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We study a null controllability problem for a convection-diffusion equation with the control contained in Fourier boundary condition to identify M parameters of pollution. The results are achieved by means of an observability inequality derived from new Carleman estimates for the boundary null controllability and the construction of a boundary sentinel to identify the parameters. [ABSTRACT FROM AUTHOR]

This research focuses on topology optimization of fluid-structure interaction (FSI) problems using the level set method. To couple the fluid and structure equations, the Arbitrary Lagrangian-Eulerian (ALE) description is employed within a monolithic formulation. The use of ALE in FSI problems, while eliminating numerical instabilities caused by the convective term, enhances the speed and accuracy of finite element solutions in fluid-structure interaction. Additionally, considering the fluid in the unsteady state allows for the interpretation of optimal topology at any given moment of the analysis. The objective function of the optimal topology design problem is to minimize the structural compliance in the dry state, subject to a fixed volume of the design domain. To determine the normal velocity in the reaction-diffusion equation (RDE), adjoint sensitivity analysis based on pointwise gradients is used. The results obtained from this approach, compared to other topology optimization methods in the literature, demonstrate higher accuracy and clearer definition of structural boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global regularity for the two-dimensional incompressible magneto-micropolar equations with almost Laplacian magnetic diffusion and Laplacian micro-rotational diffusion, but without kinematic dissipation. The key arguments are based on the maximal regularity property of the generalized heat operators and a combined quantity. For the two-dimensional incompressible magneto-micropolar equations with only Laplacian magnetic diffusion and Laplacian micro-rotational diffusion, we derive an improved regularity criterion which is less restrictive than the classical Beale-Kato-Majda regularity criterion. Consequently, our results improve and generalize the previous works. [ABSTRACT FROM AUTHOR]

We introduce a new inequality similar to the fractional Poincar$ \acute{e} $ inequality and obtain the continuous data assimilation and feedback control of fractional reaction-diffusion equations. The feedback control scheme has finite number of determining parameters. The continuous data assimilation is obtained based on finite-dimensional feedback controls. [ABSTRACT FROM AUTHOR]

We study the existence of solutions to the problem - Δ u + u p - M ∇ u q = 0 in Ω , (1) u = μ on ∂ Ω in a bounded domain Ω, where p > 1, 1 < q < 2, M > 0, μ is a nonnegative Radon measure in ∂Ω, and the associated problem with a boundary isolated singularity at a ∈ ∂Ω, - Δ u + u p - M ∇ u q = 0 in Ω , (2) u = 0 on ∂ Ω \ α. The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to (1) is obtained under a capacitary condition μ K ≤ c min cap 2 p , p ′ ∂ Ω , cap 2 - q q , q ′ ∂ Ω for\;all\;compacts\; K ⊂ ∂ Ω. Problem (2) depends on several critical exponents on p and q as well as the position of q with respect to 2 p p + 1 . [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) have gained significant attention in diverse domains, ranging from urban planning to pandemic management. Ensuring both accuracy and robustness in GNNs remains a challenge due to insufficient quality data that contains sufficient features. With sufficient training data where all spatiotemporal patterns are well-represented, existing GNN models can make reasonably accurate predictions. However, existing methods fail when the training data are drawn from different circumstances (e.g., traffic patterns on regular days) than test data (e.g., traffic patterns after a natural disaster). Such challenges are usually classified under domain generalization. In this work, we show that one way to address this challenge in the context of spatiotemporal prediction is by incorporating domain differential equations into graph convolutional networks (GCNs). We theoretically derive conditions where GCNs incorporating such domain differential equations are robust to mismatched training and testing data compared to baseline domain agnostic models. To support our theory, we propose two domain-differential-equationinformed networks: Reaction-Diffusion Graph Convolutional Network (RDGCN), which incorporates differential equations for traffic speed evolution, and the Susceptible-Infectious-Recovered Graph Convolutional Network (SIRGCN), which incorporates a disease propagation model. Both RDGCN and SIRGCN are based on reliable and interpretable domain differential equations that allow the models to generalize to unseen patterns. We experimentally show that RDGCN and SIRGCN are more robust with mismatched testing data than state-of-the-art deep learning methods. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two‐dimensional thin domain that presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios such as quasi‐periodic or almost‐periodic oscillations. We then prove that the family of solutions converges to the solution of a one‐dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasi‐periodic context. [ABSTRACT FROM AUTHOR]

