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Results on continuous dependence on parameters, as well as on regularization, of solutions to linear systems of parabolic partial differential equations of second order with delay are given. One of the main features is that the topology on the zero order and delay coefficients is the weak-* topology on an L ∞ space, whereas the assumptions on the second- and first-order coefficients are slightly stronger. This is important in the context of generation of linear skew-product dynamical systems by such equations. [ABSTRACT FROM AUTHOR]

One standard and two nonstandard finite difference methods are used to solve a convective predator-prey model consisting of cross-diffusion terms for which no exact solution is known. The model involves a system of strongly coupled nonlinear parabolic partial differential equations. Through some numerical experiments, we observe that reasonable numerical solutions are obtained when the temporal step size satisfies $ k \le 0.0005 $ k ≤ 0.0005 with spatial step size $ h = 0.1 $ h = 0.1. We construct a nonstandard method using ideas from Anguelov et al. and the scheme is denoted by NSFD1. We obtain four conditions for NSFD1 to replicate the positivity property of the continuous model, but these conditions consist of many parameters. Therefore, an unconditionally positivity-preserving scheme is considered which is denoted as NSFD2. However, we found that NSFD2 is in general not consistent. We obtain conditions under which NSFD2 is both consistent and bounded and some numerical results are presented when these conditions are satisfied. [ABSTRACT FROM AUTHOR]

We consider the finite element approximation of a coupled fluid‐structure interaction (FSI) system, which comprises a three‐dimensional (3D) Stokes flow and a two‐dimensional (2D) fourth‐order Euler–Bernoulli or Kirchhoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface which is assumed to be fixed. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, since it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to generate some numerical results concerning the approximate solutions to the FSI model. For this, we propose a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This article explores the dissipative control for a class of nonlinear DP-CPS (distributed parameter cyber physical system) within a finite-time interval. By utilizing a Takagi-Sugeno (T-S) fuzzy model to represent the system's nonlinear aspects, the studied system is formulated as a class of fuzzy parabolic partial differential equation (PDE). In order to optimize network resources, both the system state and input signal are subjected to quantization using dynamic quantizers. Subsequently, a dynamic state control strategy is proposed, taking into account potential DoS attack. The finite-time boundedness of the fuzzy parabolic PDE is analyzed, with respect to the influence of quantization, through the construction of an appropriate Lyapunov functional. The article then presents the conditions for finite-time dissipative control design, alongside the adjustment parameters for the dynamic quantizers within the fuzzy closed-loop system. Furthermore, the decoupling of interlinked nonlinear terms in the control design conditions is achieved by using an arbitrary matrix. Finally, an example is provided and the simulation results indicate the effectiveness of the dissipative control method proposed. [ABSTRACT FROM AUTHOR]

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent and L1-data of the type... where λ > 0 and T is positive constant. The main contribution of our work is to prove the existence of a renormalized solution. The functional setting involves Lebesgue-Sobolev spaces with variable exponents. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments. [ABSTRACT FROM AUTHOR]

This paper presents an analytical investigation of the solutions to a control-volume model for liquid films flowing down a vertical fibre. The evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film's radius and axial velocity. We demonstrate the existence of weak solutions to this coupled system by applying a priori estimates derived from energy-entropy functionals. Additionally, we establish the existence of traveling wave solutions for the system. To illustrate our analytical findings, we present numerical studies that showcase the dynamic solutions of the partial differential equations as well as the traveling wave solutions. [ABSTRACT FROM AUTHOR]

The global-in-time existence of weak solutions to a degenerate Cahn–Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic fourth-order partial differential equations and describes the evolution of the fiber phase volume fraction and the solute concentration, modeling the pre-patterning of lymphatic vessel morphology. The fiber phase fraction satisfies the segregation property if this holds initially. The existence proof is based on a three-level approximation scheme and a priori estimates coming from the energy and entropy inequalities. While the free energy is nonincreasing in time, the entropy is only bounded because of the cross-diffusion coupling. [ABSTRACT FROM AUTHOR]

In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme’s application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented. [ABSTRACT FROM AUTHOR]

