Browsing by Author "Khalil, Hammad"

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Citation Count: Khalil, H...et al. (2016). Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions. Advance in Difference Equations. http://dx.doi.org/ 10.1186/s13662-016-0910-7
Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions
(Springer International Publishing, 2016) Khalil, Hammad; Khan, Rahmat Ali; Baleanu, Dumitru; Saker, Samir H.
This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrices that we will developed in this paper have the ability to convert fractional differential equations together with its nonlocal boundary conditions to a system of easily solvable algebraic equations. Some test problems are presented to illustrate the efficiency, accuracy, and applicability of the proposed method.
Citation Count: Khalil, Hammad...et al. (2019). "Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains", Computers & Mathematics With Applications, Vol. 78, No. 6, pp. 1826-1837.
Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains
(Pergamon-Elsevier Science LTD, 2019) Khalil, Hammad; Khan, Rahmat Ali; Baleanu, Dumitru; Rashidi, Mohammad Mehdi; 56389
Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditions on both ends of the domain. The proposed method is based on spectral method. We construct some new operational matrices which have the ability to handle integral type non-local boundary constrains. These operational matrices can be effectively applied to convert the FPDEs together with its integral types boundary conditions to easily solvable matrix equation. The accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems. The proposed method has the ability to solve non-local FPDEs with high accuracy and low computational cost. Different aspects of presented approach are compared with two other recently developed methods, Haar wavelets collocation method and a family of collocation methods which are based on Radial base functions. It is observed that the proposed method is highly accurate, robust, efficient and stable as compared to these methods. (C) 2016 Elsevier Ltd. All rights reserved.