Department of Mathematics

Dr. Md. Abdul Hakim Khan

Professor
Specialization

Approximation Methods, Fluid Dynamics, Heat and Mass Transfer,

Academic Biography

Dr. Md. Abdul Hakim Khan is a distinguished mathematician who has been a vital member of the Department of Mathematics at Bangladesh University of Engineering and Technology (BUET) since February 15, 1992. With a career spanning several decades, Dr. Khan has established himself as a leading figure in the field.
With a wide-ranging research portfolio, Dr. Khan has made significant contributions to numerous areas of mathematics and applied sciences. His research expertise includes Approximant methods, Computational Fluid Dynamics (CFD), Stability of flows, Fluid-Thermal-Acoustic models, Natural ventilation in buildings, Traffic flow dynamics, and Bio-fluid Dynamics. His work seamlessly bridges theoretical mathematics with practical applications, resulting in numerous publications and pioneering contributions.
Dr. Khan's commitment extends beyond research; he is equally passionate about teaching. Over the years, he has imparted knowledge to a new generation of mathematicians, teaching a diverse range of subjects that include Differential Calculus, Integral Calculus, Vector Analysis, Series Solution, Algebra, Geometry, Numerical Analysis, Differential Equations, Matrices, Linear Algebra, Fluid Dynamics, Advanced Numerical Methods, and Nonlinear Dynamical Systems. His dedication to educating and mentoring students has been a cornerstone of his academic career.
Dr. Md. Abdul Hakim Khan holds a Ph.D. in Mathematics from the University of Bristol, Bristol, UK, which he earned in 2001. He was also recognized for his academic excellence with the prestigious Commonwealth Academic Staff Scholarship in 1998, awarded by the University of Bristol, UK
His proficiency in using software tools such as Fortran, Mathematica, Mathlab, and Comsol Multiphysics has enhanced his research and modeling capabilities.
With more than three decades of unwavering commitment to mathematics education, pioneering research, and academic leadership, Dr. Khan continues to shape the landscape of mathematics education and research at BUET and beyond, leaving a lasting impact on the field and the next generation of mathematicians.

Contact
Educational Information
  • Ph.D., University of Bristol, Bristol, UK , 2001
  • M.Phil., Department of Mathematics, BUET, 1998
  • M.Sc. (Thesis), University of Dhaka, 1987
  • B.Sc. (Hons) , University of Dhaka, 1986
Research Accounts
Honors And Awards
  • Commonwealth Academic Staff Scholarship 1998
  • DU Merit Scholarship 1986 - 1987
Professional Memberships
  • Editor in Chief, GANIT-J. of Bangladesh Mathematical Society. (2020-2021, 2022-2024)
  • Math. Consultant, For the New curriculum 2023-2025 NCTB
  • Member, American Society of Thermal and Fluids Engineers (ASTFE) 2018
  • Executive member, Bangladesh Mathematical Society. 2018-2019
  • Vice-president, Bangladesh Mathematical Society (2014-2015, 2016-2017)
  • Convener, BUET-Taletalk Inter University Mathematics Olympiad Committee. 2009
  • Convener, Dhaka South Regional Undergraduate Mathematics Olympiad Committee. 2015
  • Moderator, National Undergraduate Mathematics Olympiad Committee. 2016-2017
  • Initiator, Prof. Dr. Syed Ali Afzal Computational Mathematics Laboratory. BUET
Research Interests
  • Computational Fluid Dynamics
  • Theoretical and Computational Biology
All Publications
Mahtab Uddin, Mohammad Monir Uddin, Md. Abdul Hakim Khan
SVD-Krylov-based sparsity-preserving techniques to optimally stabilize the incompressible Navier–Stokes flows
Springer Nature, International Journal of Dynamics and Control, 2023
Publication Type: Journal
, Publication Link
