Related papers: Solutions and Green's function of the first order …
Here we present a new approach to deal with first order ordinary differential equations (1ODEs), presenting functions. This method is an alternative to the one we have presented in [1]. In [2], we have establish the theoretical background…
This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear fractional differential equation with mixed boundary value conditions. Riemann-Liouville fractional derivative is considered. An…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
First-order learning involves finding a clause-form definition of a relation from examples of the relation and relevant background information. In this paper, a particular first-order learning system is modified to customize it for finding…
Linear and nonlinear optical effect has been widely discussed in large quantity of materials using theoretical or experimental methods. Except linear optical conductivity, higher-order nonlinear responses are not studied fully. Starting…
In this work the implicit function theorem is used for searching local symbolic resolution of differential equations. General results of existence for first order equations are proven and some examples, one relative to cavitation in a…
It is shown that the Green's function on a finite lattice in arbitrary space dimension can be obtained from that of an infinite lattice by means of translation operator. Explicit examples are given for one- and two-dimensional lattices.
During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the…
This article deals with the second order linear differential equations with entire coefficients. We prove some results involving conditions on coefficients so that the order of growth of every non-trivial solution is infinite.
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the…
Starting with the Green's functions found for normal diffusion, we construct exact time-dependent Green's functions for subdiffusive equation (with fractional time derivatives), with the boundary conditions involving a linear combination of…
We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear"…
Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green…
We study the one-dimensional Schr\"odinger equation and derive exact expressions for the Green function in terms of reflection coefficients which are defined for semi-infinite intervals. We also discuss the relation between our results and…
We derive equations of motion for higher order density response functions using the theory of thermodynamic Green's functions. We also derive expressions for the higher order generalized dielectric functions and polarization functions.…
In this paper we develop a way of obtaining Green's functions for partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative…
Classically, a mainstream approach for solving a convex-concave min-max problem is to instead solve the variational inequality problem arising from its first-order optimality conditions. Is it possible to solve min-max problems faster by…
This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.