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Green's functions are highly useful in analyzing the dynamical behavior of polynomials in their escaping set. The aim of this paper is to construct an analogue of Green's functions for planar quasiregular mappings of degree two and constant…
We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The…
In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are…
Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in physics. The prior knowledge for specific physical systems routinely offers essential regularization schemes for solving the ill-posed…
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They…
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…
We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are…
Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritt's pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the…
In this work, we study nonlocal differential equations with particular focus on those with reflection in their argument and piecewise constant dependence. The approach entails deriving the explicit expression of the solution to the linear…
Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter (e.g., the temperature) that depends on integrals of the unknown…
In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In…
So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well…
In this article we obtain two-sided estimates for the Greens function of fractional boundary value problems on $\mathbb R_+ \times \mathbb R_+ \times \mathbb R^d$ of the form \[(-{}_{t_1}D^\beta_{0+*} - {}_{t_2}D^\gamma_{0+*})u(t_1, t_2, x)…
We consider a two-dimensional diffusion process in a two-layered plane, governed by distinct covariance matrices in the upper and lower half-planes and by two drift vectors pointed away from the $x$-axis. We first analyze the case where the…
Neural operators, which learn mappings between the function spaces, have been applied to solve boundary value problems in various ways, including learning mappings from the space of the forcing terms to the space of the solutions with the…
By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation ($BV$) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$…
This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The…
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms…
Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs…
This paper studies a new class of integration schemes for the numerical solution of semi-explicit differential-algebraic equations of differentiation index 2 in Hessenberg form. Our schemes provide the flexibility to choose different…