Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies
Abstract
1. Introduction
2. Preliminaries
2.1. First-Order Time-Delayed Differential Equation
- Roots of the Equation: Solving for s yields the roots. Complex roots typically signify oscillatory dynamics, while the real part of each root, , plays a critical role in stability analysis. A system is stable if and unstable if .
- Delay Effects: The delay term significantly influences root locations and can lead to changes in system stability and behavior, such as bifurcations.
- Parameter Influence: The parameter a affects the feedback within the system, influencing the roots and dynamics. Larger values of a may amplify feedback effects.
2.2. Fractional Caputo Time-Delayed Differential Equation
2.3. Higher-Order Fractional Differential Equation with Time Delays
3. Time-Delayed Bulk Viscosity
3.1. Time-Delayed Bulk Viscosity
3.2. Linearization
3.3. Error Estimation
3.4. Discussion
4. Time-Delayed Bulk Viscosity in Fractional Cosmology
4.1. Problem Setting
- : order of the fractional derivative;
- a: constant coefficient of the delayed term;
- b: constant coefficient of the linear term;
- T: time delay.
- Start with the differential equation
- Apply the Laplace transform:
- Laplace transform of the Caputo fractional derivative:where is the Laplace transform of and , are the initial conditions.
- Laplace transform of the delayed term:
- Laplace transform of the linear response term:
- Substitute into the original equation:
- Combine terms:
- Characteristic equation:
- Inverse Transforms: Inverse transforms, such as the inverse Laplace transform, are powerful tools for solving differential equations. They convert complex differential equations into simpler algebraic forms, making them easier to analyze. Once solutions are obtained in the transformed domain, inverse transforms convert them back to the original domain. This approach helps in understanding the system’s behavior.
- Characteristic Equation: The characteristic equation is derived from the differential equation governing the system. It encapsulates the system’s key properties and helps determine its stability, oscillatory behavior, and response to external stimuli. By examining the roots of the characteristic equation, we gain insights into the system’s dynamics and can predict its long-term behavior.
- Physical Application: Time-Delayed Bulk Viscosity Cosmology. Consider applying the model to time-delayed bulk viscosity cosmology as a practical example. Bulk viscosity refers to the resistance of cosmic fluids to compression, affecting the Universe’s expansion rate. Time delays account for the finite response time of these fluids to changes in pressure and density.
- Capture Delayed Reactions: Time delays introduce memory effects, meaning the system’s current state depends on its past states. This is crucial for modeling realistic physical systems where changes do not happen instantaneously.
- Analyze stability: The characteristic equation provides information about the stability of the cosmological model. By examining the roots, we can determine whether the Universe’s expansion will be stable, oscillatory, or exhibit other behaviors.
- Predict Cosmic Evolution: We can predict how the Universe’s expansion rate evolves by solving the time-delayed differential equations. This can help address unresolved issues in cosmology, such as the nature of dark energy and the mechanisms driving accelerated expansion.
4.2. Solution
4.3. Error and Smooth Transition Correction
- : Integer division of t by T, which determines the upper bound of summation.
- : Heaviside step function, restricting contributions to cases where .
- : A rapidly decaying term as k increases, primarily driven by factorial growth in the denominator.
- : An exponentially growing term, significant for large j.
- : A power-law term dependent on , which decays when .
- : An exponential factor determined by , growing or decaying based on .
- : A power-law term influenced by :
- -
- If , this term increases with .
- -
- If , it decreases with .
- -
- If , it remains constant.
- : A polynomial decay factor in the denominator, which is subleading.
- : A dominant exponential decay term in the denominator for large j.
- Smooth Transition: The exponential decay gradually introduces the correction as t exceeds .
- Bounded Between 0 and 1: The function transitions smoothly from 0 at and asymptotically approaches 1 as t increases.
- Gaussian-Like Decay: The squared term in the exponent resembles a Gaussian mollifier, controlling the transition rate.
- Discontinuity Prevention: This function ensures corrections are applied smoothly rather than abruptly, preserving continuity.
4.4. Numerical Solution
4.5. Generalization
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Lambert W Function
Appendix B. Mittag–Leffler Functions
Appendix C. Laplace Transform of Time-Delayed Function
Appendix D. Laplace Transform of the Caputo Derivative
Appendix E. Numerical Considerations and Forward Difference Formulation
Appendix F. Optimized Algorithms
Appendix F.1. Optimized Algorithm to Implement Exact Solutions for (53), (58) Together with (60), and (59) Together with (61), as Given in Section 3.2
Algorithm A1 Optimized Algorithm to Implement Exact Solutions for (53), (58) Together with (60), and (59) Together with (61), as Given in Section 3.2 |
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Appendix F.2. Algorithm for Computing H(t) Using Equation (97) with the Mollifier (96)
Algorithm A2 Algorithm for Computing Using Equation (97) with the Mollifier (96) |
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Appendix F.3. Algorithm Implementing the Numerical Procedure (108)–(114)
Algorithm A3 Algorithm Implementing the Numerical Procedure (108)–(114) |
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Appendix F.4. Algorithm Implementing the Fractional Nonlinear Scheme (117) and (118)
Algorithm A4 Algorithm Implementing the Fractional Nonlinear Scheme (117) and (118) |
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Micolta-Riascos, B.; Droguett, B.; Mattar Marriaga, G.; Leon, G.; Paliathanasis, A.; del Campo, L.; Leyva, Y. Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies. Fractal Fract. 2025, 9, 318. https://doi.org/10.3390/fractalfract9050318
Micolta-Riascos B, Droguett B, Mattar Marriaga G, Leon G, Paliathanasis A, del Campo L, Leyva Y. Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies. Fractal and Fractional. 2025; 9(5):318. https://doi.org/10.3390/fractalfract9050318
Chicago/Turabian StyleMicolta-Riascos, Bayron, Byron Droguett, Gisel Mattar Marriaga, Genly Leon, Andronikos Paliathanasis, Luis del Campo, and Yoelsy Leyva. 2025. "Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies" Fractal and Fractional 9, no. 5: 318. https://doi.org/10.3390/fractalfract9050318
APA StyleMicolta-Riascos, B., Droguett, B., Mattar Marriaga, G., Leon, G., Paliathanasis, A., del Campo, L., & Leyva, Y. (2025). Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies. Fractal and Fractional, 9(5), 318. https://doi.org/10.3390/fractalfract9050318