby applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, themth derivative of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμis considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions$\mathcal{I}_{\nu ,\lambda }^{\mu }$Iν,λμdefined by$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$Iν,λμ(z)=i−2λ−νJν,λμ(iz)of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$Iν,λμ(z)are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

International audience; The paper formulates the concept of persistence of excitation for discrete-time linear switched systems, and provides sufficient conditions for an input signal to be persistently exciting. Persistence of excitation is formulated as a property of the input signal, and it is not tied to any specific identification algorithm. The results of the paper rely on realization theory and on the notion of Markov-parameters for linear switched systems.

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. An erratum is appended at the end of the tex., The main goal of this submission is to add an erratum to the old paper published in 2000

B -convexity is defined as a suitable Peano-Kuratowski limit of linear convexities. An alternative idempotent convex structure called inverse B -convexity was recently proposed in the literature. This paper continues and extends some investigation started in these papers. In particular we focus on the Ky-Fan inequality and prove the existence of a Nash equilibrium for inverse B -convex games. This we do by considering a suitable “harmonic” topological structure which allows to establish a KKM theorem as well as some important related properties. Among other things a coincidence theorem is established. The paper also establishes fixed point results and Nash equilibriums properties in the case where two different convex topological structures are merged. It follows that one can consider a large class of games where the players may optimize their payoff subject to different forms of convexity. Among other things an inverse B -convex version of the Debreu-Gale-Nikaido theorem is proposed.

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.

In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature.

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned nonlinearities are known as the Kerr law, power, dual power law, triple power law, quadratic–cubic law and anti-cubic law. The dark, bright, singular and combinations of these solutions are retrieved using an efficient, simple integration scheme. These solutions suggest that this method is more simple, straightforward and reliable compared to existing methods in the literature. The novelty of this paper is that the perturbed nonlinear Schrödinger equation is investigated in different nonlinear media using a novel derivative operator. Furthermore, the numerical simulation for certain solutions is also presented.

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

We study the properties of electronic circuits after linearization around a fixed operating point in the context of closed-loop stability analysis. When distributed elements, like transmission lines, are present in the circuit it is known that unstable circuits can be created without poles in the complex right half-plane. This undermines existing closed-loop stability analysis techniques that determine stability by looking for right half-plane poles. We observed that the problematic circuits rely on unrealistic elements with an infinite bandwidth. In this paper, we therefore define a class of realistic linearized components and show that a circuit composed of realistic elements is only unstable with poles in the complex right half-plane. Furthermore, we show that the amount of right half-plane poles in a realistic circuit is finite, even when distributed elements are present. In the second part of the paper, we provide examples of component models that are realistic and show that the class includes many existing models, including ones for passive devices, active devices and transmission lines.

This article uses fractional calculus to create novel links between the well-known Mittag-Leffler functions of one, two, three, and four parameters. Hence, this paper studies several new analytical properties using fractional integration and differentiation for the Mittag-Leffler function formulated by confluent hypergeometric functions. We construct a four-parameter integral expression in terms of one-parameter. The paper explains the significance and applications of each of the four Mittag-Leffler functions, with the goal of using our findings to make analyzing specific kinds of experimental results considerably simpler.

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

In this paper, we determine the wave front sets of solutions to Schrodinger equations of a harmonic oscillator with sub-quadratic perturbation by using the representation of the Schrodinger evolution operator of a harmonic oscillator introduced in Kato, Kobayashi and Ito (2012) via the wave packet transform. In the previous paper (2011, 2014), the authors have studied the wave front sets for Schrodinger equations with a free particle and a harmonic oscillator and Schrodinger equations with sub-quadratic potential.

In this paper, we consider the model of the Landau-Lifshitz-Maxwell system coupled with spin accumulation. This system consists of the Landau-Lifshitz equation, Maxwell equation describing the magnetic field and a quasi-linear parabolic equation describing the process of spin accumulation. This paper establishes the existence of weak solutions ( M , S , H , E ) to the coupled system in three dimensions and shows the magnetization vector field M is Holder continuous except a closed set Σ, which has locally finite 3-dimensional parabolic Hausdorff measure. Finally, we deduce the Holder continuity of ∇M away from Σ in two special cases.

