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A fractional study of mhd casson fluid motion with thermal radiative flux and heat injection/suction mechanism under ramped wall condition: application of rabotnov exponential kernel

Rehman Aziz Ur, Jarad Fahd, Riaz Muhammad Bilal

Acta Mechanica et Automatica

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2024

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Vol. 18, no. 1

84--92

EN

The primary objective of this research is to extend the concept of fractionalized Casson fluid flow. In this study, a comprehensive analysis of magnetohydrodynamic (MHD) natural convective flow of Casson fluid is conducted, focusing on obtaining analytical solutions using the non-integer-order derivative known as the Yang–Abdel-Aty–Cattani (YAC) operator. The YAC operator utilized in this research possesses a more generalized exponential kernel. The fluid flow is examined in the vicinity of an infinitely vertical plate with a characteristic velocity denoted as 𝑢0. The mathematical modelling of the problem incorporates partial differential equations, incorporating Newtonian heating and ramped conditions. To facilitate the analysis, a suitable set of variables is introduced to transform the governing equations into a dimensionless form. The Laplace transform (LT) is then applied to the fractional system of equations, and the obtained results are presented in series form and also expressed in terms of special functions. The study further investigates the influence of relevant parameters, such as 𝛼, 𝛽, 𝑃𝑟, 𝑄, 𝐺𝑟, 𝑀, 𝑁𝑟 and 𝐾, on the fluid flow to reveal interesting findings. A comparison of different approaches reveals that the YAC method yields superior results compared to existing operators found in the literature. Graphs are generated to illustrate the outcomes effectively. Additionally, the research explores the limiting cases of the Casson and viscous fluid models to derive the classical form from the YAC fractionalized Casson fluid model.

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Approximate solution of painlevé equation i by natural decomposition method and laplace decomposition method

Amir Muhammad, Haider Jamil Abbase, Ahmad Shahbaz, Ashraf Asifa, Gul Sana

Acta Mechanica et Automatica

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2023

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Vol. 17, no. 3

417--422

EN

The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply. Novelty: One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.

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An investigation of the complexities of a malignant tumor's fractional-order mathematical model

Rakhi Singh, Gupta Vijay Kumar, Mishra Jyoti

Annals of Computer Science and Information Systems

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2022

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Vol. 33

207--211

EN

In this paper, model of a malignant tumor \& associated problems are examined using fractional-order method. We consider a case where the malignant tumor cells' net death rate is solely time-dependent. Fractional homotopy decomposition method (HDM) has been applied to determine model's series solution. The answer to the HDM is given using the Maclaurin expansion. This method's use of the Mathematica software package allows for fast and simple computation of series solutions, which is one of its benefits.

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On the contact problem in piezoelectroelasticity

Kaliński W., Rogowski B.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 1

157-174

EN

The problem of electroelasticity for piezoelectric materials is considered. The fundamental solutions for the axi-symmetric problem of piezoelasticity are utilized to solve a smooth contact problem. Exact solutions are obtained for elastic and electric fields in the contact problem. If the contact region is an annular three-part then the mixed boundary value problem is considered. In this case the solution is approximated as series solution.