On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in t...

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Main Author: Clement McCalla
Format: Article
Language:English
Published: MDPI AG 2020-08-01
Series:Mathematics
Subjects:
fundamental solution
kernel matrix
Borel measures
characteristic matrix
loop-digraph
Online Access:https://www.mdpi.com/2227-7390/8/9/1418
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spelling doaj-524364e2573340cfb43cf14703846dab2020-11-25T03:43:34ZengMDPI AGMathematics2227-73902020-08-0181418141810.3390/math8091418On Fundamental Solution for Autonomous Linear Retarded Functional Differential EquationsClement McCalla039 Girard St Marlboro, NJ 07746, USAThis document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.https://www.mdpi.com/2227-7390/8/9/1418fundamental solutionkernel matrixBorel measurescharacteristic matrixloop-digraph
collection DOAJ
language English
format Article
sources DOAJ
author Clement McCalla
spellingShingle Clement McCalla
On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
Mathematics
fundamental solution
kernel matrix
Borel measures
characteristic matrix
loop-digraph
author_facet Clement McCalla
author_sort Clement McCalla
title On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
title_short On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
title_full On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
title_fullStr On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
title_full_unstemmed On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
title_sort on fundamental solution for autonomous linear retarded functional differential equations
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2020-08-01
description This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.
topic fundamental solution
kernel matrix
Borel measures
characteristic matrix
loop-digraph
url https://www.mdpi.com/2227-7390/8/9/1418
work_keys_str_mv AT clementmccalla onfundamentalsolutionforautonomouslinearretardedfunctionaldifferentialequations
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