In this paper, motivated by a biodegradable stent problem, we consider a parabolic differential equation on graphs. This kind of equation describes the stent material degradation in time. Since stents are mesh-like structures with thin struts, they c...
This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial deriv...
This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of...
A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such...
In this work, the homotopy analysis method (HAM) is applied to solve the fourth-order parabolic partial differential equations. This equation practically arises in the transverse vibration problems. The proposed iterative scheme finds the solution wi...
This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the...
Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has suc...
Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a c...
In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmet...
The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a po...
We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approxim...
In this paper, we study a new type of optimal control problem subject to a parabolic uncertain partial differential equation where the expected value criterion is adopted in the objective function. The basic idea of Haar wavelet transformation is to...
In this paper we study deep neural network algorithms for solving linear and semilinear parabolic partial integro-differential equations with boundary conditions in high dimension. Our method can be considered as an extension of the deep splitting me...
Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by...
A family of strongly nonlinear nonstationary equations of mathematical physics with three independent variables is investigated, which contain an arbitrary degree of the first derivative with respect to time and a quadratic combination of second deri...
We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by t...
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...
This paper investigates the spatial asymptotic behavior of solutions to a class of nonlinear parabolic equations defined on an exterior region in
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 paper is dedicated to the acoustic inversion of the vertical sound speed profiles (SSPs) in the underwater marine environment. The method of automatic differentiation is applied for the first time in this context. Representing the finite-differe...
The parabolic quasi-Sturmian approach, recently introduced for the calculation of ion–atom ionizing collisions, is adapted and applied here to the single ionization of helium induced by an intermediate-energy proton impact. Within the method, t...
When a sample of semiconducting material is illuminated by monochromatic light, in which the photon energy is higher than the energy gap of the semiconductor, part of the absorbed electromagnetic energy is spent on the generation of pairs of quasi-fr...
In a previous investigation (Bigerelle and Iost, 2004), the authors have proposed a physical interpretation of the instability λ = Δt/Δx2 > 1/2 of the parabolic partial differential equations when solved by finite differences. However, our results...
In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS)...
In this paper, an efficient iterative method based on the generalized multiscale finite element (Iter-GMs) method is applied to solve the distributed optimal control problems governed by stochastic parabolic equations. Within this framework, we first...
In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the...
The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the...
Chaotic nonlinear dynamical systems, in which the generated time series exhibit high entropy values, have been extensively used and play essential roles in tracking accurately the complex fluctuations of the real-world financial markets. We are conce...
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...
The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially i...
We study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak m...
The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provid...
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...
We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method,...
Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati’s variational principle. This variational principle both in its differential (local) and in integral (global) f...
In this paper, we consider the solution of the initial-value problem in thermodiffusion in a solid body in three-dimensional space. Not only do we prove the behavior of the solution over time, but we identify some of its technical aspects as well. St...
15 November 2026
In this paper, the fault-tolerant tracking problem is studied for parabolic PDE multi-agent systems with unknown distributed-parameter faults. A physics-informed neural network (PINN) reconstructs the states and faults from sparse interior measuremen...
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...
We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semi...
The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the M...
This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the stud...
A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution...
The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the part...
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...
We study the solvability of the Ionkin problem for some differential equations with one space variable. These equations include parabolic and quasiparabolic, hyperbolic and quasihyperbolic, pseudoparabolic and pseudohyperbolic, elliptic and quasielli...
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...
Parabolic trough solar collectors (PTSCs) are generally utilized to reach high temperatures in solar-thermal applications. The current work investigates entropy production analysis and the influence of nano solid particles on a parabolic trough surfa...
We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems,...
We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented...
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 empl...
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