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Analysis of Delay Differential Equations Using the Offset Linear Canonical Transform

This paper proposes a novel framework utilizing the Offset Linear Canonical Transform (OLCT) to solve delay differential equations by converting them into Volterra-type integral equations, demonstrating through analytical and numerical results that this approach offers enhanced accuracy and effectiveness compared to traditional Fourier transform methods.

Original authors: Gita Rani Mahato Manab Kundu

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gita Rani Mahato Manab Kundu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Many systems in the natural world do not react instantly to changes. A plant does not grow the moment it receives water; an economy does not shift the instant a policy is announced. These systems carry a memory, where the present state is shaped by what happened in the past. Scientists describe this behavior using a specific type of mathematical equation known as a delay differential equation. These equations are essential for modeling everything from the spread of infectious diseases to the stability of engineering controls, because they account for the time lag between a cause and its effect. However, solving these equations is notoriously difficult. The delay introduces a complexity that standard mathematical tools often struggle to handle, frequently leading to oscillations or instability that are hard to predict or control.

For decades, researchers have relied on a powerful mathematical tool called the Fourier transform to analyze such systems. This tool acts like a prism, breaking down complex signals into their simpler, underlying frequencies, making it easier to see how a system behaves over time. While effective, the Fourier transform is a fixed instrument; it has a specific way of looking at the world that cannot be easily adjusted. In recent years, mathematicians have developed a more flexible version of this tool, known as the offset linear canonical transform. Think of the Fourier transform as a standard camera lens, while this new tool is a lens with adjustable knobs that allow the user to shift, scale, and tilt the view of the data before analyzing it. This added flexibility offers a new way to look at old problems, potentially revealing details that the standard approach might miss.

In a recent study, researchers Gita Rani Mahato and Manab Kundu explored whether this more flexible mathematical lens could provide a better framework for solving delay differential equations. Their goal was to see if the offset linear canonical transform could not only solve these equations but also offer a unified way to understand them, bridging the gap between different mathematical methods. They focused on a specific class of these equations where the delay is constant, a common scenario in real-world modeling. By applying their new method, they transformed the difficult differential equation, which involves rates of change and past values, into a different type of problem known as an integral equation. This new form is often easier to handle numerically, allowing the researchers to build a solution step-by-step, moving forward in time intervals just as one would walk up a staircase.

The team found that this approach worked remarkably well. They demonstrated that the new method could produce accurate solutions that matched those obtained using the traditional Fourier transform. In fact, they showed that the traditional Fourier method is actually just a special, less flexible case of their new, broader framework. When they tested their method with different values for the delay and other parameters, the results were consistent. They observed that as the delay in the system increased, the behavior of the solution became calmer and more stable, with the oscillations dying down more quickly. Conversely, when the delay was short, the system showed more vigorous and rapid changes. The researchers also noted that the sign of a key parameter in their equation determined whether the delayed effect supported or opposed the current state, influencing how quickly the system settled down.

To ensure their findings were solid, the researchers compared their results against established solutions derived by other scientists using the Fourier transform. The comparison revealed a high degree of agreement. The real parts of the solutions from both methods overlapped closely, confirming that the new approach was reliable. The only notable difference was that their new method produced a complex result with both real and imaginary components, whereas the traditional method yielded only a real component. This extra information provided by the new method is a natural consequence of its more complex mathematical structure, offering a richer description of the system's behavior without contradicting the known results. The study also included a detailed look at how errors behaved, showing that the differences between the methods were small and occurred primarily near the peaks of the oscillations, further validating the accuracy of the new technique.

Ultimately, this work establishes a promising new foundation for tackling these complex time-delay problems. The researchers showed that by using the offset linear canonical transform, they could convert a difficult delay differential equation into a form that is both analytically tractable in special cases and numerically robust for general cases. While this study focused on linear equations, the author suggests that the flexibility of their method and the properties of the resulting integral equation make it a strong candidate for future work on even more difficult nonlinear problems, which often involve transcendental terms that are currently very hard to solve. By proving that this generalized transform can handle delay equations effectively, the study opens the door for more precise modeling of systems where the past constantly shapes the present.

