Browsing by Author "Talib, I."

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Coupled Lower and Upper Solution Approach for the Existence of Solutions of Nonlinear Coupled System With Nonlinear Coupled Boundary Conditions
(Universidad Catolica del Norte, 2016) Talib, I.; Asif, N.A.; Tunc, C.
The present article investigates the existence of solutions of the following nonlinear second order coupled system with nonlinear coupled boundary conditions (CBCs), where f1, f2: [0, 1] × R → R, μ: R6 → R2 and v: R2 → R2 are continuous functions. The results presented in [γ, 11] are extended in our article. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder's fixed point theorem play an important role in establishing the arguments. Some examples are taken to ensure the validity of the theoretical results.
Exploring the Lower and Upper Solutions Approach for Abc-Fractional Derivative Differential Equations
(Springer, 2024) Talib, I.; Riaz, M.B.; Batool, A.; Tunç, C.
The lower and upper solution approach has been widely employed in the literature to ensure the existence of solutions for integer-order boundary value problems. Therefore, in this proposed study, our primary objective is to extend this method to establish the existence results for Atangna-Baleanu-Caputo (ABC) fractional differential equations of order 0<γ<1, with generalized nonlinear boundary conditions. We propose a generalized approach that unifies the existence criteria for certain specific boundary value problems formulated using the ABC fractional-order derivative operator, particularly addressing periodic and anti-periodic cases as special instances. The framework of the proposed generalized approach relies heavily on the concept of coupled lower and upper solutions together with certain fixed point results, including Arzela-Ascoli and Schauder’s fixed point theorems. By means of the generalized approach, we first define appropriate lower and upper solutions that bound the potential solution. We then construct a modified problem that incorporates these bounding solutions, ensuring the existence of a solution to the original problems without relying on iterative techniques. This approach involves verifying that the lower solution is less than or equal to the upper solution, and that both satisfy the given boundary conditions, thus guaranteeing the existence of a solution within the specified bounds. The inclusion of the specific examples with periodic and anti-periodic boundary conditions further reinforces the validity and relevance of our theoretical results. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2024.
Extension of Lower and Upper Solutions Approach for Generalized Nonlinear Fractional Boundary Value Problems
(Taylor and Francis Ltd., 2022) Batool, A.; Talib, I.; Riaz, M.B.; Tunç, C.
Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. © 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
The New Soliton Configurations of the 3d Fractional Model in Arising Shallow Water Waves
(Springer, 2023) Alam, M.N.; Talib, I.; Tunç, C.
This paper utilizes the analytical technique (AT) to investigate the new soliton configurations (NSCs) of the nonlinear 3D fractional model (NL3DFM) with its spatial and temporal variables (STVs) which is stretch for the KdV model. The procedure gives a diversity of NSCs to the scientific literature. A lot of new computing soliton configurations have been taken. Some soliton configurations are shaped in density plots (DPs), contour plots (CPs), 2D plots (2DPs) and 3D plots (3DPs) that represent the unidirectional generation of short amplitude deep waves on the surface of acoustic and hydro-magnetic fluctuations in a channel especially for shallow water (SW). Moreover, some effects are manifested via numerical simulations (NSs) that confirm the originality of our task. Besides, we note that totally the NSCs are novel and an outstanding influence to solitary wave theory's existing literature. The chief knowledge of this investigation understanding novel NSCs and hence NSCs of this novel perception through its STVs and NL3DFM. The created process can be implemented to numerous other NL3DFMs. Comparing our studied answers and that got in earlier written scientific papers offers the novelty of our examination. The AT could also be implemented to get NSCs for other 3D-fractional models. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.
Nonlinear Fractional Partial Coupled Systems Approximate Solutions Through Operational Matrices Approach
(Cambridge Scientific Publishers, 2019) Talib, I.; Belgacem, F.B.M.; Khalil, H.; Tunc, C.
In this article, the numerical method based on operational matrices of fractional order derivatives and integrals in the Caputo and Riemann-Liouville senses of two-parametric orthogonal shifted Jacobi polynomials is proposed for studying the approximate solutions for a generalized class of fractional order partial differential equations. The technique is extended herein to generalized classes of fractional order coupled systems having mixed partial derivatives terms. One salient aspect of this article is the development of a new operational matrix for mixed partial derivatives in the sense of Caputo. Validity of the method is established by comparing our simulated results with literature solutions obtained otherwise, yielding negligible errors. Furthermore, as a result of the comparative study, some results presented in the literature are extended and improved in the investigation herein. © 2019, Cambridge Scientific Publishers.