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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.

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R 3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in L q ( R 3 ) for 2 ≤ q ≤ ∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate ( 1 + t ) − 3 4 + 1 4 q which is faster than the rate ( 1 + t ) − 3 4 + 1 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate ( 1 + t ) − 3 4 + 3 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q which is slower than the rate ( 1 + t ) − 3 4 + 3 4 q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.

In (Euzebio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincare disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

We prove unexpected decay of the L2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps., Comment: 21 pages

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

This work proposes an approach to using the non-cooperative game theory for solving finite element problems. For this, the concept of generalized Nash equilibrium is applied to finite elements implying that each element is treated as a non-cooperative “player” in a larger finite element “game”. The aim of the approach is for all players to reach a Nash equilibrium ensuring that the entire discretization is at a minimum with respect to the decision variables considered. The approach is numerically demonstrated by investigating a nonlinear elasticity problem formulated as a finite element game. It is shown that the approach matches analytical solutions in linear elasticity and is convergent to a prescribed precision for two-player nonlinear problems.

A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with spatial inhomogeneity coefficient K ( x ) behaves like | x | − b for 0 b min ⁡ { N 2 , 4 } . We show the local well-posedness in the whole H s -subcritical case, with 0 s ≤ 2 . The difficulties of this problem come from the singularity of K ( x ) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.

In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.

In this paper, we introduce a multi-domain Muntz-polynomial spectral collocation method with graded meshes for solving second kind Volterra integral equations with a weakly singular kernel. This method is particularly suitable for problems whose solutions contain non-integer exponent factors. We carry out a rigorous error analysis of hp-version in the L ∞ - and weighted L 2 -norms. Several numerical examples are presented to demonstrate the efficiency and accuracy of the method.

The current article is to study the solvability of Nirenberg problem on S 2 through the so-called Gaussian curvature flow. We aim to propose a unified method to treat the problem for candidate functions without sign restriction and non-degenerate assumption. As a first step, we reproduce the following statement: suppose the critical points of a smooth function f with positive critical values are non-degenerate. Then the required solution exists, if the difference between the number of the local maximum points with positive values and the number of the saddle points with positive critical values as well as negative Laplace is not equal to 1. This statement has been proved for nearly thirty years through different methods.

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

In this paper, we propose approximate solvers for two-phase fluids with general equations of state (EOS) in high dimension. The standard finite volume scheme is used for each fluid away from material interface. The two-phase interfaces are captured by the level set method coupled with a multi-resolution algorithm. For the Riemann solver of two-phase fluids with general equations of state, we construct an iterative approximation method by the solver for binomial equations of state. The velocity of the interface and the interface exchange fluxes are obtained precisely. With the help of the adaptive multi-resolution algorithms, we extend the method to three space dimensions conveniently. Numerical examples are carried out to demonstrate the strength and robustness of this method.

In this paper, an adaptive element free Galerkin (EFG) method is presented to solve Poisson equation. In general, element free Galerkin method using moving least square (MLS) approximation needs a background mesh for integration. With the arbitrary polygonal influence domain technique, the shape function of MLS has almost interpolation property, and the Gaussian quadrature points in the background integration element only contribute to the vertices of that element, which enables us to compute the residual based on the background integration element just like the finite element method (FEM). The adaptive procedure based on triangular or tetrahedral background integration elements is then developed for EFG method, in which the residual-based a posteriori error estimation of FEM is used. Numerical examples are provided to illustrate the efficiency of the proposed adaptive EFG method.

By employing the kernel functions with the form of piecewise polynomials in the Sobolev reproducing kernel Hilbert spaces (RKHSs), a globally superconvergent numerical technique is proposed to solve the second kind linear integral equations of Fredholm type. This method has an order of global convergence O ( h 4 ) and O ( h 6 ) based on the kernel functions in the Sobolev RKHSs H 1 and H 2 , respectively. Three linear Fredholm integral equations, one Volterra-Fredholm integral equation and one nonlinear Fredholm integral equation are numerically solved by the present approach to verify the superconvergence and effectiveness.

As a mesoscopic approach, the lattice Boltzmann method has achieved considerable success in simulating fluid flows and associated transport phenomena. The calculation, however, suffers from a massive amount of computing resources. A predictive model, to reduce the computing cost and accelerate the calculations, is proposed in this work. By employing an artificial neural network, composed of convolution layers and convolution long short-term memory layers, the model is an equivalent substitution of multiple time steps. A physical informed training loss function is introduced to improve the model predictive accuracy; and for the two-dimensional driven cavity problem, the mean square error of the prediction is less than 1.5 × 10 − 6 . For non-stationary flow, a time-dependent computing structure based on the current model is established. Nine iterative model calculations are performed consecutively for a two-dimensional driven cavity model, and the results are validated by comparing with the original (serial) lattice Boltzmann algorithm. Generally, in the case of training Reynolds number, for velocity and speed, the mean and the maximum absolute errors are lower than 0.012 and 0.12. Similarly, in the generalizing case, the mean and the maximum absolute errors are lower than 0.017 and 0.012. Besides, the current model's efficiency is about 15 times higher than that of the original lattice Boltzmann method.

