Related papers: Explicit Representations of Green's Function for L…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
In this paper we will show several properties of the Green's functions related to various boundary value problems of arbitrary even order. In particular, we will write the expression of the Green's functions related to the general…
In this work we have presented a rather general and easy-to-apply method for discrete Hilbert space representation of quantum mechanical Green's operators. We have shown that if in some discrete Hilbert space basis representation the…
In this paper we obtain the explicit expression of the Green's function related to a general $n$ order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, a $n$ dimensional parameter…
In this work we revise the most recent developments concerning the study of first order problems regarding differential equations with involutions. We take into account two cases: the case of initial conditions and constant coefficients and…
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…
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's…
Field-theoretic construction of functional representations of solutions of stochastic differential equations and master equations is reviewed. A generic expression for the generating function of Green functions of stochastic systems is put…
We look at estimates for the Green's function of time-fractional evolution equations of the form $D^{\nu}_{0+*} u = Lu$, where $D^{\nu}_{0+*}$ is a Caputo-type time-fractional derivative, depending on a L\'evy kernel $\nu$ with variable…
We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover,…
A new analytical operator method is discussed which solves linear ordinary differential equations with regular singularities. Solutions are obtained in analytic series form and also in Mellin-Barnes-type contour integral form. Exact series…
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 construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary…
An estimate of Green's function of the bounded solutions problem for the ordinary differential equation $x'(t)-Bx(t)=f(t)$ is proposed. It is assumed that the matrix coefficient $B$ is triangular. This estimate is a generalization of the…
In this article we use linear algebra to improve the computational time for the obtaining of Green's functions of linear differential equations with reflection (DER). This is achieved by decomposing both the `reduced' equation (the ODE…
We introduce more general concepts of Riemann-Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value…
The Laplace transform method for solving of a wide class of initial value problems for fractional differential equations is introduced. The method is based on the Laplace transform of the Mittag-Leffler function in two parameters. To extend…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…
This introduction to Green's functions is based on their role as kernels of differential equations. The procedures to construct solutions to a differential equation with an external source or with an inhomogeneity term are put together to…