In this work, an efficient sparsity-preserving two-sided projection technique through reduced-order modeling is proposed to optimally stabilize the incompressible Navier–Stokes flows. In most of the contemporary reduced-order modeling approaches, the authors used to employ classical sparsity exploiting balanced truncation (BT) and iterative rational Krylov algorithm (IRKA) methods. Among them, the BT method is very time-demanding due to the computation of the large-dimensional system Gramians, whereas in IRKA attaining the solution of the Riccati equation is uncertain and the stability of the reduced-order models is not guaranteed. To refrain from the above-mentioned adversities and take their pragmatic amenities by assembling a compact algorithm for reduced-order modeling through the matrix projections, the required projectors are constructed through singular-value decomposition (SVD) and the iterative Krylov subspace approach. Improved forms of the low-rank alternating direction implicit (LR-ADI) and IRKA techniques are implemented to find the left and right projectors, respectively. As a result, reduced-order feedback matrices can be attained from the solutions of the reduced-order Riccati equations, and hence the desired optimal feedback matrices to stabilize the target models can be gained by the inverse projection procedure. To validate the adaptability and accuracy of the proposed techniques, transient behaviors of the target systems are investigated through numerical simulations via MATLAB. Also, H2 error norm optimality of the proposed techniques will be verified for the target models. In the end, to exhibit the improvement in the proposed techniques, the present work will be compared with some existing works.
Mahtab Uddin, Mohammad Monir Uddin, Md. Abdul Hakim Khan
A frequency-restricted approach for reduced-order modelling of second-order linear time-invariant systems through the two-sided iterative algorithm
A F Mujibur Rahman-Bangladesh Mathematical Society National Mathematics Conference, January, 2023
Publication Type: Conference
, Publication Link
In this work, we are proposing a reduced-order modelling technique for the second-order linear time-invariant systems within a restricted frequency. In the proposed technique an updated form of through two-sided iterative algorithm is utilized with keeping the second-order system structure unaltered. Structure-preserving Sylvester equations having sparse-dense composition are solved without converting to equivalent first-order systems. The compatibility and exactness of the proposed techniques are validated through the approximation of some real-world second-order linear time-invariant systems. Numerical computation is done via MATLAB and comparative analysis is manifested graphically. Comparison is made between with and without frequency-restricted reduced-order models. Moreover, a speedup analysis is provided to confirm the amenity of the structure-preserving formulation.
Khalilur Rahman, Salma Parvin, Abdul Hakim Khan
Analysis of two-phase flow in the porous medium through a rectangular curved duct.
Research Square, Experimental and Computational Multiphase Flow, 2023
Publication Type: Journal
, Publication Link
For the best structure and reliable maintenance, two- or multiphase flow is becoming more and more essential in engineering systems. However, a variety of different of biological organisms and natural phenomena that exhibit two-phase situations can be explored to improve our knowledge of this. This is so because current industrial technology does not place these limitations. This research aims to investigate the effect of an external magnetic field on unsteady laminar incompressible two-phase flow in a porous medium via a rectangular curved duct. The relevant governing equations are represented by the Navier-Stokes equations and by the Level set equation with boundary conditions. Fluid flow through curved rectangular ducts behaves differently from fluid flow through straight ducts due to the centrifugal action generated by duct curvature. Within curved ducts, centrifugal force is generated secondary flow vortices and spiraling fluid motion. This analysis graphically depicts the fluid phase distribution, the Dean vortex, velocity contours, and fluid volume fractions. Furthermore, displayed are the effects of the aspect ratio, porosity, Dean number, radius of curvature, and Hartmann number. Additionally, a comparison of two-phase flow between various fluids is presented.
