Browsing by Author "Asif, Muhammad"

Now showing 1 - 4 of 4

Citation - WoS: 15
Citation - Scopus: 19
On a New Method for Finding Numerical Solutions To Integro-Differential Equations Based on Legendre Multi-Wavelets Collocation
(Elsevier, 2022) Asif, Muhammad; Amin, Rohul; Al-Mdallal, Qasem; Jarad, Fahd; Khan, Imran
In this article, a wavelet collocation method based on linear Legendre multi-wavelets is proposed for the numerical solution of the first as well as higher orders Fredholm, Volterra and Volterra-Fredholm integro-differential equations. The presented numerical method has the capability to tackle the solutions of both linear and nonlinear problems of these model equations. In order to endorse accuracy and efficiency of the method, it is tested on various numerical problems from literature with the aid of maximum absolute errors and rates of convergence. L-infinity norms are used to compare the numerical results with other available methods such as Multi-Scale-Galerkin's method, Haar wavelet collocation method and Meshless method from literature. The comparability of the presented method with other existing numerical methods demonstrates superior efficiency and accuracy. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.
Citation - WoS: 11
Citation - Scopus: 18
A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models
(Wiley, 2022) Jarad, Fahd; Rana, Gule; Al-Mdallal, Qasem; Asif, Muhammad; Haider, Nadeem; Bilal, Rubi; Ahsan, Muhammad
In this paper, a new numerical technique is proposed for the simulations of advection-diffusion-reaction type elliptic and parabolic interface models. The proposed technique comprises of the Haar wavelet collocation method and the finite difference method. In this technique, the spatial derivative is approximated by truncated Haar wavelet series, while for temporal derivative, the finite difference formula is used. The diffusion coefficients, advection coefficients, and reaction coefficients are considered discontinuously across the fixed interface. The newly established numerical technique is applied to both linear and nonlinear benchmark interface models. In the case of linear interface models, Gauss elimination method is used, whereas for nonlinear interface models, the nonlinearity is removed by using the quasi-Newton linearization technique. The L & INFIN; errors are calculated for different number of collocation points. The obtained numerical results are compared with the immersed interface method. The stability and convergence of the method are also discussed. On the whole, the numerical results show more efficiency, better accuracy, and simpler applicability of the newly developed numerical technique compared to the existing methods in literature.
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, Fahd
Parabolic 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.
Citation - WoS: 3
Citation - Scopus: 2
Robust Numerical Techniques for Modeling Telegraph Equations in Multi-Scale and Heterogeneous Environments
(Springer Heidelberg, 2025) Asif, Muhammad; Bilal, Faisal; Haider, Nadeem; Jarad, Fahd
The 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.