Analysis of the Fractional‐Order Delay Differential Equations by the Numerical Method

Abstract

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

“…In order to model these complex problems, many authors have used a new approach known as fractional differential equations (FDEs). FDEs are widely used in the mathematical modeling of real-life physical problems, and this is as a result of their several applications in real-world science and engineering problems, such as solid mechanics, economics, oscillation of earthquakes, continuum and statistical mechanics, anomalous transport, rough substrates, dynamics of interfaces between soft nanoparticles and solid mechanics, fluid-dynamic traffic models and bio-engineering and colored noise (see [29] and the references therein).…”

“…DDEs are suitable for physical systems, depend on past data and simplify the ordinary differential equation. FDDs have been applied in different areas of mathematical modelings, such as epidemiology, population dynamics, physiology, immunology and neural networks [29].…”

Genetic Algorithm Based Optimization on Approximate Fractional Delay Differential Equations With Conformal Operator

“…The equations discussed in this context find utility in a wide range of academic disciplines and practical domains. These applications encompass hydrology, signal processing, control theory, medical sciences, networks, cell biology, climate models, infectious diseases, navigation prediction, circulating blood, population dynamics, oncolytic virotherapy, delayed plant disease model, the body's response to carbon dioxide, and various other fields 3,4,5,6 . The best-known fractional pantograph delay differential equation is defined as 6 :…”

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

“…In recent decades, fractional delay differential equations have had an important role in engineering and natural sciences. Applications of these equations include hydrology, signal processing, control theory, medical sciences, networks, cell biology, climate models, infectious diseases, navigation prediction, circulating blood, population dynamics, oncolytic virotherapy, delayed plant disease model, the body reaction to carbon dioxide, and many others [3][4][5][6].…”