Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article The Analytical Analysis of Time-Fractional Fornberg-Whitham Equations(MDPI AG, 2020) Baleanu, Dumitru; Shah, Rasool; Aly, Shaban; Khan, Hassan; Alderremy, A.A.Article Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay(MDPI AG, 2019) Baleanu, Dumitru; Shah, Rasool; Arif, Muhammad; Khan, Hassan; Kumam, PoomArticle A Novel Fractional Case Study of Nonlinear Dynamics Via Analytical Approach(Zhejiang Univ Press, 2024) Khan, Hassan; Khan, Adnan; Shah, Rasool; Baleanu, DumitruThe present work describes the fractional view analysis of Newell-Whitehead-Segal equations, using an innovative technique. The work is carried with the help of the Caputo operator of fractional derivative. The analytical solutions of some numerical examples are presented to confirm the reliability of the proposed method. The derived results are very consistent with the actual solutions to the problems. A graphical representation has been done for the solution of the problems at various fractional-order derivatives. Moreover, the solution in series form has the desired rate of convergence and provides the closed-form solutions. It is noted that the procedure can be modified in other directions for fractional order problems.Article Citation - WoS: 49Citation - Scopus: 61Laplace Decomposition for Solving Nonlinear System of Fractional Order Partial Differential Equations(Springer, 2020) Shah, Rasool; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; Khan, HassanIn the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.Article Citation - WoS: 17Citation - Scopus: 17The Analytical Analysis of Nonlinear Fractional-Order Dynamical Models(Amer inst Mathematical Sciences-aims, 2021) Khan, Hassan; Shah, Rasool; Alderremy, A. A.; Aly, Shaban; Baleanu, Dumitru; Xu, JiabinThe present research paper is related to the analytical solution of fractional-order nonlinear Swift-Hohenberg equations using an efficient technique. The presented model is related to the temperature and thermal convection of fluid dynamics which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In this work Laplace Adomian decomposition method is implemented because it require small volume of calculations. Unlike the variational iteration method and Homotopy pertubation method, the suggested technique required no variational parameter and having simple calculation of fractional derivative respectively. Numerical examples verify the validity of the suggested method. It is confirmed that the present method's solutions are in close contact with the solutions of other existing methods. It is also investigated through graphs and tables that the suggested method's solutions are almost identical with different analytical methods.Article The analytical analysis of nonlinear fractional-order dynamical models(Amer Inst Mathematical Sciences-AIMS, 2021) Xu, Jiabin; Khan, Hassan; Shah, Rasool; Alderremy, A. A.; Aly, Shaban; Baleanu, DumitruThe present research paper is related to the analytical solution of fractional-order nonlinear Swift-Hohenberg equations using an efficient technique. The presented model is related to the temperature and thermal convection of fluid dynamics which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In this work Laplace Adomian decomposition method is implemented because it require small volume of calculations. Unlike the variational iteration method and Homotopy pertubation method, the suggested technique required no variational parameter and having simple calculation of fractional derivative respectively. Numerical examples verify the validity of the suggested method. It is confirmed that the present method's solutions are in close contact with the solutions of other existing methods. It is also investigated through graphs and tables that the suggested method's solutions are almost identical with different analytical methods.Article Citation - WoS: 24Citation - Scopus: 25Modified Modelling for Heat Like Equations Within Caputo Operator(Mdpi, 2020) Khan, Adnan; Al-Qurashi, Maysaa; Shah, Rasool; Baleanu, Dumitru; Khan, HassanThe present paper is related to the analytical solutions of some heat like equations, using a novel approach with Caputo operator. The work is carried out mainly with the use of an effective and straight procedure of the Iterative Laplace transform method. The proposed method provides the series form solution that has the desired rate of convergence towards the exact solution of the problems. It is observed that the suggested method provides closed-form solutions. The reliability of the method is confirmed with the help of some illustrative examples. The graphical representation has been made for both fractional and integer-order solutions. Numerical solutions that are in close contact with the exact solutions to the problems are investigated. Moreover, the sample implementation of the present method supports the importance of the method to solve other fractional-order problems in sciences and engineering.Article Citation - WoS: 41Citation - Scopus: 47An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems(Mdpi, 2020) Khan, Adnan; Ali, Izaz; Al Qurashi, Maysaa; Khan, Hassan; Shah, Rasool; Baleanu, Dumitru; Qin, Ya; Qurashi, Maysaa AlMostly, it is very difficult to obtained the exact solution of fractional-order partial differential equations. However, semi-analytical or numerical methods are considered to be an alternative to handle the solutions of such complicated problems. To extend this idea, we used semi-analytical procedures which are mixtures of Laplace transform, Shehu transform and Homotopy perturbation techniques to solve certain systems with Caputo derivative differential equations. The effectiveness of the present technique is justified by taking some examples. The graphical representation of the obtained results have confirmed the significant association between the actual and derived solutions. It is also shown that the suggested method provides a higher rate of convergence with a very small number of calculations. The problems with derivatives of fractional-order are also solved by using the present method. The convergence behavior of the fractional-order solutions to an integer-order solution is observed. The convergence phenomena described a very broad concept of the physical problems. Due to simple and useful implementation, the current methods can be used to solve problems containing the derivative of a fractional-order.Article Citation - WoS: 28Citation - Scopus: 29The Analytical Analysis of Time-Fractional Fornberg-Whitham Equations(Mdpi, 2020) Khan, Hassan; Shah, Rasool; Aly, Shaban; Baleanu, Dumitru; Alderremy, A. A.This article is dealing with the analytical solution of Fornberg-Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.Article Citation - WoS: 32Citation - Scopus: 38A Novel Method for the Analytical Solution of Fractional Zakharov-Kuznetsov Equations(Springer, 2019) Khan, Hassan; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; Shah, RasoolIn this article, an efficient analytical technique, called Laplace-Adomian decomposition method, is used to obtain the solution of fractional Zakharov- Kuznetsov equations. The fractional derivatives are described in terms of Caputo sense. The solution of the suggested technique is represented in a series form of Adomian components, which is convergent to the exact solution of the given problems. Furthermore, the results of the present method have shown close relations with the exact approaches of the investigated problems. Illustrative examples are discussed, showing the validity of the current method. The attractive and straightforward procedure of the present method suggests that this method can easily be extended for the solutions of other nonlinear fractional-order partial differential equations.