In this article, we consider parabolic partial differential equations in (2+1) dimensions, with a focus on optimal convergence for solutions that are smooth in time but have corner singularities in space owing to the non-smoothness of the spatial boundary. The method employs geometric mesh along with a nonconforming least squares spectral element approximation in space coupled with the implicit Crank-Nicolson finite difference scheme in time. The unconditional stability of the scheme is rigorously proved and the error estimate which is almost optimal second order accurate in time and exponentially accurate in space is established. At each time instant, the subdivided problem is solved in parallel via the preconditioned conjugate gradient method in different processors by using an almost optimal preconditioner, in the sense that the condition number of the resulting system is O ((l n N) 2) (N being the degree of spatial approximation), without having to store any stiffness (or mass) matrix or load vector and the inter-element communication is established through Message Passing Interface (MPI). Specific numerical examples are presented with regard to the performance of the scheme. [ABSTRACT FROM AUTHOR]

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers' type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme. [ABSTRACT FROM AUTHOR]

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton–Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ε$$ \varepsilon $$‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ε$$ \varepsilon $$‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments. [ABSTRACT FROM AUTHOR]

This study presents an (ε,μ) -uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's l/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be secondorder accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature. [ABSTRACT FROM AUTHOR]

In this research, the (2+1)‐dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much‐improved results. This analysis includes computing one‐dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment. [ABSTRACT FROM AUTHOR]

This paper is concerned with a parabolic evolution equation of the form A(ut)+B(u)=f$A(u_t) + B(u) = f$, settled in a smooth bounded domain of Rd$\mathbb {R}^d$, d≥1$d\ge 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, −B$-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the m$m$‐Laplacian for suitable m∈(1,∞)$m\in (1,\infty)$, the "variable‐exponent" m(x)$m(x)$‐Laplacian, or even some fractional order operators. The operator A$A$ is assumed to be in the form [A(v)](x,t)=α(x,v(x,t))$[A(v)](x,t)=\alpha (x,v(x,t))$ with α$\alpha$ being measurable in x$x$ and maximal monotone in v$v$. The main results are devoted to proving existence of weak solutions for a wide class of functions α$\alpha$ that extends the setting considered in previous results related to the variable exponent case where α(x,v)=|v(x)|p(x)−2v(x)$\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called Δ2$\Delta _2$‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators A$A$, B$B$) to which the result can be applied. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence. [ABSTRACT FROM AUTHOR]

We investigate the problem of the Lévy flight foraging hypothesis in an ecological niche described by a bounded region of space, with either absorbing or reflecting boundary conditions. To this end, we consider a forager diffusing according to a fractional heat equation in a bounded domain and we define several efficiency functionals whose optimality is discussed in relation to the fractional exponent s 2 (0, 1) of the diffusive equation. Such an equation is taken to be the spectral fractional heat equation (with Dirichlet or Neumann boundary conditions). We analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, we show that if the target is close enough to the forager, then the most rewarding search strategy will be in a small neighborhood of s D 0. Interestingly, we show that s D 0 is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, we deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent, thus the forager may opt for a less performant, but safer, hunting method. However, the biological literature has already collected several pieces of evidence of foragers diffusing with very low Lévy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a high-risk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a backward differentiation formula in time. Within this paper, we focus on the second-order case, resulting in a |$3$| -step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically. [ABSTRACT FROM AUTHOR]

We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak formulation and provide numerical simulations to demonstrate the practicability of the proposed discretization. [ABSTRACT FROM AUTHOR]

We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

The object of this study is the process of liquid combustion in a spill, and the subject of research is the distribution of temperature along the wall and roof of a vertical steel tank that is heated under the thermal influence of a spill fire. The heat balance equation for the wall and roof of the tank with oil product was constructed. The assumption of a small thickness of the wall and roof of the tank relative to its linear dimensions makes it possible to move to two-dimensional differential equations of the parabolic type. The equations take into account the radiative heat exchange with the flame, the environment, the internal space of the tank, as well as the convective heat exchange with the surrounding air, the vapor-air mixture, and the liquid inside the tank. Using the methods of similarity theory, estimates of the coefficients of convection heat exchange of the outer surface of the tank with the surrounding air and the inner surface with the vapor-air mixture and liquid in the tank in the conditions of free convection were constructed. The application of the finite difference method for solving the heat balance equations has made it possible to derive the temperature distribution on the surface of the tank at an arbitrary moment in time. It is shown that the value of the coefficient of convection heat exchange of the liquid exceeds the corresponding value for the air-vapor mixture by (1÷2) orders of magnitude. As a result, the part of the wall located below the oil product level is heated to a temperature of (80÷230) °C depending on the viscosity of the liquid. This occurs despite the fact that the value of the mutual radiation coefficient reaches its maximum value at the lower part of the wall. From a practical point of view, this means that the part of the wall above the level of the oil product in the tank can reach dangerous temperature values, and it should be cooled first. The constructed model of tank heating also enables determining the limit time for the start of cooling of the tank [ABSTRACT FROM AUTHOR]