This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain. [ABSTRACT FROM AUTHOR]

An approach to solvability of certain quasilinear parabolic equations is presented by approximating the quasilinear equation under consideration with a parameter family of semilinear problems with stronger linear fractional diffusion term. Defined on arbitrarily long time intervals, solutions to the original problem are found as a suitable limit of global solutions to those semilinear approximations. The method is applied to nonlinear parabolic Kirchhoff equation, quasilinear reaction–diffusion equation, and critical 2D surface quasi‐geostrophic equation associated with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

This work concerns with a novel computational approach to solve a two‐dimensional Caputo time‐fractional convection reaction–diffusion (CTFCRD) equation with the weak singularity at the initial time. In this method, the L1$$ L1 $$‐scheme on graded mesh is used to approximate the Caputo temporal derivative, while the spatial derivatives are approximated by using a compact finite difference method (CFDM) coupled with alternating direction implicit (ADI) scheme. We present a methodology to examine the optimal error estimates of the proposed scheme, in terms of L2$$ {L}&#x0005E;2 $$‐norm. Convergence and stability results are proved. It is shown that the suggested method yields an optimal order, that is, min(2‐ β$$ \beta $$, aβ$$ a\beta $$, 2 β$$ \beta $$+1) in time direction, where β$$ \beta $$ is the order of time fractional derivative and a$$ a $$ is the grading parameter and has a fourth‐order of convergence in space direction. Finally, two numerical examples are provided to verify the theoretical estimates and to illustrate the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen--Cahn equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka--Volterra equations for d ∈ { 1, 2, 3, 4} and the Brusselator for d ∈ { 1, 2, 3} . Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on It\^o's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new Lζ-coercivity/dissipativity conditions and the development of an Lp(Lq )- framework for systems of reaction-diffusion equations, which are needed when treating dimensions d ∈ { 2, 3} in the case of cubic or higher order nonlinearities. [ABSTRACT FROM AUTHOR]

Reaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field u(x,t)$u({x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of u$u$, which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity D(u)=(u−a)(u−b)$D(u) = (u - a)(u - b)$ that is negative for u∈(a,b)$u\in (a,b)$. We use a nonclassical symmetry to construct analytic receding time‐dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, and we convert them to single‐valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan‐like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the u=0$u = 0$ and u=1$u = 1$ constant solutions, and prove for certain a$a$ and b$b$ that receding traveling waves are spectrally stable. In addition, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well‐known equal‐area rule, but for nonsymmetric diffusivity it results in a different shock position. [ABSTRACT FROM AUTHOR]

We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology. [ABSTRACT FROM AUTHOR]

A one-dimensional space-time fractional reaction-diffusion problem is considered. We present a complete theory for the solution of the time-space fractional reaction-diffusion model, including existence and uniqueness in the case of using the spectral representation of the fractional Laplacian operator. An optimal error estimate is presented for the Galerkin spectral approximation of the problem under consideration. [ABSTRACT FROM AUTHOR]

We consider a diffusion equation on the real line with growth rates (reaction terms) being negative in a bounded unfavorable region and bistable on two sides. The equation can be used to model a species living in a habitat with a polluted or hunting zone but still tries to survive or even spread to the whole space. Using the zero number argument, we first show the general convergence to stationary solution for any nonnegative global solutions, and then we prove a spreading–transition–vanishing trichotomy result for the asymptotic behavior of global solutions. The key point is to find a quite special stationary solution to distinguish the spreading and transition solutions. Finally, we construct precise upper and lower solutions to estimate the asymptotic speed for the spreading solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of N 1 and N 2 increases for the three choices of and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables. [ABSTRACT FROM AUTHOR]

