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Spectral Methods to Study the Numerical Solutions of Fractional Order Differential Equations in Caputo Sense

This topic explores the application of spectral methods in solving fractional-order differential equations (FODEs), specifically in the Caputo sense. Fractional-order differential equations are a generalization of classical integer-order differential equations, offering a more accurate representation of memory and hereditary properties in dynamic systems. These equations have gained significant attention in various fields such as physics, engineering, and finance, as they can better model complex processes involving non-local behavior and memory effects.

The Caputo definition of fractional derivatives is one of the most widely used in numerical methods, as it provides a clear physical interpretation and makes the associated boundary conditions more straightforward. However, solving FODEs numerically remains challenging due to the complexity introduced by fractional derivatives.

Spectral methods, which involve expanding the solution in terms of a series of basis functions (often orthogonal polynomials or trigonometric functions), provide an efficient way to approximate the solution to FODEs. These methods are highly accurate and can handle irregularities in the solution space with fewer grid points, making them particularly useful for solving problems involving fractional derivatives.

This study focuses on developing and applying spectral methods for the numerical solution of fractional-order differential equations in the Caputo sense. By using appropriate basis functions, we aim to derive high-precision approximations of the solutions and analyze their accuracy and efficiency in comparison to traditional methods. The goal is to provide a robust framework for tackling complex fractional-order problems and contribute to the broader understanding and application of fractional calculus in real-world scenarios.

Abstract

The aim of this study is to solve numerically the Caputo fractional initial value problems. For this purpose, we extend the Spectral methods to fully operational matrices approach which is indeed a new approach to solve Caputo fractional initial value problems. The developed approach is based on the operational matrices of fractional Legendre function vectors which are the fractional extensions of classical Legendre polynomials. Some new generalized operational matrices which are based on fractional operators and fractional Legendre functions vectors have also been developed in the present study. In addition to that, the developed approach has ability to transform the fractional problems into a Matrix Equations of Sylvester type which are then solved by using the software MATLAB. Finally, the solution is approximated as basis vectors of fractional Legendre function vectors. Several examples are taken to demonstrate the applicability of the developed approach.

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Abstract

Preventing DoS attacks are critical to avoiding disruptions of the regular services over networks, by being able to identify the attacks when they occur. In the literature there are several methods which enable the detection and prevention of such an attack. However, with the traditional techniques, the attacks with distributed and low-rate strategies are often successful in the absence of control that is not aware of spatio-temporal characteristics of network traffic. In such cases, the attackers attain the total disruption of the services or the network components, or consumption of the available network resources in bandwidth. Traditional DoS detection controls are easily fooled by attack scenarios if the attackers use non-attack traffic patterns or payloads that are spatially distributed. Even the mechanisms involving traffic patterns or payload contents for detecting the attacks cannot safeguard the networks and servers from complicated DoS attacks. These issues are addressed by this research proposal through a proposed usage of DoS detection techniques and Spatio-temporal network traffic features. Knowledge of Spatio-temporal features enables detection controls to determine the attacks that cannot normally be detected using conventional methods including jamming attacks in the wireless networks. Thus, based on the characteristics of the traffic information, MR can use a hybrid model integrating sequential and convolutional machines to extract the corresponding spatio-temporal features. Different from more traditional use such as seasonal probability and long and short term forecasting. The above mentioned methods shall give the prediction of the temporal-spatial characteristics regarding the delay, the pure throughput and the probability density of the jitter. Hence the main objective of this research is to develop an enhanced procedure to identify DoS attacks particularly in cases of low data rate attacks with the availably of Spatio-temporal features of network traffic.

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