Browsing by Author "Talib, Imran"

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Existence of Solution for First-Order Coupled System With Nonlinear Coupled Boundary Conditions
(Springeropen, 2015) Asif, Naseer Ahmad; Talib, Imran; Tunc, Cemil
In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.
Existence of Solutions To Second-Order Nonlinear Coupled Systems With Nonlinear Coupled Boundary Conditions
(Texas State Univ, 2015) Talib, Imran; Asif, Naseer Ahmad; Tunc, Cemil
In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations u ''(t) = f(t, v(t)), t is an element of[0,1], v '' (t) = g(t, u(t)), t is an element of [0, 1], with nonlinear coupled boundary conditions phi(u(0), v(0), u(1), v(1), u'(0), v'(0)) = (0, 0), psi(u(0), v(0), u(1),v(1),u'(1), v'(1)) = (0,0), where f, g: [0, 1] x R -> R and phi, psi : R-6 -> R-2 are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder's fixed point theorem. The results presented in this article extend those in [1, 3, 15].
New Operational Matrices of Orthogonal Legendre Polynomials and Their Operational
(Taylor & Francis Ltd, 2019) Talib, Imran; Tunc, Cemil; Noor, Zulfiqar Ahmad
Many conventional physical and engineering phenomena have been identified to be well expressed by making use of the fractional order partial differential equations (FOPDEs). For that reason, a proficient and stable numerical method is needed to find the approximate solution of FOPDEs. This article is designed to develop the numerical scheme able to find the approximate solution of generalized fractional order coupled systems (FOCSs) with mixed partial derivative terms of fractional order. Our main objective in this article is the development of a new operational matrix for fractional mixed partial derivatives based on the orthogonal shifted Legendre polynomials (SLPs). The fractional derivatives are considered herein in the sense of Caputo. The proposed method has the advantage to reduce the considered problems to a system of algebraic equations which are simple in handling by any computational software. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some examples are included to demonstrate the accuracy and validity of the proposed method.
New Results and Applications on the Existence Results for Nonlinear Coupled Systems
(Springer, 2021) Talib, Imran; Abdeljawad, Thabet; Alqudah, Manar A.; Tunc, Cemil; Ameen, Rabia
In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153-160, 2003; Franco et al. in Appl. Math. Comput. 153:793-802, 2004; Franco and O'Regan in Arch. Inequal. Appl. 1:423-430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.