Biological models for cardiac regeneration and remodeling, along with the effects of cytokines or chemokines during the therapy with mesenchymal stem cells after a myocardial infarction, are of crucial importance for understanding the complex underlying mechanisms. This paper presents a mathematical model composed of three coupled partial differential equations that describes the dynamics of stem cells, nutrients and chemokines, highlighting the fundamental role of the chemokines during the myocardial tissue regeneration process. The system is solved numerically using mimetic difference operators and the MOLE library for MATLAB. The results show the tissue regeneration process in the necrotic part closest to the cell implantation area. [ABSTRACT FROM AUTHOR]

We obtain a contrast-structure type solution of a system of equations for the baretting effect that include a nonlinear singularly perturbed parabolic equation and an additional nonlocal integral relation. We prove the existence of the solution with an internal transition layer and construct the asymptotic approximation of this solution. We obtain estimates of the main physical model parameters, which coincide with experimental data and the estimates obtained previously by other methods. [ABSTRACT FROM AUTHOR]

The existence of stationary solutions of singularly perturbed systems of reaction–diffusion–advection equations is studied in the case of fast and slow reaction–diffusion–advection equations with nonlinearities containing the gradient of the squared sought function (KPZ nonlinearities). The asymptotic method of differential inequalities is used to prove the existence theorems. The boundary layer asymptotics of solutions are constructed in the case of Neumann and Dirichlet boundary conditions. The case of quasimonotone sources and systems without the quasimonotonicity requirement is also considered. [ABSTRACT FROM AUTHOR]

In this work, we examine a quadratic thin film equation with a constant negative absorption term. This equation extends a broad variety of the famous scalar reaction-diffusion equations appearing in nonlinear sciences and is derived from the estimations of lubrication theory to represent thin films of a Newtonian liquid dominated by surface tension effects. It is typically used to describe the behavior of light when it interacts with thin films, such as coatings on lenses or mirrors. The connection between thin film equations and optical quantum mechanics lies in the microscopic interactions between photons and the electrons in the thin film material. Employing the invariant subspace approach, we obtain explicit fractional exact solutions for the time-fractional case of the model containing the Riemann-Liouville derivative operator. Furthermore, we illustrate 3-D and 2-D plots of the obtained exact solutions for a better understanding of the physical phenomena. [ABSTRACT FROM AUTHOR]

Inspired by a recent study associating shifting temperature conditions with changes in the efficiency with which predators convert prey to offspring, we propose a predator-prey model of reaction-diffusion type to analyze the consequence of such effects on the population dynamics and spread of the predator species. In the model, the predator conversion efficiency is represented by a spatially heterogeneous function depending on the variable ξ = x - c1t for some given c1 > 0. Using the Hamilton-Jacobi approach, we provide explicit formulas for the spreading speed of the predator species. When the conversion function is monotone increasing, the spreading speed is determined in all cases and non-local pulling is possible. When the function is monotone decreasing, we provide formulas for the spreading speed when the rate of shift of the conversion function is sufficiently fast or slow. [ABSTRACT FROM AUTHOR]

We prove the existence of an optimal treatment combining chemotherapy and a lactate targeting drug for a mathematical model for high grade gliomas and lactate kinetics. A necessary condition for a treatment to be optimal is also devised, allowing to design numerically the proper therapy. The analytical results are validated by the numerical simulations; as clinical experience proves, the simulations show that the effect of chemotherapy is enhanced by the other drug whose sole administration is, in turn, almost irrelevant. [ABSTRACT FROM AUTHOR]

