Browsing Mathematics by Title
Now showing items 4160 of 195

Detailed error analysis for a fractional adams method with graded meshesWe consider a fractional Adams method for solving the nonlinear fractional differential equation $\, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0$, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil 1$. Here $\alpha$ may be an arbitrary positive number and $ \lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption $\, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T]$, Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{2})$ if $\alpha > 1$ and $O(N^{1\alpha})$ if $\alpha \leq 1$. They also showed that if $\, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T]$, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $ 0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if $\, ^{C}_{0}D^{\alpha}_{t} y$ behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Determining control parameters for dendritic cellcytotoxic T lymphocyte interactionDendritic cells (DC) are potent immunostimulatory cells facilitating antigen transport to lymphoid tissues and providing efficient stimulation of T cells. A series of experimental studies in mice demonstrated that cytotoxic T lymphocytes (CTL) can be efficiently induced by adoptive transfer of antigenpresenting DC. However, the success of DCbased immunotherapeutic treatment of human cancer, for example, is still limited because the details of the regulation and kinetics of the DCCTL interaction are not yet completely understood. Using a combination of experimental mouse studies, mathematical modeling, and nonlinear parameter estimation, we analyzed the population dynamics of DCinduced CTL responses. The model integrates a predatorpreytype interaction of DC and CTL with the nonlinear compartmental dynamics of T cells. We found that T cell receptor avidity, the halflife of DC, and the rate of CTLmediated DCelimination are the major control parameters for optimal DCinduced CTL responses. For induction of high avidity CTL, the number of adoptively transferred DC was of minor importance once a minimal threshold of approximately 200 cells per spleen had been reached. Taken together, our study indicates that the availability of high avidity T cells in the recipient in combination with the optimal application regimen is of prime importance for successful DCbased immunotherapy.

Developing A Highperformance Liquid Chromatography Method for Simultaneous Determination of Loratadine and its Metabolite Desloratadine in Human Plasma.Allergic diseases are considered among the major burdons of public health with increased prevalence globally. Histamine H1receptor antagonists are the foremost commonly used drugs in the treatment of allergic disorders. Our target drug is one of this class, loratadine and its biometabolite desloratadine which is also a non sedating H1 receptor antagonist with antihistaminic action of 2.5 to 4 times greater than loratadine. To develop and validate a novel isocratic reversedphase high performance liquid chromatography (RPHPLC) method for rapid and simultaneous separation and determination of loratadine and its metabolite, desloratadine in human plasma. The drug extraction method from plasma was based on protein precipitation technique. The separation was carried out on a Thermo Scientific BDS Hypersil C18 column (5µm, 250 x 4.60 mm) using a mobile phase of MeOH : 0.025M KH2PO4 adjusted to pH 3.50 using orthophosphoric acid (85 : 15, v/v) at ambient temperature. The flow rate was maintained at 1 mL/min and maximum absorption was measured using PDA detector at 248 nm. The retention times of loratadine and desloratadine in plasma samples were recorded to be 4.10 and 5.08 minutes respectively, indicating a short analysis time. Limits of detection were found to be 1.80 and 1.97 ng/mL for loratadine and desloratadine, respectively, showing a high degree of method sensitivity. The method was then validated according to FDA guidelines for the determination of the two analytes in human plasma. The results obtained indicate that the proposed method is rapid, sensitive in the nanogram range, accurate, selective, robust and reproducible compared to other reported methods. [Abstract copyright: Copyright© Bentham Science Publishers; For any queries, please email at epub@benthamscience.net.]

The diffusiondriven instability and complexity for a singlehanded discrete Fisher equationFor a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusiondriven instability/Turing instability for a singlehanded discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2periodic patterns have been observed. Motivated by these pattern formations, the existence of 2periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.

Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equationsIn this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in $t$ of degree at most $q1, q \geq 1$, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.

A discrete mutualism model: analysis and exploration of a financial applicationWe perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system’s positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach.

Distributed order equations as boundary value problemsThis preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputotype conditions are given away from the origin.

DOMestic Energy Systems and Technologies InCubator (DOMESTIC) and indoor air quality of the built environmentOral presentation at RMetS Students and Early Career Scientists Conference 2020 on research project DOMESTIC (DOMestic Energy Systems and Technologies InCubator), which aims to build a facility for the demonstration of domestic technologies and design methodologies (i.e. air quality, energy efficiency).

Double Bordered Constructions of SelfDual Codes from Group Rings over Frobenius RingsIn this work, we describe a double bordered construction of selfdual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F2 + uF2 and F4 + uF4. We demonstrate the importance of this new construction by finding many new binary selfdual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables

A DufortFrankel Difference Scheme for TwoDimensional SineGordon EquationA standard CrankNicolson finitedifference scheme and a DufortFrankel finitedifference scheme are introduced to solve twodimensional damped and undamped sineGordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictorcorrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Dynamics of shadow system of a singular GiererMeinhardt system on an evolving domainThe main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular GiererMeinhardt model on an isotropically evolving domain. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the GiererMeinhardt model is reduced to a single though nonlocal equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blowup results for this nonlocal equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusiondriven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusiondriven blowup, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of globalintime solutions towards nonconstant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

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 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.