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Explicit representation of solutions to a linear wave equation with time delay

Javad A. Asadzade, Jasarat J. Gasimov, Nazim I. Mahmudov, Ismail T. Huseynov

TL;DR

The paper tackles the explicit representation of solutions to a one-dimensional linear wave equation with a constant time delay on a bounded interval, under non-homogeneous Dirichlet boundary data and a prescribed history. It develops an explicit spectral approach that combines separation of variables with Sturm–Liouville theory, reducing the PDE to a countable family of scalar delay differential equations solved via delay-dependent fundamental solutions $C_{\tau}^{a,b}$ and $S_{\tau}^{a,b}$ to yield a Fourier-series solution. Key contributions include formal representations for both homogeneous and non-homogeneous dynamics, rigorous convergence and admissibility results for term-by-term spatial differentiation, and a numerical example illustrating practical truncation and visualization. The work extends explicit-representation techniques to broader delayed-operator wave models and provides a usable framework for analysis and simulation of delayed wave phenomena with nontrivial boundary and history data.

Abstract

This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a prescribed history function. To accommodate the loss of global smoothness in time caused by delay terms, solutions are understood in a \textit{stepwise classical sense}, allowing jump discontinuities in the second time derivative at multiples of the delay while maintaining continuity of the solution and its first time derivative. By combining separation of variables with Sturm-Liouville expansions, the delayed PDE is reduced to a family of scalar second-order delay differential equations. Using delay-dependent fundamental solutions, we derive closed-form representation formulas for the modal dynamics and reconstruct the PDE solution as a Fourier series. Convergence conditions guaranteeing uniform convergence and admissibility of termwise differentiation in space are established. A numerical example demonstrates the practical computation of truncated series solutions and their visualization.

Explicit representation of solutions to a linear wave equation with time delay

TL;DR

The paper tackles the explicit representation of solutions to a one-dimensional linear wave equation with a constant time delay on a bounded interval, under non-homogeneous Dirichlet boundary data and a prescribed history. It develops an explicit spectral approach that combines separation of variables with Sturm–Liouville theory, reducing the PDE to a countable family of scalar delay differential equations solved via delay-dependent fundamental solutions and to yield a Fourier-series solution. Key contributions include formal representations for both homogeneous and non-homogeneous dynamics, rigorous convergence and admissibility results for term-by-term spatial differentiation, and a numerical example illustrating practical truncation and visualization. The work extends explicit-representation techniques to broader delayed-operator wave models and provides a usable framework for analysis and simulation of delayed wave phenomena with nontrivial boundary and history data.

Abstract

This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a prescribed history function. To accommodate the loss of global smoothness in time caused by delay terms, solutions are understood in a \textit{stepwise classical sense}, allowing jump discontinuities in the second time derivative at multiples of the delay while maintaining continuity of the solution and its first time derivative. By combining separation of variables with Sturm-Liouville expansions, the delayed PDE is reduced to a family of scalar second-order delay differential equations. Using delay-dependent fundamental solutions, we derive closed-form representation formulas for the modal dynamics and reconstruct the PDE solution as a Fourier series. Convergence conditions guaranteeing uniform convergence and admissibility of termwise differentiation in space are established. A numerical example demonstrates the practical computation of truncated series solutions and their visualization.
Paper Structure (19 sections, 10 theorems, 239 equations, 3 figures)

This paper contains 19 sections, 10 theorems, 239 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Analytical simplification of the wave equation
  3. Homogeneous equation
  4. Representation of solution of \ref{['eq98']}
  5. The solution of homogeneous equation
  6. Non-homogeneous equation
  7. General case solution
  8. Convergence of the Fourier series
  9. Numerical Example
  10. Conclusion
  11. Proof of Theorem \ref{['combined1']}
  12. Proof of Theorem \ref{['c1']}
  13. Proof of Theorem \ref{['t2']}
  14. Preliminaries
  15. Analytical simplification of the wave equation
  16. ...and 4 more sections

Key Result

Theorem 1

Let $a,b\in\mathbb{R}$ and $\tau>0$. Then the functions $C_{\tau}^{a,b}$ and $S_{\tau}^{a,b}$ defined in def:Ctau--def:Stau are stepwise classical solutions (in the sense of Definition def:eq98-solution with $f\equiv 0$) of the homogeneous delay equation and satisfy the initial conditions Moreover, for each integer $n\ge 0$, the identities hold pointwise for all $t\in(n\tau,(n+1)\tau)$.

Figures (3)

  • Figure 1: Truncated approximation $u_N(t,x)$ for the numerical example. Left: 3D surface plot of $u_N(t,x)$ over $(t,x)\in[0,T]\times[0,\pi]$. Right: 2D heatmap (top view) of the same data, where colors encode the solution magnitude. For visualization we use percentile clipping and a symmetric-log color normalization to reveal the structure in the presence of large-amplitude peaks.
  • Figure 2: Delayed Hyperbolic Cosine
  • Figure 3: Delayed Hyperbolic Sine

Theorems & Definitions (29)

  • Definition 1: Stepwise (piecewise-in-time) classical solution
  • Remark 1
  • Definition 2: Stepwise solution of \ref{['eq98']}
  • Definition 3: Delayed perturbation functions
  • Theorem 1: Fundamental solutions of the homogeneous delayed equation
  • proof : Proof sketch (full proof in Appendix)
  • Theorem 2: Cauchy formula for the homogeneous problem
  • proof : Proof sketch (full proof in Appendix)
  • Theorem 3
  • proof : Proof sketch
  • ...and 19 more