The paper deals with the well-posedness and long-time behavior of solutions for a class of viscous Camassa-Holm equations with fractional diffusion on bounded domains via the global attractor approach. We first prove the existence and uniqueness of weak solutions. Next, we point out the existence of a finite fractal dimensional global attractor and derive upper bounds for the fractal dimension of the global attractor. Finally, the number of determining modes is also studied here. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability analysis for a contaminant convection-reaction-diffusion model of the recovered fracturing fluid (RFFM, for short), which couples a nonlinear and non-smooth stationary incompressible Navier-Stokes equation with a multivalued frictional boundary condition, and a nonlinear reaction-diffusion equation with mixed Neumann boundary conditions. First, we introduce a family of perturbation problems corresponding to (RFFM), and present the variational formulation of perturbation problem which is a perturbation elliptic hemivariational inequality driven by a perturbation nonlinear variational equation. Then, the existence of solutions and the uniform bound of the solution set to the perturbation problem are obtained. Finally, it is established that, as the perturbation parameter tends to zero, the solution set of the perturbation problems converge to the solution set of (RFFM) in the sense of the Kuratowski upper limit. This shows that (RFFM) is stable with respect to the perturbation data. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations often arise in many real-world problems. The complexity of solving nonlinear systems arises from the strong interdependence between the variables of the system and the boundary conditions. An immobilized glucose isomerase-based mathematical model for the enzymatic isomerization process that converts glucose to fructose is presented. The model's kinetic mechanism is stated using the nonlinear reaction-diffusion equation for MichalisMenten kinetics. The general approximate analytical formulas for the glucose molar concentration and flux inside packed-bed reactors are determined by solving the nonlinear equation using Akbari-Ganji's method. The effects of the kinetic parameters and pore-level Thiele modulus on concentration and flux were discussed. Estimating the kinetic parameters from current density is suggested. It has been shown that this method reduces processing time without affecting the quality of the solution. This method works well for many different types of nonlinear systems, making it useful in engineering and other fields of science. [ABSTRACT FROM AUTHOR]

Although there is currently no unified standard theoretical formula for calculating the contact stress of cylindrical gears with a circular arc tooth trace (referred to as CATT gear), a mathematical model for determining the contact stress of CATT gear is essential for studying how parameters affect its contact stress and building the contact stress limit state equation for contact stress reliability analysis. In this study, a mathematical relationship between design parameters and contact stress is formulated using the KrigingMetamodel. To enhance the model's accuracy, we propose a new hybrid algorithm that merges the genetic algorithm with the Quantum Particle Swarm optimization algorithm, leveraging the strengths of each. Additionally, the "parental inheritance+self-learning" optimization model is used to fine-tune the KrigingMetamodel's parameters. Following this, amathematicalmodel for calculating the contact stress of Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT) gears using the optimized Kriging model was developed. We then examined how different gear parameters affect the VH-CATT gears' contact stress. Our simulation results show: (1) Improvements in R², RMSE, and RMAE. R² rose from0.9852 to 0.9974 (a 1.22% increase), nearing 1, suggesting the optimized Kriging Metamodel's global error is minimized. Meanwhile, RMSE dropped from3.9210 to 1.6492, a decline of 57.94%. The global error of the GA-IQPSO-Kriging algorithm was also reduced, with RMAE decreasing by 58.69% from 0.1823 to 0.0753, showing the algorithm's enhanced precision. In a comparison of ten experimental groups selected randomly, the GA-IQPSO-Kriging and FEM-based contact analysis methods were used to measure contact stress. Results revealed a maximum error of 12.11667 MPA, which represents 2.85% of the real value. (2) Several factors, including the pressure angle, tooth width, modulus, and tooth line radius, are inversely related to contact stress. The descending order of their impact on the contact stress is: tooth line radius>modulus>pressure angle>tooth width. (3) Complex interactions are noted among various parameters. Specifically, when the tooth line radius interacts with parameters such as pressure angle, tooth width, and modulus, the resulting stress contour is nonlinear, showcasing amultifaceted contour plane. However, when tooth width, modulus, and pressure angle interact, the stress contour is nearly linear, and the contour plane is simpler, indicating a weaker coupling among these factors. [ABSTRACT FROM AUTHOR]

Constructing a higher order collocation framework for solving the Caputo multi-term time-fractional advection–diffusion and diffusion-type problems is the primary objective of this work, which has influenced the field of scientific disciplines. Advection–diffusion and reaction–diffusion equations were developed by modeling scientific phenomena in fluid flow issues, solid oxide fuel cells, and solvent diffusion into heavy oils. As a result, numerical solutions to these problems have garnered significant attention. The L 1 - 2 approximation approach approximates the fractional derivatives of orders η , η i ∈ (0 , 1) that are present in the considered problem. This approach provides a higher accuracy of O (k 3 - max { η , η i }) in time direction. Fourth-order convergence in space is achieved by employing a spline collocation technique with trigonometric quintic splines. Results from applying the suggested computational approach to four test examples have demonstrated its superiority and validity. [ABSTRACT FROM AUTHOR]