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.

In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂ u ∂ t = − v and ∂ ϕ ∂ t = ∂ 2 ψ ∂ x 2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O ( Δ t 2 + h 2 ) and O ( Δ t 2 + h 4 ) in L ∞ - and L 2 -norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes.

Making a rank-one modification on the classical BFGS (Broyden–Fletcher–Goldfarb–Shanno) updating formula, we develop a class of augmented BFGS methods. The suggested formula can be considered as a hybridization of the basic BFGS updating formula for Hessian with an additional rank-one term embedded to guarantee a general modified secant equation. By using the well-known Sherman–Morrison formula, the inverse of a memoryless version of the given updating formula is computed to be applied for solving large-scale problems. Convergence analysis is concisely carried out as well. At last, the practical merits of the method are investigated by numerical tests on a set of CUTEr problems as well as the well-known compressed sensing problem. Results show the computational efficiency of the given method.

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

We study a filtered Lie splitting scheme for the cubic nonlinear Schrödinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in H s H^s with 0 > s > 1 0>s>1 overcoming the standard stability restriction to smooth Sobolev spaces with index s > 1 / 2 s>1/2 . More precisely, we prove convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 at this level of regularity.

PurposeThe purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.FindingsThe IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.Originality/valueThe IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

This work presents a numerical study of the simulation of bi-phasic flow for a bubble rising in the viscous liquid. A 2D model is employed via Smoothed Particle Hydrodynamics (SPH). The liquid/bubble capillary interaction must be taken into account, implementing adequate properties of density, viscosity, and surface tension at the interface. Likewise, the simulation results are evaluated with numerical and experimental measurements from the literature in terms of terminal bubble morphology. These results are characterized by the variation of the dynamic parameters, such as the Reynold number (Re ≤ 50 ), Eotvos number (Eo ≤ 200), density ratio ( Φ ≤ 1000 ), SPH resolution size ( Δ x ≤ 1 / 500 ), and the ratio between cavity width and the bubble diameter ( β ≤ 8 ). Consequently, it is explored the independence of sidewalls on a single bubble rising, these results display different shapes and vortex regimes between the circular, ellipsoidal, and cap morphologies. Furthermore, The behavior of the bubble deformations during the high or low influence of sidewall is shown throughout cases with lateral instabilities, such as tails, break-up, and satellite bubble formation. Finally, It is demonstrated that the bi-phase SPH algorithm is robust enough, providing qualitative and quantitative information about global typical characteristics of a single rising bubble.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities (FNLS) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N p 2 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 p 1 p 2 and 4 s N ≤ p 2 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 p 2 4 s N − 2 s , or a = 1 , 0 p 1 4 s N p 2 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

In this paper we consider a fluid-fluid system coupled with a friction-type interface condition which is described by a jump of the velocities along the tangential interface. Based on domain decomposition, we propose a Schwarz type iterative method to decouple the different physical process in each subdomain occupied by a single fluid, which allows solving in each subdomain only the physical process described by a Stokes problem. Using energy estimate, we prove that the algorithm converges for any traction coefficient. A more detailed analysis based on Fourier analysis shows that the convergence rate depends on both the fluids' properties, the traction coefficient and the time stepsize, but is independent of the spatial mesh. We finally use numerical examples to illustrate the theoretical results.

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallee Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In previous works this has already been proved under different sufficient conditions. Here, we complete the study by stating also the necessary conditions to get it. Several numerical experiments are also given to test the theoretical results and make comparisons to Lagrange interpolation at the same nodes.

We study the uniform weighted resolvent estimates of Schrodinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on R 2 , we prove some new uniform weighted resolvent estimates for this 2D Schrodinger operator and, as applications, we show local smoothing estimates for the Schrodinger equation in this setting.

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation x ′ ( t ) + c ( t ) x ( τ ( t ) ) = 0 , t ≥ 0 , where c is locally integrable of any sign, τ ( t ) ≤ t is Lebesgue measurable, lim t → ∞ ⁡ τ ( t ) = ∞ , we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the 3/2-stability criterion to the case of measurable parameters, improving 1 + 1 / e to the sharp 3/2 constant.

In this paper, we study the Morse theory on uniform random attractors for non-autonomous random dynamical systems. To handle the non-invariance of uniform random attractors, we construct the Morse sets of uniform random attractor as the projections of the Morse sets on pullback attractor of the skew product semiflow. First, we obtain the Morse decomposition of uniform random attractors in probability. Second, we describe a decaying energy level on such attractor by Lyapunov function with probability one. Furthermore, we study the stability of Morse decompositions of deterministic uniform attractor under a small random disturbance. Finally, we discuss and collect the results on the Morse decomposition under random setting.

