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From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems in light of the Adomian decomposition method. Accordingly, we also generate another general solution of the singular FoLTI system with the use of the same methodology. Several more numerical examples are given to illustrate the core points of the perturbations of the considered singular FoLTI systems that can ultimately generate a variety of corresponding solutions. [ABSTRACT FROM AUTHOR]

In the context of reaching the best way to control the movement of autonomous cars linearly and angularly, making them more stable and balanced on different roads and ensuring that they avoid road obstacles, this manuscript chiefly aims to reach the optimal approach for a fractional-order PID controller (or P I γ D ρ -controller) instead of the already classical one used to provide smooth automatic parking for electrical autonomous cars. The fractional-order P I γ D ρ -controller is based on the particle swarm optimization (PSO) algorithm for its design, with two different approximations: Oustaloup's approximation and the continued fractional expansion (CFE) approximation. Our approaches to the fractional-order PID using the results of the PSO algorithm are compared with the classical PID that was designed using the results of the Cohen–Coon, Ziegler–Nichols and bacteria foraging algorithms. The scheme represented by the proposed P I γ D ρ -controller can provide the system of the autonomous vehicle with more stable results than that of the PID controller. [ABSTRACT FROM AUTHOR]

The two approaches to solving nonlinear Caputo time-fractional wave-like equations with variable coefficients are examined in this study. The Homotopy perturbation transform method and the Yang transform decomposition method are the names of these two techniques. Three separate numerical examples are provided to demonstrate the effectiveness and precision of the suggested methods. The results were acquired to demonstrate the effectiveness and power of the two approaches, providing estimates with better precision and closed form solutions. The solutions to these kinds of equations can be found using the suggested methods as infinite series, and when these series are in closed form, they provide the exact solution. The suggested techniques have been demonstrated to be effective and efficient in their application. Three numerical examples are used to examine the methods accuracy and effectiveness. [ABSTRACT FROM AUTHOR]

This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coeffcients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coeffcients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models. [ABSTRACT FROM AUTHOR]

This paper solves a fractional system of non-linear Whitham-Broer-Kaup equations using a natural decomposition technique with two fractional derivatives. Caputo-Fabrizio and Atangana-Baleanu fractional derivatives were applied in a Caputo-manner. In addition, the results of the suggested method are compared to those of well-known analytical techniques such as the Adomian decomposition technique, the Variation iteration method, and the optimal homotopy asymptotic method. Two non-linear problems are utilized to demonstrate the validity and accuracy of the proposed methods. The analytical solution is then utilized to test the accuracy and precision of the proposed methodologies. The acquired findings suggest that the method used is very precise, easy to implement, and effective for analyzing the nature of complex non-linear applied sciences. [ABSTRACT FROM AUTHOR]

In light of the COVID-19 pandemic, many patients have suffered from Acute Respiratory Distress Syndrome (ARDS) in Intensive Care Units (ICUs) around the world. In the medical field, it is known that the so-called artificial ventilation device, which has become the mainstay of treatment of such syndrome, decreases mortality in critically ill COVID-19 patients. Due to the high reliability of this device, there is an emergency need to follow up the progress made on designing a robust controller for improving its performance. From this perspective, this work introduces different control design schemes for obtaining an optimal Fractional-order PID controller (or simply PIρDμ-controller) of the Artificial Ventilation (AV) system through two optimization algorithms: the Bacteria Foraging Optimization (BFO) and the Particle Swarm Optimization (PSO) algorithms. The realization of the controller is accomplished using four approximations: Oustaloup's approximation, the Continued Fractional Expansion (CFE) approximation and the 1st- and 2nd-order El-Khazali approximations. The validation of the controller design and the AV system behavior are verified via numerical simulation in order to demonstrate the effectiveness and the potency of all proposed schemes. [ABSTRACT FROM AUTHOR]

In this article, we solved pantograph delay differential equations by utilizing an efficient numerical technique known as Chebyshev pseudospectral method. In Caputo manner fractional derivatives are taken. These types of problems are reduced to linear or nonlinear algebraic equations using the suggested approach. The proposed method's convergence is being studied with particular care. The suggested technique is effective, simple, and easy to implement as compared to other numerical approaches. To prove the validity and accuracy of the presented approach, we take two examples. The solutions we obtained show greater accuracy as compared to other methods. Furthermore, the current approach can be implemented for solving other linear and nonlinear fractional delay differential equations, owing to its innovation and scientific significance. [ABSTRACT FROM AUTHOR]

With effective techniques like the homotopy perturbation approach and the Adomian decomposition method via the Yang transform, the time-fractional vibration equation's solution is found for large membranes. In Caputo's sense, the fractional derivative is taken. Numerical experiments with various initial conditions are carried out through a few test examples. The findings are described using various wave velocity values. The outcomes demonstrate the competence and reliability of this analytical framework. Figures are used to discuss the solution of the fractional vibration equation using the suggested strategies for different orders of memory-dependent derivative. The suggested approaches reduce computation size and time even when the accurate solution of a nonlinear differential equation is unknown. It is helpful for both small and large parameters. The results show that the suggested techniques are trustworthy, accurate, appealing and effective strategies. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function's behavior according to different fractional-order values of α numerically. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability. [ABSTRACT FROM AUTHOR]

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vriesmodified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems. [ABSTRACT FROM AUTHOR]

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop analytical and approximate solutions in quick convergent series types by utilizing the idea of the limit with less effort and time than the residual power series method. The given model is tested and simulated to confirm the proposed technique's simplicity, performance, and viability. The results show that the above-mentioned technique is simple, reliable, and appropriate for investigating nonlinear engineering and physical problems. [ABSTRACT FROM AUTHOR]

The objective of this study are to apply the OHAM to find approximate solutions of singular two-point boundary value problems comparisons with exact solutions and other method like spline method were made. The results of equations studied using OHAM solutions were significantly reliable. [ABSTRACT FROM AUTHOR]