Related papers: Fractional Dissipative PDEs
The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…
In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the Conservation-dissipation Formalism of irreversible thermodynamics. As governing…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
In these notes, we describe the strategy for the derivation of the hydrodynamic limit for a family of long range interacting particle systems of exclusion type with symmetric rates. For $m \in \mathbb{N}:=\{1, 2, \ldots\}$ fixed, the…
Fractional calculus is an effective tool in incorporating the effects of non-locality and memory into physical models. In this regard, successful applications exist rang- ing from signal processing to anomalous diffusion and quantum…
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a…
Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…
This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we…
Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…
In the present work, we investigate the potential of fractional derivatives to model atmospheric dispersion of pollutants. We propose simple fractional differential equation models for the steady state spatial distribution of concentration…
In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven…
In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest.…
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…