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On the $q$-analogue of Duhamel's principle

Mohammed Elamine Sebih, Serikbol Shaimardan, Irfan Ali

Abstract

In this paper, we revisit the classical Duhamel's principle and provide a self-contained proof of this fundamental tool for linear evolution equations and systems of coupled equations. Moreover, we establish a $q$-analogue of Duhamel's principle for $q$-evolution equations of order $k\geq 1$ generated by Jackson's $q$-difference operator.

On the $q$-analogue of Duhamel's principle

Abstract

In this paper, we revisit the classical Duhamel's principle and provide a self-contained proof of this fundamental tool for linear evolution equations and systems of coupled equations. Moreover, we establish a -analogue of Duhamel's principle for -evolution equations of order generated by Jackson's -difference operator.
Paper Structure (14 sections, 10 theorems, 75 equations)

This paper contains 14 sections, 10 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Duhamel's principle
  4. First order Duhamel's principle
  5. Second order Duhamel's principle
  6. $k$-order Duhamel's principle
  7. Duhamel's principle for systems of coupled equations
  8. Systems of coupled evolution equations of first order
  9. Systems of coupled evolution equations of second order
  10. Systems of coupled evolution equations of mixed orders
  11. $q$-analogue of Duhamel's principle
  12. First order Duhamel's principle for $q$-evolution equations
  13. Second order Duhamel's principle for $q$-evolution equations
  14. $k$-order Duhamel's principle for $q$-evolution equations

Key Result

Lemma 2.1

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that $D_{q,x}f(x,y)$ exists. Then, the function $F$ defined on $\mathbb{R}$ by $F(x)=\int_{0}^{x}f(x,t)\mathrm d_{q} t$ is $q$-differentiable and we have or equivalently

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • ...and 13 more