Khalilur Rahman, Salma Parvin, Abdul Hakim Khan
Analysis of Two-phase Flow through a Rectangular Curved Duct
Hill Publication, Journal of Applied Mathematics and Computation, 2023
Publication Type: Journal
, Publication Link
The present work mainly focuses on analyzing an unsteady laminar incompressible two-phase flow in a rectangular curved duct. The corresponding governing equations are represented by the Navier-Stokes equations and Level set equation with the boundary conditions. Fluid flow through curved rectangular ducts is in-fluenced by the centrifugal action arising from duct curvature and has a unique behavior different from fluid flow through straight ducts. Centrifugal force-induced secondary flow vortices and produce spiraling fluid motion within curved ducts. This paper shows the vector plot of the field flow, velocity contours, axial flow velocity and fluid volume fractions visualization graphically. The effect of curvature, Dean number and aspect ratio is also displayed. A comparison of two-phase flow between different fluids is also shown. The results reveal that the unstable behavior of the flow reduces for the increasing values of curvature, Dean number, and high viscosity flow.
Md Khalilur Rahman, Salma Parvin, Md. Abdul Hakim Khan
Analysis of two-Phase Flow in Male Medium Through a Rectangular Curved Duct
SSRN, 13th International Conference on Marine Technology (MARTEC 2022), 2022
Publication Type: Conference
, Publication Link
For their optimal structure and secure maintenance, two- or multiphase flow is becoming more and more significant in engineering systems. To better appreciate, however, two-phase situations can be observed in a broad range of biological organisms and natural phenomena. It's because modern industrial technology does not impose such limitations. This study's major objective was to investigate how well a porous medium impacted an unsteady laminar incompressible two-phase flow through a rectangular curved duct. The Navier-Stokes equations and Level set equation with the boundary conditions represent the corresponding governing equations of physical problem. Due to the centrifugal force generated by the duct's curvature, fluid flow across curved rectangular ducts behaves differentially from fluid flow across straight ducts. Centrifugal force fluids to whirl, generating new flow vortices through curved ducts. This paper shows the vector plot of the field flow, velocity contours, and fluid volume fractions graphically. The effect of curvature, Dean number and porosity also displayed. The results reveal that the unstable behavior of the flow reduces for the increasing values of curvature, Dean number, and high viscosity flow.
Md. Toriqul Islam, Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan, Md. Tanzim Hossain
Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow
Hindawi, Mathematical Problems in Engineering, 2022
Publication Type: Journal
, Publication Link
In this paper, we explore the Riccati-based boundary feedback stabilization of the incompressible Navier–Stokes flow via the Krylov subspace techniques. Since the volume of data derived from the original models is gigantic, the feedback stabilization process through the Riccati equation is always infeasible. We apply a H2 optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system. An extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA) is implemented, where the computation of an equivalent Sylvester equation is included for minimizing the computation time and enhancing the stability of the reduced-order models with satisfying the Wilson conditions. Inverse projection approaches are applied to get the optimal feedback matrix from the reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing works.
Md. Khalilur Rahman, Salma Parvin, Md. Abdul Hakim Khan
Magnetohydrodynamic Effect on Two-Phase Flow in the porous medium through a Rectangular Curved Duct
Research Square, International Journal of Applied and Computational Mathematics, 2022
Publication Type: Journal
, Publication Link
For the best structure and reliable maintenance, two- or multiphase flow is becoming more and more essential in engineering systems. However, a variety of different of biological organisms and natural phenomena that exhibit two-phase situations can be explored to improve our knowledge of this. This is so because current industrial technology does not place these limitations. This research aims to investigate the effect of an external magnetic field on unsteady laminar incompressible two-phase flow in a porous medium via a rectangular curved duct. The relevant governing equations are represented by the Navier-Stokes equations and by the Level set equation with boundary conditions. Fluid flow through curved rectangular ducts behaves differently from fluid flow through straight ducts due to the centrifugal action generated by duct curvature. Within curved ducts, centrifugal force is generated secondary flow vortices and spiraling fluid motion. This analysis graphically depicts the fluid phase distribution, the Dean vortex, velocity contours, and fluid volume fractions. Furthermore, displayed are the effects of the aspect ratio, porosity, Dean number, radius of curvature, and Hartmann number. Additionally, a comparison of two-phase flow between various fluids is presented.