The first part of the paper concerns with infinite symmetric block Jacobi matrices J with p × p -matrix entries. We present new conditions for general block Jacobi matrices to be self-adjoint and have discrete spectrum. In the second part of the paper a special classes of block Jacobi matrices J X , α are investigated. Our approach here substantially relies on a close connection between Jacobi matrices J X , α from this class and symmetric 2 p × 2 p Dirac operators D X , α with point interactions in L 2 ( ( a , b ) ; C 2 p ) established in our previous papers. In particular, their deficiency indices are related by n ± ( D X , α ) = n ± ( J X , α ) and under a simple additional assumption they are discrete only simultaneously. For block Jacobi matrices of this class we present several conditions ensuring equality n ± ( J X , α ) = k with any k ≤ p . It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality n ± ( J X , α ) = p for J X , α it is first established for Dirac operator D X , α . We also find several conditions for matrix Schrodinger and Dirac operators with point interactions on finite or infinite intervals to have discrete spectrum.

This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the solution to the sNLS equation. In the paper we first carry out a detailed analysis of the properties of the solution which lays down a theoretical foundation and guidance for numerical analysis, we then present a family of three-parameters fully discrete finite element methods which differ mainly in their time discretizations and contains many well-known schemes (such as the explicit and implicit Euler schemes and the Crank-Nicolson scheme) with different combinations of time discetization strategies. The prototypical \begin{document}$ \theta $\end{document}-schemes are analyzed in detail and various stability properties are established for its numerical solution. An extensive numerical study and performance comparison are also presented for the proposed fully discrete finite element schemes.

This paper continues the development of regularity results for quasilinear elliptic equations − div ( A ( x , ∇ u ) ) = μ in Ω , and u = 0 on ∂ Ω , in Lorentz and Lorentz–Morrey spaces, where Ω ⊂ R n ( n ≥ 2 ); A is a monotone Caratheodory vector valued operator acting between W 0 1 , p ( Ω ) and its dual W − 1 , p ′ ( Ω ) ; and μ is a datum in some Lebesgue space L m ( Ω ) , for m p ′ . It emphasizes that in this paper, we restrict our study to the case of ‘very singular’ when 1 p ≤ 3 n − 2 2 n − 1 , and under mild assumption that the p -capacity uniform thickness condition is imposed on the complement of domain Ω . There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the ‘very singular’ case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces.

In this paper, the local stabilization problem of continuous-time nonhomogeneous Markovian jump fuzzy systems with incomplete transition descriptions is investigated using a mode-dependent nonquadratic Lyapunov function. Different from other existing studies, this paper proposes a method that can improve the feasibility of local stabilization conditions (i) by not assuming the maximum allowable bound of the time-derivative of the fuzzy basis functions, and (ii) by utilizing hidden information of the partial derivative of the fuzzy weighting functions with respect to the state variables. Furthermore, since the problem of estimating the attraction region is addressed with consideration of energy-bounded disturbances, the proposed method can be directly applied to the H ∞ control problem without any other assumptions on the magnitude of disturbances.

We investigate the density of compactly supported smooth functions in the Sobolev space W k , p on complete Riemannian manifolds. In the first part of the paper, we extend to the full range p ∈ [ 1 , 2 ] the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when k = 2 ) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order k − 3 (when k > 2 ). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every n ≥ 2 and p > 2 we construct a complete n -dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in W k , p does not hold for any k ≥ 2 . We also deduce the existence of a counterexample to the validity of the Calderon–Zygmund inequality for p > 2 when Sec ≥ 0 , and in the compact setting we show the impossibility to build a Calderon–Zygmund theory for p > 2 with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.

The paper is concerned with the study of the limit behaviour of the sequences of the positive linear functionals and operators associated with integrated generalized means defined with respect to a given probability Borel measure in the framework of Borel convex subsets of a Hilbert space. The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space. Several applications are shown in the special cases of bounded and unbounded real intervals which involve the most common integrated means. Furthermore, some consequences concerning the convergence in distribution, and hence stochastic, of generalized means of vector-valued random variables are also presented. Finally the paper ends with an application related to the so-called box integral problem which refers to the problem to evaluate the limit behaviour as n → ∞ of the average distance between two points of [ 0 , 1 ] n randomly chosen according to a given distribution on [ 0 , 1 ] n .

