Browsing by Author "Asif, Muhammad"

Now showing 1 - 2 of 2

A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models
(2022) Jarad, Fahd; Asif, Muhammad; Haider, Nadeem; Bilal, Rubi; Ahsan, Muhammad; Al-Mdallal, Qasem; Jarad, Fahd; 234808
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∞ 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.
On a new method for finding numerical solutions to integro-differential equations based on Legendre multi-wavelets collocation
(2022) Jarad, Fahd; Asif, Muhammad; Amin, Rohul; Al-Mdallal, Qasem; Jarad, Fahd; 234808
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∞ 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.