Related papers: Green's function estimates for time fractional evo…
In previous paper derivations of the Green function have been given for 5D off-shell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter $\tau$). In this…
Based on results of Digne-Michel-Lehrer (2003) we give two formulae for two-variable Green functions attached to Lusztig induction in a finite reductive group. We present applications to explicit computation of these Green functions, to…
Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $\psi:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $\xi,g:D\to\mathbb{R}_+$ be locally bounded Borel…
We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional…
On the unit square, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator with either periodic or homogeneous Neumann boundary conditions. This method involves analytically…
The Green's function method which has been originally proposed for linear systems has several extensions to the case of nonlinear equations. A recent extension has been proposed to deal with certain applications in quantum field theory. The…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of…
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain…
We study the following Cauchy problems for semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu (-\Delta)^\delta u_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x) \end{equation*}…
In this study, we address the challenge of obtaining a Green's function operator for linear partial differential equations (PDEs). The Green's function is well-sought after due to its ability to directly map inputs to solutions, bypassing…
The evolution of a quantity, described by a function of space and time, relates the first derivative in time of this function to a spatial operator applied to the function. The initial value of the function at time $t=0$ is given. The…
Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on…
We derive a method to efficiently compute the Green function of on arbitrary Hamiltonians defined on semi-infinite and periodic quasi-one-dimensional lattices. Computing the Green function is the backbone of quantum transport, electronic…
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary…
We prove new estimates of the Caputo derivative of order $\alpha \in (0,1]$ for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on…
We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times \Omega$, with $I \subset \mathbb{R}$ and $\Omega \subset\mathbb{R}^n$. The operators $L$ we consider are…
Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a…