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions. [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]

Purpose: Singular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis. Design/methodology/approach: We design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme. Findings: Consistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2 , where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction. Originality/value: Parabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem's solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates. [ABSTRACT FROM AUTHOR]

Initial-boundary value problems are considered for homogeneous parabolic systems with Dini-continuous coefficients and zero initial conditions in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps, on which general boundary conditions with variable coefficients are given. A theorem on unique classical solvability of these problems in the space of functions that are continuous and bounded together with their first spatial derivatives in the closure of the domain is proved by applying the boundary integral equation method. A representation of the resulting solutions in the form of vector single-layer potentials is given. [ABSTRACT FROM AUTHOR]

This research focuses on the identification of failure times in thermal systems governed by partial differential equations, a task known for its complexity. A new model‐based diagnostic approach is presented that aims to accurately identify failing heat sources and accurately determine their failure times, which is crucial when multiple heat sources fail and there is a delay in detection by distant sensors. To validate the effectiveness of the approach, a comparative analysis is carried out with an established method based on a Bayesian filter, the Kalman filter. The aim is to provide a comprehensive analysis, highlighting the advantages and potential limitations of the methodology. In addition, a Monte Carlo simulation is implemented to assess the impact of sensor measurements on the performance of this new approach. [ABSTRACT FROM AUTHOR]

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation ∂ t u + (− Δ) 2 u = ∇ ⋅ F (∇ u) , where F satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case F (ξ) = ± | ξ | 2 ξ , we also examine the large time behaviour and stability of global solutions for arbitrary and small initial data in VMO, respectively. [ABSTRACT FROM AUTHOR]

We prove that there is a unique solution for a system of diffusion-reaction equations, which occur when simulating microbiological growth at the pore scale with a high enough spatial resolution. Moreover, we show that the solution depends continuously on initial data. The diffusion for each component of the system is either coercive on Ω, only elliptic on a subset Ωi (and zero elsewhere), or zero everywhere. This yields a noncoercive diffusion operator for the system of partial differential equations. The reaction is assumed to be Lipschitz continuous. [ABSTRACT FROM AUTHOR]

We consider a nonlinear parabolic partial differential equation in the case where the unknown function depends on two spatial variables and time, which is a generalization of the well-known Kuramoto–Sivashinsky equation. We consider homogeneous Dirichlet boundary-value problems for this equation. We examine local bifurcations when spatially homogeneous equilibrium states change stability. We show that post-critical bifurcations are realized in the boundary-value problems considered. We obtain asymptotic formulas for solutions and examine the stability of spatially inhomogeneous solutions. [ABSTRACT FROM AUTHOR]

In this work, we construct a structure-preserving Galerkin reduced-order model for the resolution of parametric cross-diffusion systems. Cross-diffusion systems are often used to model the evolution of the concentrations or volumic fractions of mixtures composed of different species, and can also be used in population dynamics (as for instance in the SKT system). These systems often read as nonlinear degenerated parabolic partial differential equations, the numerical resolutions of which are highly expensive from a computational point of view. We are interested here in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. In this work, we propose a new reduced-order modelling method, based on a reduced basis paradigm, for the resolution of parameter-dependent cross-diffusion systems. Our method preserves, at the level of the reduced-order model, the main mathematical properties of the continuous solution, namely mass conservation, non-negativeness, preservation of the volume-filling property and entropy–entropy dissipation relationship. The theoretical advantages of our approach are illustrated by several numerical experiments. [ABSTRACT FROM AUTHOR]