This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo. The 2-dim Gauss-Legendre quadrature rule together with the Lagrange multipliers method are utilized to find the solutions of the given 2DNVOFOCP. The convergence analysis of the presented method is investigated. The examined numerical examples manifest highly accurate results.

In this research, the localized singular boundary method (LSBM) is proposed to solve the Laplace and Helmholtz equations in 2D arbitrary domains. In the traditional SBM, the resultant matrix system is a dense matrix, and it is unsuited for solving the large-scale problems. As a localized domain-type meshless method, a local subdomain for every node can be composed by its own and several nearest nodes. To each of the subdomains, the SBM formulation is applied to derive an implicit expression of the variable at each node in conjunction with the moving least-square approximation. To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained. Thus, the numerical solutions at all nodes can be achieved by solving it. Compared with the traditional SBM, the LSBM involves only the origin intensity factor on a circular boundary associated with Dirichlet boundary conditions. It can also effectively avoid the boundary layer effect in the conventional SBM. Furthermore, the proposed LSBM requires less memory storage and computational cost due to the sparse and banded matrix system. Several numerical examples are tested to verify the accuracy and stability of the proposed LSBM.

Spherulites are growth patterns of average spherical form which may occur in the polycrystallization of binary mixtures due to misoriented angles at low grain boundaries. The dynamic growth of spherulites can be described by a phase field model where the underlying free energy depends on two phase field variables, namely the local degree of crystallinity and the orientation angle. For the solution of the phase field model we suggest a splitting scheme based on an implicit discretization in time which decouples the model and at each time step requires the successive solution of an evolutionary inclusion in the orientation angle and an evolutionary equation in the local degree of crystallinity. The discretization in space is done by piecewise linear Lagrangian finite elements. The fully discretized splitting scheme amounts to the solution of two systems of nonlinear algebraic equations. For the numerical solution we suggest a predictor-corrector continuation method with the discrete time as a parameter featuring constant continuation as a predictor and a semismooth Newton method for the first system and the classical Newton method for the second system as a corrector. This allows an adaptive choice of the time steps. Numerical results are given for the formation of a Category 1 spherulite.

We find an explicit form of solution of a convolution type integro-differential equation of the first order (1) ∂ Q ( t , z ) ∂ z = α ( t , z ) − ∫ 0 t β ( t − u ) Q ( u , z ) d u , ( t , z ) ∈ ( 0 , T ) × ( 0 , T ) , where T > 0 , and α , β are two given locally integrable functions and Q satisfies a boundary condition Q ( t , 0 ) = γ ( t ) for a locally integrable function γ. To achieve this goal we introduce the notion of a biconvolution algebra of locally integrable functions on R + 2 . We investigate the properties of the biconvolution algebra and study the Volterra integral equations of the second kind associated with the biconvolution operation. Finally, we present an explicit form of solution of integro-differential equation associated with a linear operator (2) Λ n = ∂ n + 1 ∂ z ∂ t n + ∑ i = 0 n − 1 λ i ∂ i ∂ t i , λ i ∈ R , i ≤ n − 1 , n ∈ N .

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition., Comment: arXiv admin note: text overlap with arXiv:1209.5458 by other authors

We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP-convergence, i.e. convergence in the sense of the energy-dissipation principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics. Deutsche Forschungsgemeinschafthttps://doi.org/10.13039/501100001659

We investigate the stability of solutions of perturbed set differential equations with causal operators in regard to their corresponding unperturbed ones considering the difference in initial conditions (time and position) by utilizing Lyapunov functions and Lyapunov functionals.

By the classical Li–Yau inequality, an immersion of a closed surface in ℝ n {\mathbb{R}^{n}} with Willmore energy below 8 ⁢ π {8\pi} has to be embedded. We discuss analogous results for curves in ℝ 2 {\mathbb{R}^{2}} , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

In this paper, we study some properties of multi-solitons for the non-linear Schrodinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in pape...

In the current study, an Immersed Boundary Method for simulating cavitating flows with complex or moving boundaries is presented, which follows the discrete direct forcing approach. Although the Immersed Boundary Methods are widely used in various applications of single phase, multiphase and particulate flows, either incompressible or compressible, and numerous alternative formulations exist, to the best of the authors' knowledge, a handful of computational works employ such methodologies on cavitating flows. The herein proposed method, following the works of the author's group1,2,3, tries to fill this gap and to solidify the development of a computational tool of a simple formulation capable to tackle complex numerical problems of cavitation modelling. The method aims to be used in a wide range of applications of industrial interest and treat flows of engineering scales. Therefore, a validation of the method is performed by numerous benchmark test-cases, of progressively increasing complexity, from incompressible low Reynolds number to compressible and highly turbulent cavitating flows.

In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals., 14 pages. arXiv admin note: text overlap with arXiv:1811.02845

We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

We introduce a multigrid method that is homogeneous in the sense that it uses the same hybridizable discontinuous Galerkin (HDG) discretization scheme for Poisson’s equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.