Md. Toriqul Islam, Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan
Sparsity-preserving Krylov subspace-based techniques for Riccati-based feedback stabilization of incompressible Navier-Stokes models
1st International Conference on Frontier in Sciences, November, 2022
Publication Type: Conference
, Publication Link
To develop a Krylov subspace-based two-sided iterative algorithm (TSIA) to solve continuous-time algebraic Riccati equation (CARE) arising from incompressible Navier-Stokes models, and hence stabilize the target systems utilizing the reduced-order models (ROM). We propose an extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA). The necessary ROM will be attained by the sparsity-preserving TSIA approach and hence reduced-order feedback matrix will be attained. The attainment of the desired optimal feedback matrix will be followed the classical inverse projection scheme and hence the objective incompressible Navier-Stokes models will be optimally stabilized. The H_2 norm optimality will be applied to justify the accuracy of the offered strategies. The validity of the intended methodologies will be legitimated by the numerical investigation of the transient behaviors of the target models arising from engineering applications.
Mahtab Uddin, Md. Toriqul Islam, Mohammad Monir Uddin, Md. Abdul Hakim Khan
Riccati-based feedback stabilization of incompressible Navier-Stokes models via reduced-order modelling by iterative rational Krylov algorithm
IEEE, 25th International Conference of Computer and Information Technology (ICCIT), 2022
Publication Type: Conference
, Publication Link
Due to the system structure and memory requirement, stabilization of higher dimensional incompressible NavierStokes models is always a very challenging task. The conventional simulation techniques are time laborious, sparsity exploiting, and contain complex matrix-vector formulations. To bypass those infeasibilities, we are proposing a modified sparsity-preserving Iterative Rational Krylov Algorithm (IRKA) based projection technique for Riccati-based feedback stabilization of incompressible Navier-Stokes models via reduced-order modelling. In this work, reduced-order matrices and hence reduced-order feedback matrices will be used to attain the optimal feedback matrix for the target models through the inverse projection scheme. In the numerical validation, graphical approaches to the measures of transient behaviors are included. The accuracy of the work is justified by the H 2 -norm optimality.
Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan
Computationally Efficient Optimal Control for Unstable Power System Models
Hindawi, Mathematical Problems in Engineering, 2021
Publication Type: Journal
, Publication Link
In this article, the focus is mainly on gaining the optimal control for the unstable power system models and stabilizing them through the Riccati-based feedback stabilization process with sparsity-preserving techniques. We are to find the solution of the Continuous-time Algebraic Riccati Equations (CAREs) governed from the unstable power system models derived from the Brazilian Inter-Connected Power System (BIPS) models, which are large-scale sparse index-1 descriptor systems. We propose the projection-based Rational Krylov Subspace Method (RKSM) for the iterative computation of the solution of the CAREs. The novelties of RKSM are sparsity-preserving computations and the implementation of time-convenient adaptive shift parameters. We modify the Low-Rank Cholesky-Factor integrated Alternating Direction Implicit (LRCF-ADI) technique-based nested iterative Kleinman–Newton (KN) method to a sparse form and adjust this to solve the desired CAREs. We compare the results achieved by the Kleinman–Newton method with that of using the RKSM. The applicability and adaptability of the proposed techniques are justified numerically with MATLAB simulations. Transient behaviors of the target models are investigated for comparative analysis through the tabular and graphical approaches.
Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan, M. Tanzim Hossain
SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models
Journal of Engineering Advancements, August, 2021
Publication Type: Journal
, Publication Link
We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approaches do not preserve the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from the left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized computation time and ℌ2 -norm of the error system for reduced-order models of the target models.
Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan
Effect of Shift Parameters in Rational Krylov Subspace Method for Solving Riccati Equations Arise from Power System Models
Universe Publishing Group, International Journal of Material and Mathematical Sciences, 2021
Publication Type: Journal
, Publication Link
The simulation of continuous-time algebraic Riccati equations (CARE) derived from the very large power system models is a highly laborious task and most cases infeasible due to the sophisticated structural ingredients. The computation is very time costly and the rate of convergence can be severely affected in the direct solvers. To overcome those adversities, an iterative approach Rational Krylov Subspace Method (RKSM) is introduced to deal with those large-scale CAREs. The solutions of those CAREs and hence the optimal feedback matrices can be efficiently explored by the RKSM approach to stabilize the power system models of unstable categories. In this approach, shift parameters play a vital role in the convergence of the computations and size of the solution spaces. The goal of the work is to investigate the effect of different types of shift parameters on the stabilization process. To attain the mentioned objective, a modified version of the iterative RKSM algorithm is proposed by employing two types of shift parameters, namely, the adaptive ADI shifts and heuristic shifts. Qualitative discussions for the outcomes for those shift parameters are narrated by tabular and figurative methods.
Mahtab Uddin, M. Monir Uddin, Md. Abdul Hakim Khan, Md Motlubar Rahman
Interpolatory projection technique for Riccati-based feedback stabilization of index-1 descriptor systems
IOPscience, International Conference on Recent Innovations in Engineering and Technology (ICRIET), 2020
Publication Type: Conference
, Publication Link
The work aims to stabilize the unstable index-1 descriptor systems by Riccati-based feedback stabilization via a modified form of Iterative Rational Krylov Algorithm (IRKA), which is a bi-tangential interpolation-based technique. In the basic IRKA, for the stable systems the Reduced Order Models (ROMs) can be found conveniently, but it is unsuitable for the unstable ones. In the proposed technique, the initial feedback is implemented within the construction of the projectors of the IRKA approach. The solution of the Riccati equation is estimated from the ROM achieved by IRKA and hence the low-rank feedback matrix is attained. Using the reverse projecting process, for the full model the optimal feedback matrix is retrieved from the low-rank feedback matrix. Finally, to validate the aptness and competency of the proposed technique it is applied to unstable index-1 descriptor systems. The comparison of the present work with two previous works is narrated. The simulation is done by numerical computation using MATLAB, and both the tabular method and graphical method are used as the supporting tools of comparative analysis.
Mahtab Uddin, Md. Abdul Hakim Khan, M. Monir Uddin
Efficient computation of Riccati-based optimal control for power system models
IEEE, 22nd International Conference of Computer and Information Technology (ICCIT), 2019
Publication Type: Conference
, Publication Link
The computational technique for solving continuous algebraic Riccati equations governed from a very large dimensional power system with sophisticated ingredients requires highly expensive time dealings and invade by the infeasible rate of convergence. The aim of the work is mainly focused on acquiring the optimal control for the large-scale power system model and stabilizing the corresponding system through the Riccati based feedback stabilization. To achieve the desired goal, a nested iterative Kleinman-Newton (K-N) method is proposed by means of Alternative Direction Implicit (ADI) technique. The proposed algorithm will allow the structure-preserving simulations and can be efficiently applied to the perturbed systems with proper adaptation.
Mahtab Uddin, Md. Abdul Hakim Khan, M. Monir Uddin
Riccati based optimal control for linear quadratic regulator problems
IEEE, 5th International Conference on Advances in Electrical Engineering (ICAEE), 2019
Publication Type: Conference
, Publication Link
The computational approach to the continuous algebraic Riccati equations arise from large-scale power system with various components, are time expensive with the inaccurate rate of convergence. In this article, attention is mainly focused on finding Riccati based optimal control for linear quadratic regulator problem subject to the time-invariant continuous-time linear system applying rational Krylov subspace method. A novel algorithm is proposed to solve very large Riccati equation by means of nested iterative techniques. The computations will allow the sparsity pattern and can be applied within closed-loop simulations.
