Mathematical Modelling and Applied Computation (MMAC)

Head of the Research Group

Dr. Ali Saleh Alshomrani

Deanship of Scientific Research
King Abdulaziz University, Jeddah, Saudi Arabia
Email: aszalshomrani@kau.edu.sa

Members


Academic Title Names of the group members (Full Name) Email
Associate Professor Sivasankaran Sivanandam Email: smsivanandam@kau.edu.sa
Assistant Professor Ramandeep Behl Email: rlal@kau.edu.sa
Assistant Professor Malik Zaka Ullah Email: zmalek@kau.edu.sa
Assistant Professor Abdulh Khamees Alzahrani Email: akalzahrani@kau.edu.sa
Asocciate professor Faris Saeed Alzahrani falzahrani1@kau.edu.sa
Assistant Professor Fouad Mallawi fmallawi@stu.kau.edu.sa
Assistant Professor Majed Alshaeri malshaere@kau.edu.sa
Assistant Professor Metib Alghamdi dr.matabalghamdi@kau.edu.sa
Assistant Professor Noufe H. Aljahdaly nhaljahdaly@kau.edu.sa
Assistant Professor Hessah Fiehan Al-Qahtani hfalqahtani@kau.edu.sa
Assistant Professor Sudip Samanta hhaalsalmi@kau.edu.sa

Abstract

To meet the challenges of modern time new inventions in all fields of life are required where mathematics has basic role. Here the application of modern approximation techniques; to explain how, why, and when they can be expected to work; and to provide a foundation for further study whether it is industry, engineerin, physics, biology, chemistry or medical sciences using numerical analysis and scientific computing will be required. One of the most fundamental and earliest problems of numerical methods concerned with the solution of non-linear models from various desciplines in science and engineering. Analytical method for solving such complicated problems are almost nonexistent or very rare in literature. We must turn towards the iterative procedures for their solutions. In addition, these numerical techniques have the capability to handle real life problems e.g., aerodynamics, nanotechnology, civil engineering, chemical and food processing, heat and mass transfer, biology, etc. These problems has been solved by either modified or standard numerical techniques (e.g., finite difference, finite element, Newton’s and control volume methods, Lattice Boltzmann method, etc). Funding is key factor for smooth running of the research group. Research capacity grant is required for basic research conducted by research teams composed of local Saudi Arabian and international universities. This grant will be motivated by the desired to provide a stable source of funding to promising teams of Saudi and international researchers, who can advances in science & technology to achieve Saudi Arabia’s broader national objective.

1- Description of the Proposed Research Group:

Mathematics "the mother of sciences" has lead science and engineering from centuries. Improvement in this subject to meet the challenges from time to time was required. Analytical solutions were computed for deep understanding. But when these analytical solutions were impossible to find or found to be difficult/cumbersome for analysis, mathematicians decided to go for approximation techniques for their solutions. In this regard the invention of electronic computer supported a lot to obtain these approximate solutions for practical problems.

