Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

1995 ◽  
Vol 49 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Jan Veit
Keyword(s):  
Difference Equation ◽  
Fundamental Solution ◽  
Matrix Coefficients ◽  
Multidimensional Difference Equation

scholarly journals The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones

2020 ◽  
pp. 187-196
Author(s):  
Alexander P. Lyapin ◽  
Sreelatha Chandragiri
Keyword(s):  
Cauchy Problem ◽  
Difference Equation ◽  
Fundamental Solution ◽  
Difference Equations ◽  
Generating Functions ◽  
Cauchy Data ◽  
The Cauchy Problem ◽  
Constant Coefficients ◽  
Lattice Cones ◽  
Multidimensional Difference Equation

We consider a variant of the Cauchy problem for a multidimensional difference equation with constant coefficients, which connected with a lattice path problem in enumerative combinatorial analysis. We obtained a formula in which generating function of the solution to the Cauchy problem is expressed in terms of generating functions of the Cauchy data and a formula expressing solution to the Cauchy problem through its fundamental solution and Cauchy data

scholarly journals Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation

2021 ◽  
Vol 31 (3) ◽  
pp. 414-423
Author(s):  
A.P. Lyapin ◽  
S.S. Akhtamova
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Difference Equation ◽  
Recurrence Relations ◽  
Multidimensional Analogue ◽  
Recurrent Relations ◽  
Generating Series ◽  
The Cauchy Problem ◽  
Constant Coefficients ◽  
Multidimensional Difference Equation

In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

scholarly journals Matrix Shanks Transformations

2019 ◽  
Vol 35 ◽  
pp. 248-265
Author(s):  
Claude Brezinski ◽  
Michela Redivo-Zaglia
Keyword(s):  
Difference Equation ◽  
Numerical Experiments ◽  
Linear Difference Equation ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Sequence Transformation ◽  
Matrix Case

Shanks' transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix $\varepsilon$-algorithm of Wynn. Then, the transformation is related to matrix Pad\'{e}-type and Pad\'{e} approximants. Numerical experiments showing the interest of this transformation end the paper.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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