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

In this paper, we study the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the Monge–Ampere equation with the lower obstacle function, which arises in the prescribed Gauss curvature with an obstacle. The main feature of this paper is that we consider the obstacle problem for Monge–Ampere operator, which is a log-concave operator, with the lower obstacle function. Generally, the problem for a convex operator with a lower obstacle or a concave operator with an upper obstacle is considered due to the definition of the solutions and the classification of the global solutions. In this paper, difficulties caused by the incongruousness of the operator and the location of the obstacle are considered in many parts such as the penalization problem, classification of the global solutions, and the directional monotonicity.

In this paper, we prove that the Schrodinger map flows from R d with d ≥ 3 to compact Kahler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper [21] where the energy critical case d = 2 was solved. In the first part of this paper, for heat flows from R d ( d ≥ 3 ) to Riemannian manifolds with small data in critical Sobolev spaces, we prove the decay estimates of moving frame dependent quantities in the caloric gauge setting, which is of independent interest and may be applied to other problems. In the second part, with a key bootstrap-iteration scheme in our previous work [21] , we apply these decay estimates to the study of Schrodinger map flows by choosing caloric gauge. This work with our previous work solves the open problem raised by Tataru.

In this present paper, first, we investigate a new generalized Pochhammer's symbol and its various properties in terms of a new symbol $(s; k)$, where $s; k > 0$. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of $(s; k)$. Also, we define a new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of $(s; k)$. We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results and properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature.

In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H1 errors can reach almost the same convergence rate of the L∞ and L2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM.

To compute the spatially distributed dielectric constant from the backscattering computationally simulated ane experimentally collected data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve this inverse problem, we establish a new version of the Carleman estimate and then employ this estimate to construct a cost functional, which is strictly convex on a convex bounded set of an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and it is the central analytical result of this paper. Minimizing this cost functional by the gradient descent method, we obtain the desired numerical solution to the coefficient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer starting from an arbitrary point of that bounded set. We also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods, which are based on the optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring a good initial guess. Results of numerical studies of both computationally simulated and experimentally collected data are presented.

In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

In this paper we discuss the model of fractional oscillator where the inertial and restoring force terms maintains their usual expression but the damping term involves a fractional derivative of Caputo type, the so called fractional Kelvin-Voigt oscillator. The transient solution of this model is given in terms of the so called bivariate Mittag-Leffler function, while the steady-state solution in response to a sinusoidal force involves a 4-variate Mittag-Leffler function. We give numerical examples comparing the solutions for different values of the order α of the fractional derivative (0

Micromechanical systems (MEMS) based on electrostatic actuators are commonly involved in driving and actuating high-speed micro-structures. For this to be efficient, these micro-actuators must produce sufficient actuating force in order to achieve the required large strokes for such operations (usually in the order of tens of microns). However, this larger amount of displacement will require high actuation electrostatic voltages (typically in the order of hundreds of volts) or larger actuator dimensions, resulting in a bulky and inefficient design. To overcome these challenges, this paper examines the possible use of the snap-through instability in an electrostatically actuated and initially curved (shallow arch) clamped-clamped microbeam actuator arrangement. An extended version of the variational Hamilton principle is applied to derive nonlinear equations governing the steady-state behavior of the structure under step DC load. These equations are solved by effectiveness meshless numerical Galerkin decomposition technique. A convergence study is performed to show the effectiveness and advantages of the applied numerical approach to the formulated nonlinear problem. The investigation inspects different initial curvature profiles (first three symmetric buckled modes: first, third, and fifth modes) for the shallow arch. It was reported that the presence of the snap-through instability, for all investigated initial profiles, in the structural behavior of the micro-actuator produces a non-trivial order of magnitude increase in its resultant displacement as compared to the classical parallel-plate actuator with identical geometrical dimensions and operating conditions. In addition, out of the assumed first three symmetric buckled modes to outline the initial curvature of the shallow arch microbeam, only the first and fifth modes showed a strong impact on the micro-actuator performances.

