We identify and quantify a seemingly overlook mechanism for energy transfer between adjacent frequency disturbances in the Nonlinear Parabolized Stability Equations method. Physically, this energy transfer results from the finite-bandwidth nature of...
In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a...
This research proposes a method for reducing the dimension of the coefficient vector for Crank–Nicolson mixed finite element (CNMFE) solutions to solve the fourth-order variable coefficient parabolic equation. Initially, the CNMFE schemes and c...
The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of frac...
In this paper, a framework is developed for classifying bay types using stability and a sediment supply source. The framework is used to classify a total of 212 headland-bay beaches in Southeast Asia. The results show that static bays, bays with no s...
This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpo...
The DuFort–Frankel scheme was introduced in the 1950s to solve parabolic equations, and has been widely used ever since due to its stability and explicit nature. However, for over seven decades, its application has been limited to Cartesian gri...
The accurate, robust, and efficient prediction of transition in viscous flows is a significant challenge in computational fluid dynamics. We present a coupled high-fidelity iterative approach that leverages the FUN3D flow solver and the LASTRAC stabi...
This article discusses the initial boundary value problem for a class of coupled systems of semi-linear pseudo-parabolic equations on a bounded smooth domain. Global solutions with exponential decay and asymptotic behavior are obtained when the maxim...
This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new m...
This study aims at constructing new and effective fully explicit numerical schemes for solving the heat conduction equation. We use fractional time steps for the odd cells in the well-known odd–even hopscotch structure and fill it with several...
In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability...
We consider a two-dimensional parabolic problem subject to both Neumann and Dirichlet boundary conditions, along with an integral constraint. Based on the integral observation, we solve the inverse problem of a recovering time-dependent right-hand si...
A qualitative study for a second-order boundary value problem with local or nonlocal diffusion and a cubic nonlinear reaction term, endowed with in-homogeneous Cauchy–Neumann (Robin) boundary conditions, is addressed in the present paper. Provi...
In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Me...
The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes...
In this paper, a structure-preserving local discontinuous Galerkin (LDG) method is proposed for parabolic stochastic partial differential equations with periodic boundary conditions and multiplicative noise. It is proven that under certain conditions...
This article investigates the spatial decay properties and continuous dependence on the basic geometric structure. Assuming that the total potential energy is bounded and the homogeneous Dirichlet condition is satisfied on the side of the solution wi...
The authors have recently investigated a type of Hyers–Ulam stability of one-dimensional time-independent Schrödinger equation with a symmetric parabolic potential wall. In this paper, we investigate a type of Hyers–Ulam stability of...
A type of Hyers–Ulam stability of the one-dimensional, time independent Schrödinger equation was recently investigated; the relevant system had a parabolic potential wall. As a continuation, we proved a type of Hyers–Ulam stability o...
The first author has recently investigated a type of Hyers-Ulam stability of the one-dimensional time independent Schrödinger equation when the relevant system has a rectangular potential barrier of finite height. In the present paper, we will i...
A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In additi...
This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution...
We consider two abstract systems of parabolic–hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that, in our framework, are produced...
This article presents the applications of continuous symmetry groups to the computational fluid dynamics simulation of gas flow in porous media. The family of equations for one-phase flow in porous media, such as equations of gas flow with the Klinke...
We consider nonlinear modes of the nonlinear Schrödinger equation with nonlocal nonlinearities and and PT -symmetric parabolic potential. We show that there exists a set of continuous families of nonlinear modes and study their linear stability...
A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matri...
This article considers an inverse problem of time fractional parabolic partial differential equations with the nonlocal boundary condition. Dirichlet-measured output data are used to distinguish the unknown coefficient. A finite difference scheme is...
We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also con...
The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stab...
In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We...
The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameter...
In this study, we research a nonautonomous, three-species, delayed reaction–diffusion predator–prey model (RDPPM). Firstly, we derive sufficient conditions to guarantee the existence of a strictly positive, spatially homogeneous periodic...
This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The con...
We consider a parabolic equation with a singular potential in a bounded domain Ω⊂Rn. The main result is a Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor f(x) of the source term R(x,t)f(x)....
2 October 2025
Stability theory offers a practical method on parametric studies that encompass scales in the boundary layer typically not captured in Large Eddy (LES) or Reynolds-Averaged Navier–Stokes (RANS) simulations. We investigated the transition modes...
This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving exis...
It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provabl...
We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the ne...
We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric ellipti...
This paper considers the question of local dynamics of the simplest non-periodic chains of nonlinear first-order equations with two-sided couplings. The main attention is paid to the study of chains with a large number N of elements. The critical cas...
In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinea...
The nonlinear aeroelastic behavior of suspension bridges, undergoing dynamical in-plain instability (galloping), is analyzed. A nonlinear continuous model of bridge is formulated, made of a visco-elastic beam and a parabolic cable, connected each oth...
The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction im...
The effect of nitrogen on deformation-induced martensitic transformation (DIMT) in metastable 301 austenitic stainless steel has been studied based on the inelastic deformation theory. DIMT is regarded here as continuous relaxation process of interna...
The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Bena...
A variety of bearing profile designs can be used to improve the performance of a rotor–bearing system in severe conditions, such as operating with a shaft misalignment. Misalignments usually occur due to a deformation of the journal, bearing we...
Modulating the boundary layer velocity profile is a very promising strategy for achieving transition delay and reducing the friction of the plate. By perturbing the flow with counter-rotating vortices that undergo transient, non-modal growth, streamw...
For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posednes...
This work investigates the inverse problem of identifying a time-dependent source term in a time-fractional semi-linear degenerate parabolic equation using integral measurement data. We establish the unique solvability of the inverse problem within a...
of 2