Related papers: Notes on solution maps of abstract FDEs
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to…
Uncertain fractional differential equation (UFDE) is a kind of differential equation about uncertain process. As an significant mathematical tool to describe the evolution process of dynamic system, UFDE is better than the ordinary…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional…
We study invariant solutions of a certain class of time-fractional diffusion-wave equations with variable coefficients via Lie symmetry analysis. In physics, the fractional diffusion equation describes transport dynamics that are governed…
Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay…
Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and…
The functional equations $ f^2+g^2=1 $ and $ f^2+2\alpha fg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic…
The concept of the derivative-dependent functional separable solution, as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on…
In this paper, we present numerical methods suitable for solving convex quadratic Fractional Differential Equation (FDE) constrained optimization problems, with box constraints on the state and/or control variables. We develop an…
We consider abstract evolution equations with on-off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the…
We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line.
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…
We show in the framework of Pfaff systems theory, the functional dependences of the general analytic solutions of a suitable system of involutive differential equations describing the differences between the analytic solutions of the…
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a…
In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…
We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE,…