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Edgebased nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic fluxcorrection schemesFor the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edgebased nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.

Entropydriven cell decisionmaking predicts "fluidtosolid" transition in multicellular systemsCellular decision making allows cells to assume functionally different phenotypes in response to microenvironmental cues, with or without genetic change. It is an open question, how individual cell decisions influence the dynamics at the tissue level. Here, we study spatiotemporal pattern formation in a population of cells exhibiting phenotypic plasticity, which is a paradigm of cell decision making. We focus on the migration/resting and the migration/proliferation plasticity which underly the epithelialmesenchymal transition (EMT) and the go or grow dichotomy. We assume that cells change their phenotype in order to minimize their microenvironmental entropy following the LEUP (Least microEnvironmental Uncertainty Principle) hypothesis. In turn, we study the impact of the LEUPdriven migration/resting and migration/proliferation plasticity on the corresponding multicellular spatiotemporal dynamics with a stochastic cellbased mathematical model for the spatiotemporal dynamics of the cell phenotypes. In the case of the go or rest plasticity, a corresponding meanfield approximation allows to identify a bistable switching mechanism between a diffusive (fluid) and an epithelial (solid) tissue phase which depends on the sensitivity of the phenotypes to the environment. For the go or grow plasticity, we show the possibility of Turing pattern formation for the "solid" tissue phase and its relation with the parameters of the LEUPdriven cell decisions.

Error estimates of highorder numerical methods for solving time fractional partial differential equationsError estimates of some highorder numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a highorder numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3 \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one and twodimensional cases are given to show that the numerical results are consistent with the theoretical results.

Error estimates of a continuous Galerkin time stepping method for subdiffusion problemA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and Ltype methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

Error estimates of a high order numerical method for solving linear fractional differential equationsIn this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a firstdegree compound quadrature formula was used to approximate the Hadamard finitepart integral and the convergence order of the proposed numerical method is O(∆t 2−α ), 0 < α < 1, where α is the order of the fractional derivative and ∆t is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a seconddegree compound quadrature formula was used to approximate the Hadamard finitepart integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.

Existence and regularity of solution for a Stochastic CahnHilliard / AllenCahn equation with unbounded noise diffusionThe CahnHilliard/AllenCahn equation with noise is a simpliﬁed mean ﬁeld model of stochastic microscopic dynamics associated with adsorption and desorptionspin ﬂip mechanisms in the context of surface processes. For such an equation we consider a multiplicative spacetime white noise with diﬀusion coeﬃcient of linear growth. Applying technics from semigroup theory, we prove local existence and uniqueness in dimensions d = 1,2,3. Moreover, when the diﬀusion coeﬃcient satisﬁes a sublinear growth condition of order α bounded by 1 3, which is the inverse of the polynomial order of the nonlinearity used, we prove for d = 1 global existence of solution. Path regularity of stochastic solution, depending on that of the initial condition, is obtained a.s. up to the explosion time. The path regularity is identical to that proved for the stochastic CahnHilliard equation in the case of bounded noise diﬀusion. Our results are also valid for the stochastic CahnHilliard equation with unbounded noise diﬀusion, for which previous results were established only in the framework of a bounded diﬀusion coeﬃcient. As expected from the theory of parabolic operators in the sense of Petrovsk˘ıı, the biLaplacian operator seems to be dominant in the combined model.

Existence of time periodic solutions for a class of nonresonant discrete wave equationsIn this paper, a class of discrete wave equations with Dirichlet boundary conditions are obtained by using the centerdifference method. For any positive integers m and T, when the existence of time mTperiodic solutions is considered, a strongly indefinite discrete system needs to be established. By using a variant generalized weak linking theorem, a nonresonant superlinear (or superquadratic) result is obtained and the AmbrosettiRabinowitz condition is improved. Such a method cannot be used for the corresponding continuous wave equations or the continuous Hamiltonian systems; however, it is valid for some general discrete Hamiltonian systems.

