Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article An Investigation of Discontinuities in Time-Dependent 2D and 3D Parabolic Partial Differential Equations Utilizing Collocation Methods: A Comparative Analysis of Complex Interface Problems(Springer Heidelberg, 2025) Faheem, Muhammad; Asif, Muhammad; Amin, Rohul; Haider, Nadeem; Jarad, FahdParabolic double interface problems have many applications in the fields such as materials science, fluid dynamics, and heat transfer. This paper presents a comparison of the Haar wavelet-based collocation method and two variants of radial basis function (RBF) method for solving 2D and 3D, linear as well as nonlinear, parabolic double interface problems. The two variants of RBF methods are the multiquadric RBF method and the integrated RBF method. For linear problems, the system of equations obtained from the integrated RBF method is solved using Moore-Penrose pseudoinverse. Error analysis is performed using L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty $$\end{document} norm error and root mean square error, and the findings are discussed in detail. The methods are compared based on their accuracy and efficiency in solving different benchmark problems. The results show that both the Haar wavelet collocation method and the integrated RBF method perform better than the conventional RBF method in terms of accuracy.Article Citation - WoS: 5Citation - Scopus: 4Robust Numerical Techniques for Modeling Telegraph Equations in Multi-Scale and Heterogeneous Environments(Springer Heidelberg, 2025) Asif, Muhammad; Bilal, Faisal; Haider, Nadeem; Jarad, FahdThe article presents an innovative concept called the hyperbolic telegraph interface model, which effectively integrates regular interfaces. This hybrid method leverages Haar wavelets in conjunction with the finite difference method to provide robust numerical solutions. It is expertly designed for both linear and nonlinear models, adeptly handling constant or variable coefficients across regular interfaces. At the heart of this technique is the approximation of spatial derivatives using truncated Haar series, while time derivatives are efficiently processed through the finite difference method. The methodology has been rigorously tested across a variety of linear and nonlinear models, demonstrating its effectiveness. In linear problems, the algebraic system is solved with precision using the Gauss elimination method. For nonlinear challenges, the Quasi-Newton linearization formula is applied to successfully eliminate non-linearity from the model. To evaluate the technique's performance, we analyze key metrics such as maximum absolute errors, root mean square errors, and computational convergence rates with varying numbers of collocation points. The proposed approach consistently outperforms existing methods, particularly in situations involving abrupt changes in the solution space or discontinuities between boundary and initial conditions, delivering stable solutions in these critical scenarios. The combination of strong theoretical foundations and computational stability, along with excellent convergence rates and comprehensive numerical studies, firmly validates the accuracy and versatility of this method, confirming its wide range of applications.Article Citation - WoS: 19Citation - Scopus: 21Bioconvection of Mhd Second-Grade Fluid Conveying Nanoparticles Over an Exponentially Stretching Sheet: a Biofuel Applications(Springer Heidelberg, 2023) Nadeem, Muhammad; Ali, Rifaqat; Jarad, Fahd; Siddique, ImranThe current research examines the role of chemical reaction, nonlinear thermal radiation and slippage impact on magnetic second-grade fluid flow with diluted dispersion of nanoparticles using a theoretical bioconvection model over an exponentially stretched sheet. There are also new characteristics such as Brownian motion and thermophoresis. In the problem formulation, the boundary layer approximation is used. Using the suitable transformations, the energy, momentum, micro-organisms and concentration equations are generated into nonlinear ordinary differential equations (ODEs). The solution to the resultant problems was calculated via the Homotopy analysis method (HAM). Environmental parameters' effects on velocity, temperature, microbes and concentration profiles are graphically displayed. When comparing the current results to the previous literature, there was also a satisfactory level of agreement. In comparison with a flow based on constant characteristics, the flow with variable thermal conductivity is shown to be significantly different and realistic. The temperature and motile density of the fluid grew in direct proportion to the thermophoresis motion, buoyancy ratio and Brownian motion parameters. Also, the motile density profile decreases down for Pe and Lb while increasing when bioconvection Rayleigh number and buoyancy ratio. This work is significant to bioinspired nanofluid enhanced fuel cells and nanomaterials production techniques, according to these research studies.Article Citation - WoS: 47Citation - Scopus: 51Fractional Proportional Differences With Memory(Springer Heidelberg, 2017) Abdeljawad, Thabet; Jarad, Fahd; Alzabut, JehadIn this paper, we formulate nabla fractional sums and differences and the discrete Laplace transform on the time scale hZ. Based on a local type h-proportional difference (without memory), we generate new types of fractional sums and differences with memory in two parameters which are generalizations to the formulated fractional sums and differences. The kernel of the newly defined generalized fractional sum and difference operators contain h-discrete exponential functions. The discrete h-Laplace transform and its convolution theorem are then used to study the newly introduced discrete fractional operators and also used to solve Cauchy linear fractional difference type problems with step 0 < h <= 1.Article Citation - WoS: 271Citation - Scopus: 298Generalized Fractional Derivatives Generated by a Class of Local Proportional Derivatives(Springer Heidelberg, 2017) Jarad, Fahd; Abdeljawad, Thabet; Alzabut, JehadRecently, Anderson and Ulness [Adv. Dyn. Syst. Appl. 10, 109 (2015)] utilized the concept of the proportional derivative controller to modify the conformable derivatives. In parallel to Anderson's work, Caputo and Fabrizio [Progr. Fract. Differ. Appl. 1, 73 (2015)] introduced a fractional derivative with exponential kernel whose corresponding fractional integral does not have a semi-group property. Inspired by the above works and based on a special case of the proportional-derivative, we generate Caputo and Riemann-Liouville generalized proportional fractional derivatives involving exponential functions in their kernels. The advantage of the newly defined derivatives which makes them distinctive is that their corresponding proportional fractional integrals possess a semi-group property and they provide undeviating generalization to the existing Caputo and Riemann-Liouville fractional derivatives and integrals. The Laplace transform of the generalized proportional fractional derivatives and integrals are calculated and used to solve Cauchy linear fractional type problems.Article Citation - WoS: 17Citation - Scopus: 20Lyapunov Type Inequalities Via Fractional Proportional Derivatives and Application on the Free Zero Disc of Kilbas-Saigo Generalized Mittag-Leffler Functions(Springer Heidelberg, 2019) Alzabut, Jehad; Abdeljawad, Thabet; Jarad, Fahd; Mallak, Saed F..In this article, we prove Lyapunov type inequalities for a nonlocal fractional derivative, called fractional proportional derivative, generated by modified conformable or proportional derivatives in both Riemann-Liuoville and Caputo senses. Further, in the Riemann-Liuoville case we prove a Lyapunov inequality for a fractional proportional weighted boundary value problem and apply it on a weighted Sturm-Liouville problem to estimate an upper bound for the free zero disk of the Kilbas-Saigo Mittag-Leffler functions of three parameters. The proven results generalize and modify previously obtained results in the literature.Article Citation - WoS: 30Citation - Scopus: 30Application of a Hybrid Method for Systems of Fractional Order Partial Differential Equations Arising in the Model of the One-Dimensional Keller-Segel Equation(Springer Heidelberg, 2019) Shah, Kamal; Al-Mdallal, Qasem M.; Jarad, Fahd; Haq, FazalIn this paper, we apply a hybrid method due to coupling the Laplace transform with the Adomian decomposition method (LADM) for solving nonlinear fractional differential equations that appear in the model of Keller-Segel equations with one dimension. We explain the adopted method is with several examples. It turns out that the reliability of LADM and the reductions in computations show that LADM is widely applicable. We also compare our results with the results of homotopy decomposition method (HDM).