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇ ( − Δ ) s ρ for some − 1 s 0 , with s = − 1 corresponding to the classical Euler–Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler–Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler–Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler–Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler–Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

Burgers’ equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high precision numerical solution. Most existed research focused on one-dimensional and two-dimensional Burgers’ systems. Seldom research studied the three-dimensional Burgers’ problem. In this paper, two space–time methods based on the polynomial particular solutions (ST-MPPS) and the polynomial used as basis functions directly, have been proposed to solve three-dimensional Burgers’ equation at different final time with different Reynolds numbers. The numerical examples show that the greatest attraction of the ST-MPPS is its high precision and robustness. When using the space–time polynomial functions method, some interesting phenomenon appears with the Reynolds number increasing.

In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).

In this paper we first establish a new critical point theorem which may produce a relationship between the sign-changing critical point, the Morse index, the upper bound of the critical value, the value of the type inf k ≤ dim ⁡ Y ∞ Y ⊂ E , ⁡ sup Y ⁡ G and the number of nodal domains. An unified approach is introduced for these composite properties. As one application of the abstract theorem, we study the existence of infinitely many sign-changing bound states for the following nonlinear stationary Schrodinger equation: − Δ u + a ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 2 , p > 2 and p 2 N / ( N − 2 ) : = 2 ⁎ when N > 2 ; the potential a ( x ) ∈ C ( R N , R ) is a nonnegative function verifying suitable decay assumptions, but being not symmetric or periodic. We obtain the information on the precise estimates of the energies, the Morse indices and the number of nodal domains for these sign-changing bound states. We believe that the abstract theorem in this article will have more applications.

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

A reaction-diffusion model which is called the field-road model was introduced by Berestycki, Roquejoffre and Rossi [9] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a heterogeneous landscape and establish the existence of the asymptotic spreading speed c ⁎ as well as its coincidence with the minimal wave speed of pulsating fronts along the road. We start with a truncated problem with an imposed Dirichlet boundary condition. We prove the existence of spreading speed c R ⁎ which coincides with the minimal speed of pulsating fronts for the truncated problem in the direction of the road. The arguments combine the dynamical system method with PDE's approach. Finally, we turn back to the original problem in the half-plane via generalized principal eigenvalue approach as well as an asymptotic method.

In this paper, we prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators whose coefficients are signed measures. Our approach is probabilistic, which also gives a representation of the solution. The heat kernel estimates and the theory of additive functionals play a crucial role in our approach.

Total Lagrangian - Smoothed Particle Hydrodynamics (TL-SPH) is a classical algorithm to improve the tensile instability encountered in the conventional Eulerian kernel-based SPH method for solid mechanics. Meanwhile, the gradient correction usually is employed in TL-SPH to enhance the consistency of approximation, which would generate unavoidable time costs. Therefore, a simplified gradient correction under the initial configuration is derived. Aiming at different particle configurations, the corresponding selection modes are provided and verified. Then the simplified gradient correction is introduced into TL-SPH, which is the Improved TL-SPH (ITL-SPH) method. Firstly, a classical example – elastic rubber collision is simulated to verify ITL-SPH has the same ability as TL-SPH to improve tensile instability. Then, the convergence and high efficiency of ITL-SPH is demonstrated via the uniaxial tension of a plate. Moreover, the plate with square and triangle holes also is simulated to show the adaptation and stability of ITL-SPH for the extreme boundary. Under the Total Lagrangian frame, the treatment to model the crack is necessary. So, this paper proposes a damage model and combines it with the ITL-SPH method for crack initiation and propagation, which is verified by the plate with holes and the notched beam examples.

Spatial memory has been considered significant in animal movement modeling. In this paper, we formulate a two-species interaction model by incorporating both random walk and spatial memory-based walk in their movement. The spatial memory-based walk, described by a chemotactic-like term, is derived by a modified Fick's law involving a directed movement toward the gradient of the density distribution function at a past time. For the proposed model, local stability and bifurcations are studied at constant steady states. Unlike a classical reaction-diffusion equation, we show that the accumulation points of eigenvalues for the model will locate at a vertical line in the complex plane, which will make the model generate spatially inhomogeneous time-periodic patterns through Hopf bifurcation. As illustrations, we apply these results to competition and cooperative models with memory-based diffusion. For the competition model, it turns out that the outcomes are far more complicated than those of classic Lotka-Volterra reaction-diffusion models, due to the consideration of memory-based diffusion. In particular, the existence of periodic oscillations is proved under weak competition. Similar conclusions hold for the cooperative model.

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.