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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]

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]

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]