We propose a Gaussian jump process model on a regular Cartesian lattice for the diffusion part of the Reaction–Diffusion Master Equation (RDME). We derive the resulting Gaussian RDME (GRDME) formulation from analogy with a kernel-based discretization scheme for continuous diffusion processes and quantify the limits of its validity relative to the classic RDME. We then present an exact stochastic simulation algorithm for the GRDME, showing that the accuracies of GRDME and RDME are comparable, but exact simulations of the GRDME require only a fraction of the computational cost of exact RDME simulations. We analyze the origin of this speedup and its scaling with problem dimension. The benchmarks suggest that the GRDME is a particularly beneficial model for diffusion-dominated systems in three dimensional spaces, often occurring in systems biology and cell biology. [ABSTRACT FROM AUTHOR]

In this paper, we present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic theory model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows us to perform a formal limit, obtaining macroscopic reaction–diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on the microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology. [ABSTRACT FROM AUTHOR]

We revisit the problem of optimising (either maximising or minimising) the total biomass or wider classes of criteria in logistic-diffusive models. Since [24] , a lot of effort has been devoted to the qualitative understanding of the following question: how should one spread resources in order to maximise or minimise the total biomass? This question was studied in detail, mostly in situations where the carrying capacity of the environment and the growth rate of the population are equal. Following recent contributions [7,11,40] , we propose a mathematical treatment of this question in contexts where the growth rate and the carrying capacity are de-correlated. We investigate the optimisation of generic criteria with respect to the carrying capacity. We settle the problem of existence by providing the proper relaxation of the optimisation problems, using homogenisation-type arguments. At a qualitative level, we obtain theoretical results backing up the observations and other results of [7,11,40] regarding the optimality of homogeneous environments. We similarly investigate in details the question of minimising the total biomass. We also provide several stationarity results (in the context of large diffusivities) for the minimisation of the biomass and related criteria. [ABSTRACT FROM AUTHOR]

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph's wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems. [ABSTRACT FROM AUTHOR]

In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τr+1−α, h2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5. [ABSTRACT FROM AUTHOR]

In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0 , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15-16] and Oustaloup and co-workers [14]. [ABSTRACT FROM AUTHOR]

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains , and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling. [ABSTRACT FROM AUTHOR]

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium (C a 2 +) and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of C a 2 + dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum C a 2 + -ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate (I P 3) receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease. [ABSTRACT FROM AUTHOR]

In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number R 0 was established, and it is proved that R 0 serves as a threshold parameter: When R 0 < 1 , the disease-free steady state is globally asymptotically stable, and the disease is extinct; when R 0 > 1 , the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as R 0 increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs. In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number was established, and it is proved that serves as a threshold parameter: When , the disease-free steady state is globally asymptotically stable, and the disease is extinct; when , the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs. [ABSTRACT FROM AUTHOR]

We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary-value problem for multidimensional quasilinear parabolic equations with unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L2-norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order. [ABSTRACT FROM AUTHOR]

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson's equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction. [ABSTRACT FROM AUTHOR]

The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and (α , η) -Reich type interpolative contraction in the framework of orthogonal F -metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence to sharp traveling waves of solutions with semi-compactly supported initial data for Burgers-Fisher-KPP equations with degenerate diffusion. We characterize the motion of the free boundary in the long-time asymptotic of the solution to Cauchy problem and the convergence to sharp traveling wave with almost exponential decay rates. Here a key difficulty lies in the intrinsic presence of nonlinear advection effect. After providing the analysis of the nonlinear advection effect on the asymptotic propagation speed of the free boundary, we construct sub- and super-solutions with semi-compact supports to estimate the motion of the free boundary. The new method overcomes the difficulties of the non-integrability of the generalized derivatives of sharp traveling waves at the free boundary. [ABSTRACT FROM AUTHOR]