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How can fractional differential equations with delay be used to model real-world phenomena?

Collocation method

Chebyshev filter

Legendre polynomials

Stability (learning theory)

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Fractional differential equations with delay can be used to model real-world phenomena by capturing long-term memory effects observed in various systems. These equations incorporate time delay and nonlocal operators, such as fractional order difference, to represent memory. The stability analysis of fractional order systems with delay is crucial but lacks detailed studies. Linear fractional difference equations with delay have been analyzed for stability conditions, with specific cases of delay term $\tau=1$ and $\tau=2$ studied in detail . Additionally, the effects of distributed delay times on dynamical systems have been examined using a kernel series framework, revealing the influence on bifurcation phase diagrams and the sensitivity of non-local properties to distributed delays . Numerical methods, such as the method of steps and shifted Legendre collocation method, have been proposed for solving fractional differential equations with delay, providing efficient numerical schemes and convergence analysis . The pantograph delay differential equation, a specific type of delay differential equation, has been analyzed using the Adomian decomposition method, leading to power series and exponential function solutions .

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Open access•Journal Article•DOI Advanced Study on the Delay Differential Equation y′(t) = ay(t) + by(ct) Aneefah H. S. Alenazy, Abdelhalim Ebaid, Ebrahem A. Algehyne, Hind K. Al-Jeaid - Show less +3 more 17 Nov 2022-Mathematics 4 Citations	Fractional differential equations with delay can be used to model real-world phenomena by incorporating both fractional derivatives and time delays, allowing for more accurate representation of systems with memory and complex dynamics.
Open access•Journal Article•DOI On Mixed Steps-Collocation Schemes for Nonlinear Fractional Delay Differential Equations 18 Mar 2023	Fractional differential equations with delay can be used to model real-world phenomena by capturing the memory effects and non-local behavior observed in various physical, biological, and engineering systems.
Open access•Posted Content•DOI Stability Analysis of Fractional Difference Equations with Delay 11 May 2023	Fractional differential equations with delay can be used to model real-world phenomena by incorporating long-term memory and nonlocal operators, which are observed in systems such as neural networks and epidemiological models.
Stability Analysis of Fractional Difference Equations with Delay D. D. Joshi, Sachin Bhalekar, Prashant M. Gade - Show less +2 more 11 May 2023	Fractional difference equations with delay can be used to model systems with long-term memory, such as neural networks and epidemiological models.

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Can the method proposed by Ahmed et al. (2021) be extended to solve higher order delay differential equations?4 answersYes, the method proposed by Ahmed et al. (2021) can be extended to solve higher-order delay differential equations. Various studies have introduced novel techniques to address different types of delay differential equations (DDEs) with higher orders, such as fractional delay differential equations (FDDEs), singularly perturbed delay differential equations, and first-order delay differential equations (DDEs) using higher-order numerical methods. These methods involve utilizing advanced mathematical tools like Laplace transforms, Sumudu transforms, and hybrid extended block Adams Moulton methods to enhance the accuracy, convergence rate, and efficiency of solving higher-order DDEs. By incorporating these innovative approaches, researchers have successfully tackled the challenges posed by higher-order delay systems, demonstrating improved performance in terms of stability, convergence, and computational efficiency.

Can differential equations be used as mathematical model?5 answersYes, differential equations can indeed be utilized as mathematical models. They serve as a powerful tool in various fields, including biological computation, mathematical theory application, natural science research, Mathematics Education for Engineering students, and explaining chemical processes. These equations simplify problem-solving processes, link mathematical theory with practical applications, and contribute to critical thinking development. Differential equations are extensively used to study evolutionary processes, establish optimal models for numerical solutions, enhance learning experiences, and explain chemical phenomena. The research conducted in different contexts highlights the versatility and effectiveness of using differential equations as mathematical models in diverse scientific and educational domains.

Can differential equations be used as Mathematical Model? How?4 answersDifferential equations are extensively utilized as mathematical models in various fields. They play a crucial role in studying biological processes, chemical phenomena, and educational approaches like in the Differential Equations course for Engineering students. These equations help in converting practical problems into mathematical language, simplifying their processing, and linking theory with practice. Mathematical models based on ordinary and partial differential equations are essential for addressing complex problems beyond experimental capabilities, such as Vicodin abuse and impact cratering. By using differential equations, researchers can develop models, analyze phenomena, and make predictions, showcasing the versatility and significance of this mathematical tool in diverse applications.

What are the potential benefits and drawbacks of implementing synchronization in time-delayed fractional order financial systems?5 answersImplementing synchronization in time-delayed fractional order financial systems can offer significant benefits such as enhanced stability and robustness against uncertainties and disturbances. By combining a fuzzy disturbance observer with an active controller, the synchronization process can effectively handle time-varying disturbances, showcasing robustness and effectiveness in maintaining system stability. However, there are potential drawbacks to consider. The synchronization process may be influenced by the order of the fractional derivative and control parameters, impacting the settling time of the system. Additionally, the complexity of the synchronization conditions and control strategies in fractional-order systems with delays and uncertainties may pose challenges in practical implementation, requiring careful design and analysis.

What are the recent advancements in fractional order mathematical model?5 answersRecent advancements in fractional order mathematical models have been focused on various areas of research. Fractional calculus has proven to be a valuable tool in describing real-world phenomena and improving the behavior of mathematical models. One area of advancement is in the approximation of fractional order functions, where the Nitisha-Pragya-Carlson (NPC) technique has shown promise in accurately approximating fractional order plant models to integer order systems. Another area of research is the application of neural network methods for solving and analyzing fractional differential equations (FDE). These methods have shown to save computing time and improve the performance of neural networks in different fields. Additionally, fractional calculus has been applied to sensing and filtering domains, leading to the development of fractional-order sensors and filters with increased design flexibility and better performance. Overall, recent advancements in fractional order mathematical models have expanded their applications and improved their accuracy and efficiency in various fields of study.

What are the physical descriptions of the non-local conditions in fractional differential equations?5 answersThe physical descriptions of the non-local conditions in fractional differential equations vary depending on the specific problem being studied. In some cases, the non-local conditions involve nonlocal initial conditions, where the solution at a given point depends on the values of the solution and its derivatives at other points in the domain. In other cases, the non-local conditions may involve nonlocal boundary conditions, where the solution at a boundary point depends on the values of the solution and its derivatives at other points in the domain. These non-local conditions introduce additional complexity to the problem and require the use of specialized techniques, such as fixed point theorems and spectral radius estimation, to establish the existence, uniqueness, and continuous dependence of solutions.

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