This article deals with a coupled system of singularly perturbed convection-diffusion parabolic partial differential equations (PDEs) possessing overlapping boundary layers. As the thickness of the layer shrinks for small diffusion parameter, efficient capturing of the solution and the diffusive flux (i.e., scaled first-order spatial derivative of the solution) leads to a difficult task. It is well-known that the classical numerical techniques have deficiencies in estimating the solution and the diffusive flux on equidistant mesh unless the mesh-size is adequately large. We aim to generate an efficient numerical approximation to the coupled system of PDEs by employing the implicit-Euler method in time and a classical finite difference scheme in space on a layer-adapted Shishkin mesh. Firstly, we discuss about parameter-uniform convergence of the numerical solution in C 0 -norm followed by the error analysis for the scaled discrete space derivative and the discrete time derivative. Subsequently, the parameter-uniform error bound is established in weighted C 1 -norm for global approximation to the solution and the space-time solution derivatives. The theoretical findings are verified by generating the numerical results for two test examples. [ABSTRACT FROM AUTHOR]

Consider an Ornstein–Uhlenbeck process that initially reverts to zero at a known mean-reversion rate β0, and then after some random/unobservable time, this mean-reversion rate is changed to β1. Assuming that the process is observed in real time, the problem is to detect when exactly this change occurs as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the process prior to the change of its mean-reversion rate. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. Allowing for both positive and negative values of β0 and β1 (including zero), the problem and its solution embed many intuitive and practically interesting cases. For example, the detection of a mean-reverting process changing to a simple Brownian motion (β0>0 and β1=0) and vice versa (β0=0 and β1>0) finds a natural application to pairs trading in finance. The formulation also allows for the detection of a transient process becoming recurrent (β0<0 and β1≥0) as well as a recurrent process becoming transient (β0≥0 and β1<0). The resulting optimal stopping problem is inherently two-dimensional (because of a state-dependent signal-to-noise ratio), and various properties of its solution are established. In particular, we find the somewhat surprising fact that the optimal stopping boundary is an increasing function of the modulus of the observed process for all values of β0 and β1. [ABSTRACT FROM AUTHOR]

This article describes a numerical method based on the dual reciprocity boundary element method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The equations include the classic and generalized Fisher's equations, Allen–Cahn equation, Newell–Whithead equation, FitzHugh–Nagumo equation, and generalized FitzHugh–Nagumo equation with time-dependent coefficients. The concept of the dual reciprocity is used to convert the domain integral to the boundary that leads to an integration-free method. We employ the time stepping scheme to approximate the time derivative, and the linear radial basis functions (RBFs) are used as approximate functions in the presented method. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in some numerical test examples. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme. [ABSTRACT FROM AUTHOR]

The potential of a finite-element technique based on an egalitarian meshing of the space–time domain D of physical problems described by parabolic or hyperbolic differential equations is explored. A least-squares minimization technique is applied in the meshed domain D to obtain stiffness-like matrices associated with various linear differential operators. Applications discussed include problems of boundary growth, and diffusive coalescence, in which D cannot be regarded as the Cartesian product of two independent domains in space and time. [ABSTRACT FROM AUTHOR]

This paper presents a study on the numerical solutions of the Burgers' equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers' equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers' problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers' equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the journal "Computational Optimization & Applications" dedicated to Fredi Tröltzsch on his 70th birthday. Tröltzsch is a mathematician known for his work in optimal control problems for partial differential equations (PDEs). His contributions include analytical and numerical advancements in the field, particularly in the area of bang-bang principles and Lagrange multipliers. Tröltzsch has also collaborated with industry partners on real-world applications of his research. The special issue features six papers written by Tröltzsch's colleagues, friends, and former students, covering various aspects of optimal control of nonlinear differential equations. [Extracted from the article]

We prove a-posteriori error-estimates for reduced-order modeling of quasilinear parabolic PDEs with non-monotone nonlinearity. We consider the solution of a semi-discrete in space equation as reference, and therefore incorporate reduced basis-, empirical interpolation-, and time-discretization-errors in our consideration. Numerical experiments illustrate our results. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a $ \mathcal{C}^2 $ boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach. [ABSTRACT FROM AUTHOR]

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]