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In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and…
Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the…
The machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a given number of spatial dimensions and a choice of coordinates the data have been collected in. This dependence…
Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty…
We consider an evolution equation involving the fractional powers, of order $s \in (0,1)$, of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order $\gamma \in (1,2]$. Since it has been…
Despite advances in deep probabilistic models, learning discrete latent representations remains challenging. This work introduces a novel method to improve inference in discrete Variational Autoencoders by reframing the inference problem…
Over short time intervals planetary ephemerides have been traditionally represented in analytical form as finite sums of periodic terms or sums of Poisson terms that are periodic terms with polynomial amplitudes. Nevertheless, this…
This paper focuses on the invariance control problem for discrete-time switched nonlinear systems. The proposed approach computes controlled invariant sets in a finite number of iterations and directly yields a partition-based invariance…
The space nonlocal Allen-Cahn equation is a famous example of fractional reaction-diffusion equations. It is also an extension of the classical Allen-Cahn equation, which is widely used in physics to describe the phenomenon of two-phase…
This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain physical systems. The differential inclusion is described by a time-dependent set-valued mapping…
Each training step for a variational autoencoder (VAE) requires us to sample from the approximate posterior, so we usually choose simple (e.g. factorised) approximate posteriors in which sampling is an efficient computation that fully…
Many real-world systems modeled using partial differential equations (PDEs) involve unknown parameters that must be estimated from limited, noisy system observations. While typically assumed to be constants, some of these unobserved…
We study the inverse boundary value problem for the wave equation using the single-layer potential operator as the data. We assume that the data have frequency content in a bounded interval. We prove how to choose classes of nonsmooth…
Physical processes evolving in both time and space are often modeled using Partial Differential Equations (PDEs). Recently, it has been shown how stability analysis and control of coupled PDEs in a single spatial variable can be more…
We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion…
Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much…
Fourier Neural Operators (FNO) have emerged as promising solutions for efficiently solving partial differential equations (PDEs) by learning infinite-dimensional function mappings through frequency domain transformations. However, the…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the…