Existence theory for a class of evolutionary equations with timelag, studied via integral equation formulationsIn discussions of certain neutral delay differential equations in Hale’s form, the relationship of the original problem with an integrated form (an integral equation) proves to be helpful in considering existence and uniqueness of a solution and sensitivity to initial data. Although the theory is generally based on the assumption that a solution is continuous, natural solutions of neutral delay differential equations of the type considered may be discontinuous. This difficulty is resolved by relating the discontinuous solution to its restrictions on appropriate (halfopen) subintervals where they are continuous and can be regarded as solutions of related integral equations. Existence and unicity theories then follow. Furthermore, it is seen that the discontinuous solutions can be regarded as solutions in the sense of Caratheodory (where this concept is adapted from the theory of ordinary differential equations, recast as integral equations).

Explosive solutions of a stochastic nonlocal reaction–diffusion equation arising in shear band formationIn this paper, we consider a nonlocal stochastic parabolic equation which actually serves as a mathematical model describing the adiabatic shearbanding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finitetime explosion for this nonlocal SPDE, corresponding to shearbanding formation, occurs. For that purpose some results related to the maximum principle for this nonlocal SPDE are derived and afterwards the Kaplan's eigenfunction method is employed.

Exponential stability in pth mean of solutions, and of convergent Eulertype solutions, of stochastic delay differential equationsThis article carries out an analysis which proceeds as follows: showing that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for pth mean stability of a solution; producing a discrete analogue of the Halanaytype theory, that permits the development of a pth mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving meansquare stability conditions for solutions and numerical solutions of a constantcoefficient linear test equation.

Extending an Established Isomorphism between Group Rings and a Subring of the n × n MatricesIn this work, we extend an established isomorphism between group rings and a subring of the n × n matrices. This extension allows us to construct more complex matrices over the ring R. We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

Finite Difference Method for TwoSided SpaceFractional Partial Differential EquationsFinite difference methods for solving twosided spacefractional partial differential equations are studied. The spacefractional derivatives are the lefthanded and righthanded RiemannLiouville fractional derivatives which are expressed by using Hadamard finitepart integrals. The Hadamard finitepart integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the spacefractional derivative with convergence order O(Δx^(3−α )),10 , where Δt,Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

A finite element method for time fractional partial differential equationsThis article considers the finite element method for time fractional differential equations.

Finitetime blowup of a nonlocal stochastic parabolic problemThe main aim of the current work is the study of the conditions under which (finitetime) blowup of a nonlocal stochastic parabolic problem occurs. We first establish the existence and uniqueness of the localintime weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noiseinduced blowup. In the second part we first prove the $C^{1}$spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (nonlocal) drift term induced blowup result. In the last part of the paper, we present a method which provides an upper bound of the probability of (nonlocal) drift term induced blowup.

Fixed point theroms and their application  discrete Volterra applicationsThe existence of solutions of nonlinear discrete Volterra equations is established. We define discrete Volterra operators on normed spaces of infinite sequences of finitedimensional vectors, and present some of their basic properties (continuity, boundedness, and representation). The treatment relies upon the use of coordinate functions, and the existence results are obtained using fixed point theorems for discrete Volterra operators on infinitedimensional spaces based on fixed point theorems of Schauder, Rothe, and Altman, and Banach’s contraction mapping theorem, for finitedimensional spaces.

Fourier spectral methods for some linear stochastic spacefractional partial differential equationsFourier spectral methods for solving some linear stochastic spacefractional partial differential equations perturbed by spacetime white noises in onedimensional case are introduced and analyzed. The spacefractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the spacetime white noise by using piecewise constant functions and obtain the approximated stochastic spacefractional partial differential equations. The approximated stochastic spacefractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in $L^{2}$ norm are obtained. Numerical examples are given.

Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noisesFourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in onedimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The spacetime noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the $L_{2}$ norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given.

Fractional boundary value problems: Analysis and numerical methodsThis journal article discusses nonlinear boundary value problems.

Fractional pennes' bioheat equation: Theoretical and numerical studiesIn this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperaturedependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bio heat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

Gcodes over Formal Power Series Rings and Finite Chain RingsIn this work, we define $G$codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$code is again a $G$code in this setting. We study the projections and lifts of $G$codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$adic codes over $R_\infty$ to $\gamma$adic $G$codes over the same ring. We also study $G$codes over principal ideal rings.