I am Dr. Danyal Ahmad, Assistant Professor at the School of Applied Sciences and Humanities (NUSASH), National University of Technology (NUTECH), Islamabad. With several years of experience in academia and research, my primary research interests include Numerical Linear Algebra, particularly in the development and analysis of numerical methods for fractional differential equations, the study of structured matrices, spectral analysis of matrix sequences, multigrid methods, and preconditioning techniques. My teaching philosophy focuses on building a strong theoretical foundation for students while equipping them with practical skills for effective problem-solving and real-world applications.
I have taught various undergraduate Mathematics courses across multiple disciplines. Below is the list of the courses I have taught
- Calculus and Analytical Geometry
- Discrete Structures
- Set Theory
- Linear Algebra
- Differential Equations
- Numerical Analysis
My research focuses on the ill-conditioned linear systems that arise from the discretization of different types of fractional differential equations. Different types of approximation approaches result in a sequence of linear systems with structured coefficients matrix that are being solved efficiently. Thus, exploiting the structure of the coefficient matrix (Toeplitz matrix), the related symbol (generating function) is used for the convergence analysis and to design an efficient multigrid solver. Such multigrid methods and other preconditioners are then used to deal efficiently with the large size of the involved full (dense) and ill-conditioned coefficient matrix using the Krylov subspace methods i.e., conjugate gradient and generalized minimal residual methods.
Publications
- Ahmad Danyal, Donatelli Marco, Mazza Mariarosa, Serra-Capizzano Stefano, Trotti Ken, “A smoothing analysis for multigrid methods applied to tempered fractional problems”, Linear and Multilinear Algebra, pp. 1-24, (2023), Taylor & Francis, 10.1080/03081087.2023.2242563.
- Ahmad Danyal, Donatelli Marco, Mazza Mariarosa, Serra-Capizzano Stefano, Trotti Ken, “A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring”, Mathematics, 12(12): pp. 1916, (2024), https://doi.org/10.3390/math12121916.
- Ghafoor Abdul, Hussain Manzoor, Ahmad Danyal, Arifeen Shams Ul, “Approximation of one and two-dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative”, Computers & Mathematics with Applications, 172: pp. 125-133, (2024), Elsevier, https://doi.org/10.1016/j.camwa.2024.07.032.
- Ahmad Danyal, Donatelli Marco, Dondl Patrick, Mazza Mariarosa, Striet Ludwig, “Multigrid Methods for Sinc Basis Discretization of Integral Fractional Laplacian Equation”. (to be submitted soon)
- Ahmad Danyal, Khan Muhammad Faisal, Serra-Capizzano Stefano, “Geometric means of more than two matrix-sequences in the case of hidden (asymptotic) structures”. (to be submitted soon)
