Related papers: Extension of the Partial Integral Equation Represe…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $\varphi$, invariant solutions satisfy the reduced system $F = \varphi = 0$. We propose a…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the…
In radio frequency applications, electric circuits generate signals, which are amplitude modulated and/or frequency modulated. A mathematical modelling yields typically systems of differential algebraic equations (DAEs). A multivariate…
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the…
Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to…
The particle-in-cell (PIC) method has been widely used for plasma simulation, because of its noise-reduction capability and moderate computational cost. The immersed finite element (IFE) method is efficient for solving interface problems on…
The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
Partial differential equations (PDEs) are central to describing and modelling complex physical systems that arise in many disciplines across science and engineering. However, in many realistic applications PDE modelling provides an…
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order…
Representing physical signals at different scales is among the most challenging problems in engineering. Several multi-scale modeling tools have been developed to describe physical systems governed by \emph{Partial Differential Equations}…
The $H_2$ norm is a commonly used performance metric in the design of estimators. However, $H_2$-optimal estimation of most PDEs is complicated by the lack of transfer function and state-space representations. To address this problem, we…
We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
The main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The…
In this work we study Lie symmetry analysis of initial and boundary value problems for partial differential equations (PDE) with Caputo fractional derivative. We give generalized definition and theorem for the symmetry method for PDE with…
Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…