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We present a general and automated approach for computing model gradients for PDE solvers built on sparse spectral methods, and implement this capability in the widely used open-source Dedalus framework. We apply reverse-mode automatic…
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure…
We present a hybrid quantum-classical framework for solving general time-dependent parabolic partial differential equations (PDEs) using quantum variational circuits. Building on the QPINN approach, this method applies broadly to parabolic…
In the present article, solvability in Sobolev spaces is investigated for a class of degenerate stochastic integro-differential equations of parabolic type. Existence and uniqueness is obtained, and estimates are given for the solution.
The question of finding solutions to given implicit differential equations (IDE) has been answered by several authors in the last few years, using different approaches, in an algebraic and also a geometric setting. Many of those results…
We investigate nodal radial solutions to semilinear problems of type \[\begin{cases}-\Delta u = f(|x|,u) \qquad & \text{ in } \Omega, \newline u= 0 & \text{ on } \partial \Omega, \end{cases} \] where $\Omega$ is a bounded radially symmetric…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the…
In this paper, we present a new SDC scheme for solving semi-explicit DAEs with the ability to be parallelized in which only the differential equations are numerically integrated is presented. In Shu et al. (2007) it was shown that SDC for…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…
In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for partial differential equations with variational structure. SAV approach…
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…
This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of…
We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the…
In the theory of neutral differential equations with pulse influence (neutral impulsive differential equations), there are many unsolved problems related to certain results in the theory of integral and integro-differential equations. In…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…