Browsing by Author "Riaz, M.B."

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Dynamical and Sensitivity Analysis for Fractional Kundu–eckhaus System To Produce Solitary Wave Solutions Via New Mapping Approach
(Taylor and Francis Ltd., 2024) Rehman, A.U.; Riaz, M.B.; Tunç, O.
The fractional Kundu–Eckhaus (FKE) equation, a nonlinear mathematical model, holds significance in assessing optical fibre communication systems. It takes into account various factors, including dispersion, noise and nonlinearity, which can impact the quality of signal and rates of data transmission in the systems of optical fibre. Utilizing the FKE model can contribute to optimizing the features of optical fibre network. In this academic investigation, an innovative mapping approach is applied to the FKE model to unveil novel soliton solutions. This is achieved through the utilization of beta derivative by employing the new mapping method and computer algebraic system such as Maple. The derived results are expressed in terms of hyperbolic and trigonometric functions. Our study elucidates a variety of soliton patterns such as periodic, dark, kink, bright, singular, dark–bright soliton solutions. To facilitate comprehension, certain solutions are visually depicted through two-dimensional, three-dimensional, and phase plots depicting bifurcation characteristics utilizing Maple software. Furthermore, the sensitivity of the model is explored across diverse initial conditions. Our study establishes a connection between computer science and soliton physics, emphasizing the pivotal role of soliton phenomena in advancing simulations and computational modelling. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
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.
Fractional Dynamics and Sensitivity Analysis of Measles Epidemic Model Through Vaccination
(Taylor and Francis Ltd., 2024) Riaz, M.B.; Raza, N.; Martinovic, J.; Bakar, A.; Kurkcu, H.; Tunç, O.
Measles is a highly contagious disease that mainly affects children worldwide. Even though a reliable and effective vaccination is available, there were 140,000 measles deaths worldwide in 2018, and most of them were children under the age five years. In this paper, we comprehensively investigate a novel fractional SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) model of the measles epidemic powered by nonlinear fractional differential equations to understand the epidemic’s dynamical behaviour. We use a non-singular Atangana-Baleanu fractional derivative to analyze the proposed model, taking advantage of non-locality. The existence, uniqueness, positivity and boundedness of the solutions are shown via concepts of fixed point theory, and we also perform the Ulam-Hyers stability of the considered model. The parameter sensitivity is discussed in the context of the variance with each parameter using 3-D graphics based on the basic reproduction number. Moreover, with the Atangana-Toufik numerical scheme, numerical findings are depicted for different fractional-order values. The presented approach produce results that are efficiently consistent and in excellent agreement with the theoretical results. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
Signature of Conservation Laws and Solitary Wave Solution With Different Dynamics in Thomas–fermi Plasma: Lie Theory
(Elsevier B.V., 2024) Fayyaz, M.; Riaz, M.B.; Rehman, M.J.U.; Tunç, O.
We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved. © 2024 The Authors
Thermal and Flow Properties of Jeffrey Fluid Through Prabhakar Fractional Approach: Investigating Heat and Mass Transfer With Emphasis on Special Functions
(Springer, 2024) Riaz, M.B.; Rehman, A.U.; Chan, C.K.; Zafar, A.A.; Tunç, O.
The core purpose of this work is the investigation, formulation and develop a mathematical model by dint of a new fractional modeling approach namely Prabhakar fractional operator to study the dynamics of Jeffrey fluid flow and heat transfer phenomena. Exact analysis of natural convective flow of the Jeffrey fluid to derive analytical solutions with the non-integer order derivative Prabhakar fractional operator with non-singular type kernel alongwith application of generalized laws namely Fick’s and Fourier’s are reported here. This fractional operator has multi parameter generalized Mittag-Leffler kernel. The fluid flow is elaborated near an infinitely vertical plate with characteristics velocity u0. The modeling of the considered problem is done in terms of the partial differential equations together with generalized boundary conditions. Introduced the appropriate set of variables to transform the governing equations into dimensionless form. Laplace transform is operated on the fractional system of equations and results are presented in series form and also presented the solution in the form of special functions. The pertinent parameter’s influence such as α, Pr, β, Gm, Sc, γ, Gr on the fluid flow is brought under consideration to reveal the interesting results. In comparison, we noticed the Prabhakar-like non integer approach shows better results than the existing operators in the literature, and graphs are drawn to show the results. Also, obtained the results in a limiting sense such as second grade fluid, Newtonian fluid in fractionalized form as well as Jeffrey and viscous fluid models for classical form from Prabhakar-like non integer Jeffrey fluid model. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2024.