Technical Summary: Analysis of Delay Differential Equations Using the Offset Linear Canonical Transform

Problem Statement
Delay differential equations (DDEs) model systems where the current state depends on past states, a phenomenon prevalent in biology, economics, and engineering. Unlike ordinary differential equations, DDEs exhibit complex dynamics, including stability issues and oscillatory behavior induced by time delays. While analytical methods like the Laplace transform and Fourier transform (FT) have been applied to linear DDEs, they often face limitations. Specifically, obtaining explicit solutions via Laplace transforms can be difficult for complex delay systems due to the challenge of computing inverse transforms in closed form, and delay terms frequently lead to transcendental characteristic equations. Although Ohira et al. [13,14] successfully utilized Fourier analysis to solve specific DDEs, the potential of more generalized integral transforms with additional degrees of freedom remains largely unexplored for this class of problems.

Methodology
The author proposes a framework based on the Offset Linear Canonical Transform (OLCT), a six-parameter integral transform that generalizes the Fourier transform (FT), fractional Fourier transform (FrFT), and linear canonical transform (LCT). The methodology proceeds through two primary approaches:

Direct OLCT Approach:
- The author applies the OLCT to a class of linear delay differential equations of the form $\frac{d}{dt}X(t) + \alpha t X(t) = \beta X(t - \tau)$ .
- By exploiting the operational properties of the OLCT (specifically shifting and differentiation properties), the original DDE is transformed into a functional differential equation in the transform domain.
- This equation is subsequently converted into a Volterra-type delay integral equation using an integrating factor.
- Due to the shift term $OM(u - a\tau)$ , a closed-form analytical solution is generally not available. Therefore, the authors employ Brunner's method of steps to construct a numerical solution iteratively over successive intervals.
- The final solution in the time domain is recovered via the inverse OLCT.
Fourier-Transform-Based OLCT Approach:
- To validate the direct method and establish a unified perspective, the author derives a solution by leveraging the known relationship between the OLCT and the FT.
- They solve the DDE in the Fourier domain (following Ohira's methodology) and map the result to the OLCT domain.
- For a special case where the parameter $a=0$ , an explicit analytical solution is derived in the OLCT domain, which is then expressed as an integral involving the history function.

Key Contributions

Framework Development: The paper establishes the first OLCT-based framework for solving delay differential equations, transforming them into Volterra-type delay integral equations.
Unified Formulation: It demonstrates that the classical Fourier-based formulation is a special case of the proposed OLCT framework, thereby providing a unified transform-domain perspective.
Analytical and Numerical Solutions: The author derives explicit analytical solutions for a special case ( $a=0$ ) and provide a robust numerical scheme (Method of Steps) for the general case.
Parameter Analysis: The study investigates the influence of key parameters—specifically the delay $\tau$ , the coefficient $\beta$ , and the damping parameter $\alpha$ —on the solution's behavior, including oscillation, decay, and stability.

Results

Consistency: Numerical simulations and graphical comparisons show that the direct OLCT solution and the Fourier-based OLCT solution yield nearly identical real and imaginary parts across various delay values ( $\tau = 4, 10, 16$ ). Small discrepancies are observed only near oscillation peaks, which diminish as the solution decays.
Parameter Effects:
- Delay ( $\tau$ ): Increasing the delay parameter reduces the amplitude of oscillations and leads to a more stable, smoother solution.
- Coefficient ( $\beta$ ): The sign of $\beta$ significantly affects the solution for small delays (negative $\beta$ opposes the solution, causing faster decay; positive $\beta$ supports it). For large delays, the magnitude of the solution becomes less sensitive to the sign of $\beta$ .
- Damping ( $\alpha$ ): Larger values of $\alpha$ result in faster decay and smoother solutions, while smaller values preserve delay-induced oscillations.
Comparison with Ohira's Work: When compared to the classical Fourier-transform solution by Ohira et al. [14], the OLCT solution matches the real component closely. However, the OLCT solution also provides an imaginary component due to its complex-valued kernel, offering additional phase and offset information not present in the purely real Fourier solution.

Significance and Claims
The paper claims that the proposed OLCT framework offers a promising foundation for analyzing delay differential equations by leveraging the enhanced flexibility of the OLCT (time shifts, frequency shifts, scaling, and chirp modulation). The author asserts that their method is accurate and reliable for linear DDEs, as validated by error analysis and consistency with existing Fourier-based results.

Crucially, the author states that this work serves as an initial step toward a broader objective: solving nonlinear delay differential equations involving transcendental terms. They argue that the convolution and correlation properties of the OLCT, combined with the structure of the resulting Volterra integral equations, possess the necessary properties to handle such nonlinearities. However, the paper explicitly limits its current scope to linear equations and does not present results for nonlinear systems, noting that extending the approach to such problems will be the focus of future research.

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