M. Borhan Uddin, M.M. Rahman, M.A.H. Khan, R. Saidur, Talaat A Ibrahim
Hydromagnetic Double-Diffusive Mixed Convection in Trapezoidal Enclosure Due to Uniform and Nonuniform Heating at the Bottom Side: Effect of Lewis number
Science Direct, Alexandria Engineering Journal, 2016
Publication Type: Journal
, Publication Link
This paper presents unsteady numerical results of double diffusive mixed convection flow in a trapezoidal enclosure with the uniform magnetic field effect applied in negative horizontal direction. At the bottom wall, the uniform and non-uniform heat and mass are applied while the heat and mass absorbed uniformly at the top wall. Other side walls are impermeable and adiabatic. The top wall moves along x-axis direction with a constant velocity. The transport phenomenon of this problem can be expressed by the coupled governing equation derived from the conservation of mass and momentum along with the energy equation for temperature and concentration. The finite element method (FEM) based on Galerkin weighted residual technique is used to compute the numerical result from these governing equations. The numerical computation is carried out for Lewis number (Le = 0.1–50) and Richardson’s number (Ri = 0.1–100). Computed numerical results of mass, temperature and velocity distribution are expressed graphically as iso-concentration lines, isotherm lines and streamlines respectively. Average Sherwood and Nusselt number values are used to show the mass and heat transfer rate from the heated and concentrated surface of the enclosure. It is found from the analysis that mass transfer strongly depends on Lewis number. Heat and mass transfer for uniformly heated and concentrated bottom wall is larger than the non-uniformly heated and concentrated bottom wall. Finally, a correlation has been done for average Nusselt and Sherwood numbers for both of the cases.
M. Borhan Uddin, M.M. Rahman, M.A.H. Khan , Talaat A Ibrahim
Effect of Buoyancy Ratio on Unsteady Thermosolutal Combined Convection in a Lid Driven Trapezoidal Enclosure in the Presence of Magnetic Field
Science Direct, Computers & Fluids, 2015
Publication Type: Journal
, Publication Link
In this paper we concentrate to investigate the double diffusive unsteady mixed convection flow in a trapezoidal enclosure in presence of magnetic field. The bottom wall of the enclosure is heated and concentrated uniformly (case-I) and non-uniformly (case-II) while the top wall is cooled and moved uniformly with a constant velocity. Both side walls are adiabatic and impermeable. The coupled governing equations for this phenomenon is solved numerically using weighted residual based Galerkin technique of finite element method (FEM) for Richardson’s number (Ri = 0.1–100) and Buoyancy ratio (Br = −10 to 10) at time τ = 1.0. Reynolds number, Prandtl number and Lewis number are fixed at 100, 0.71 and 10 respectively. Streamlines, isotherm lines and iso-concentration lines are used to show the result graphically for velocity, temperature and mass distribution respectively. Nusselt and Sherwood number values are presented graphically to show the heat and mass transfer rate from the bottom surface of the cavity.
M. Borhan Uddin, M.M. Rahman, M.A.H. Khan
Hydromagnetic Double-Diffusive Unsteady Mixed Convection in Trapezoidal Enclosure Due to Uniform and Nonuniform Heating at the Bottom Side
Taylor & Francis , Numerical Heat Transfer, Part A: Applications, 2015
Publication Type: Journal
, Publication Link
This article focuses on exploring the effect of the magnetic field on double-diffusive mixed convection in a lid-driven trapezoidal enclosure with a uniformly (case I) and nonuniformly (case II) heated and concentrated bottom wall for unsteady flow. Numerical calculation is carried out for different values of Hartmann number (0–50) and Richardson number (0.1–100) at time τ = 1 using the finite-element method (FEM). Temperature, mass, and velocity distribution are expressed graphically as heatlines, masslines, and stream function, respectively. Nusselt and Sherwood numbers are used to demonstrate heat and mass transfer graphically. Better heat and mass transfer were found for case I than case II.