Operational calculus approach to nonlocal Cauchy problems
type: Item
instance of: scholarly article
title: Operational calculus approach to nonlocal Cauchy problems
zbMATH Open document ID: 1221.44011
author: Margarita Spiridonova
author: Ivan Dimovski
published in: Mathematics in Computer Science
publication date: 2011-06-08T00:00:00Z
full work available at URL: http://hdl.handle.net/10525/3372
review text: \textit{I. Dimovski} published in 1990 his Convolutional calculus [2nd ed. Mathematics and Its Applications: East European Series, 43. Dordrecht etc.: Kluwer Academic Publishers (1990; Zbl 0685.44006)] and introduced the right inverse operator \(L\), defined by the solution \(y= Lf(x)\) of the elementary boundary value problem \(y'= f(x)\) and \(\Phi(y)= 0\) with a given non-local functional \(\Phi\) in the space \(C(\Delta)\) of continuous functions over the interval \(\Delta\) in \({\mathfrak R}\). With the convolution operator \(L(f)=\{1\}*f\), the convolution algebra \((C^*)\), and the ring \({\mathfrak M}\) of the convolution fractions \(f/g\) with \(f\in C\) and \(g\in D\), the non-empty subset of \(C\) of all nonzero non-divisors of \(0\) in \(C^*\), the most important element in \({\mathfrak M}\) is \(S= 1/\{1\}\), the algebraic inversion of \(L= \{1\}^*\). After the relationship between the derivative \(f'\) and the product \(Sf\) is established, the operational method for solving the elementary non-local Cauchy problem is extended to the general case: \(P(d/dt)y= F\) with \(P(\lambda)= a_0\lambda^n+ a_1\lambda^{n-1}+\cdots+ a_{n-1} \lambda+ a_n\) and \(\Phi\{y^{(k)}\}= \alpha_k\) with \(k= 0,1,2,\dots, n-1\), or to \(P(S)= F+ Q(S)\), equivalent to the single algebraic equation in the ring \({\mathfrak M}\) with \[ Q(S)= \sum^n_{J=1} \Biggl(\sum^n_{k=i} \alpha_{n-k} \alpha_{k-j}\Biggr)S^{j-1}. \] The solution can be written as \(y= F/P(S)+ Q(S)/P(S)\). The non-resonance case, where \(P(S)\) is a non-divisor of \(0\) in \({\mathfrak M}\), is further developed here, including the extension of the classical Duhamel principle. The resonance case, where \(P(S)\) is a divisor of \(0\) in \({\mathfrak M}\), will be the subiect of a later publication, where a new table of interpretation formulae will be included. Two examples, the \(\pi\)-periodic solution of \[ y^{(6)}+ 2y^{(4)}+ y''= F(t)\text{ with }F(t+\pi)= F(t)\quad\text{ and }\quad\int^\pi_0 F(\lambda)\,d\la…
reviewed by: Franz Selig
Mathematics Subject Classification ID: 44A40
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5905255
zbMATH Keywords: nonlocal Cauchy problem
zbMATH Keywords: non-classical convolution
zbMATH Keywords: convolution fraction
zbMATH Keywords: Duhamel principle
zbMATH Keywords: Heaviside algorithm
zbMATH Keywords: non-resonance solution
zbMATH Keywords: symbolic computation
zbMATH Keywords: periodic solution
zbMATH Keywords: antiperiodic solution
zbMATH Keywords: algorithm
zbMATH Keywords: computer algebra system Mathematica
describes a project that uses: Mathematica
MaRDI profile type: Publication
OpenAlex ID: W2014002623
cites work: Q5833361
cites work: Q3995986
cites work: Q5573683
cites work: Q3261651
cites work: Q5835915
cites work: Théorie générale des fonctions moyenne-périodiques
cites work: Q3561576
DOI: 10.1007/S11786-010-0054-1
description: scientific article; zbMATH DE number 5905255