Browsing by Author "Khan, Adnan"

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A NEW NUMERICAL TREATMENT FOR FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON NON-DISCRETIZATION OF DATA USING LAGUERRE POLYNOMIALS
(2020) Khan, Adnan; Shah, Kamal; Arfan, Muhammad; Abdeljawad, Thabet; Jarad, Fahd; 234808
In this research work, we discuss an approximation techniques for boundary value problems (BVPs) of differential equations having fractional order (FODE). We avoid the method from discretization of data by applying polynomials of Laguerre and developed some matrices of operational types for the obtained numerical solution. By applying the operational matrices, the given problem is converted to some algebraic equation which on evaluation gives the required numerical results. These equations are of Sylvester types and can be solved by using matlab. We present some testing examples to ensure the correctness of the considered techniques.
Citation Count: Khan, Hassan...et al. (2020). "An Analytical Investigation of Fractional-Order Biological Model Using an Innovative Technique", Complexity, Vol. 2020.
An Analytical Investigation of Fractional-Order Biological Model Using an Innovative Technique
(2020) Khan, Hassan; Khan, Adnan; Al Qurashi, Maysaa; Baleanu, Dumitru; Shah, Rasool; 56389
In this paper, a new so-called iterative Laplace transform method is implemented to investigate the solution of certain important population models of noninteger order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express the noninteger derivative of fractional order. The series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method, some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has a small volume of calculations; therefore, it can be applied to handle the solutions of various problems with fractional-order derivatives.
Citation Count: Khan, Hassan;...et.al. "An approximate analytical solution of the Navier–Stokes equations within Caputo operator and Elzaki transform decomposition method", Advances in Difference Equations, Vol.2020, No.1.
An approximate analytical solution of the Navier–Stokes equations within Caputo operator and Elzaki transform decomposition method
(2020) Hajira; Khan, Hassan; Khan, Adnan; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; 56389
In this article, a hybrid technique of Elzaki transformation and decomposition method is used to solve the Navier–Stokes equations with a Caputo fractional derivative. The numerical simulations and examples are presented to show the validity of the suggested method. The solutions are determined for the problems of both fractional and integer orders by a simple and straightforward procedure. The obtained results are shown and explained through graphs and tables. It is observed that the derived results are very close to the actual solutions of the problems. The fractional solutions are of special interest and have a strong relation with the solution at the integer order of the problems. The numerical examples in this paper are nonlinear and thus handle its solutions in a sophisticated manner. It is believed that this work will make it easy to study the nonlinear dynamics, arising in different areas of research and innovation. Therefore, the current method can be extended for the solution of other higher-order nonlinear problems.
Citation Count: Qin, Ya...et al. (2020). "An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems", Energies, Vol. 13, No. 11.
An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems
(2020) Qin, Ya; Khan, Adnan; Ali, Izaz; Al Qurashi, Maysaa; Khan, Hassan; Shah, Rasool; Baleanu, Dumitru; 56389
Mostly, 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.
Citation Count: Khan, Hassan...et al. (2020). "Modified Modelling for Heat Like Equations within Caputo Operator", Energies, Vol. 13, No. 8.
Modified Modelling for Heat Like Equations within Caputo Operator
(2020) Khan, Hassan; Khan, Adnan; Al-Qurashi, Maysaa; Shah, Rasool; Baleanu, Dumitru; 56389
The 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.
Citation Count: Sunthrayuth, Pongsakorn;...et.al. (2021). "Numerical Analysis of the Fractional-Order Nonlinear System of Volterra Integro-Differential Equations", Journal of Function Spaces, Vol.2021.
Numerical Analysis of the Fractional-Order Nonlinear System of Volterra Integro-Differential Equations
(2021) Sunthrayuth, Pongsakorn; Ullah, Roman; Khan, Adnan; Shah, Rasool; Kafle, Jeevan; Mahariq, Ibrahim; Jarad, Fahd; 234808
This paper presents the nonlinear systems of Volterra-type fractional integro-differential equation solutions through a Chebyshev pseudospectral method. The proposed method is based on the Caputo fractional derivative. The results that we get show the accuracy and reliability of the present method. Different nonlinear systems have been solved; the solutions that we get are compared with other methods and the exact solution. Also, from the presented figures, it is easy to conclude that the CPM error converges quickly as compared to other methods. Comparing the exact solution and other techniques reveals that the Chebyshev pseudospectral method has a higher degree of accuracy and converges quickly towards the exact solution. Moreover, it is easy to implement the suggested method for solving fractional-order linear and nonlinear physical problems related to science and engineering.
Citation Count: Khan, Adnan...et.al. (2022). "The Extended Laguerre Polynomials Aq,nalphax Involving Fqq,q>2", Journal of Function Spaces, Vol.2022, No.22, pp.1-14.
The Extended Laguerre Polynomials Aq,nalphax Involving Fqq,q>2
(2022) Khan, Adnan; Kalim, Muhammad; Akgül, Ali; Jarad, Fahd; 234808
In this paper, for the proposed extended Laguerre polynomials A α q , n x , the generalized hypergeometric function of the type F q q , q > 2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue’s formula are determined. Some corollaries are also discussed at the end.