Numerical Analysis

Numerical analysis is concerned with finding efficiently and accurately solutions of nonlinear models. Nonlinearity appears frequently in real world problems, which are modelled by different branches of science and engineering. These models try to simplify the original problem in terms of nonlinear algebraic and differential equations or system holding the main properties. So, they usually remain nonlinear. In the literature, we can find several examples where we can see the applicability of these iterative methods to the real-world problems. For example, nonlinear model like variational inequalities, the Bratu problem, a shallow arch, etc. However, most of them are phrased in terms of system of nonlinear equations. Recently, Rangan et al. [1] discussed the applicability of the nonlinear system on the problem of investigating coarse-grained dynamical properties of neuronal networks in kinetic theory. In addition, Nejat and Ollivier-Gooch [2], presented the problem to study the effect of discretization order on preconditioning and convergence of high-order Newton-Krylor unstructured flow solver in computational fluid dynamics. On the other hand, Grosan and Abraham [3] also showed the applicability of the system of nonlinear equations in neurophysiology, kinematics syntheses problem, chemical equilibrium problem, combustion problem and economics modeling problem. Very recently, Awawdeh & Tsoulos and Stavrakoudis [4-5], solved the reactor and steering problems by phrasing them in the system of nonlinear equations. Moreover, Lin et al.[6] and J.J. Moré [7] also discussed the applicability of the system of nonlinear equations in transport theory. Either lack or intractability of their analytic solutions often forced to researchers to resort an iterative method. Therefore, the use of iterative methods to approximate the solutions of these types of problems has revealed as a fruitful area of research. This is one of the main reasons that scholars from the worldwide are trying their best to resort an iterative method from the past few decades. In addition, iterative methods provide an approximated solution corrected up to a specified degree of accuracy which is further depend upon the chosen iterative methods and programming software namely, FORTRAN, Maple, MATLAB, Mathematica, etc. Moreover, researchers have to face several problems while using these iterative schemes and some of them are related to slow convergence, non-convergence, oscillation problem close to the initial guess, divergence, failure etc. (for the detail explanation of these problems please see Ostrowski 1960 [8], Traub 1964 [9], Ortega and Rheinboldt 1970 [10], Burden and Faires 2001 [11], Petkovic et al. 2012[12]). Notice that, in-particular there is a plethora of iterative methods for approximating solutions of the nonlinear models available in the literature. These results show that initial guess should be close to the required root for the convergence of the corresponding methods. But, how close initial guess should be required for the convergence of the corresponding method? These local results give no information on the radius of the ball convergence for the corresponding method. Further, most of the researchers proved the order of convergence of the corresponding iterative method with the help of Taylor series expansion under the hypotheses on the second and third-order derivative or even higher of the involved function. However, only first-order derivative of the involved function or no derivative appears in their proposed scheme. In this way, they put restriction on the applicability of the corresponding method. In many practical situations where it is preferable to avoid the calculation of derivative of the involved function F(x) with n unknown variables or complicated inverse of Jacobian matrix at different points for nonlinear because it increases the computational cost. A method is known as efficient method which consumes minimum cost but attains higher-order convergence. Therefore, our principle aim is not only to provide some new higher-order multi-point iterative methods for solving nonlinear scalar equation and system of nonlinear equations which will be better than the existing ones of the same order. But, also try to answer of the above challenges.

Therefore, we will try to propose the new schemes with the following objectives:

  • We will try to propose weak convergence conditions.
  • We will provide some counter example where the proceeding results were not suitable.
  • Try to present the convergent radii that will provide the sure convergence.
  • Try to propose the bounds of error of the function G by applying Lipschitz conditions.
  • With the help of Banach space setting and Lipchitz constants, we will suggest the local convergence of earlier scheme.
  • We will expand the suitability of iteration function by reducing the hypothesis.
  • We will also try to find the stability analysis of the proposed schemes for nonlinear equations.
  • Try to attain faster convergence and simple body structure for nonlinear systems.
  • Try to propose a new efficient family of higher-order methods for nonlinear system with only one inverse Jacobian matrix at per iteration, which convergence to the required root with minimum computational cost.
  • Try to develop some new efficient higher-order derivative free methods for nonlinear system of equations.
  • Boundary layer flow

    The boundary layer is a significant idea in aerodynamics, its discovery advanced the science of aerodynamics to a useful tool in predicting flows. Also, it is an important part of engineering now. But, boundary layer solutions are specialized to a specific type of flow, which can help a human to understand the nature of the flow better. For example, if an airfoil experiences a separation problem, a boundary layer solution may be more useful to an engineer trying to diagnose the problem, because one is more intimate with the data and the equations used to calculate them. The study of natural/mixed/forced convection flow of an incompressible fluid past a heated (moving or stationary) surface has attracted the interest of many researchers due to many applications such as cooling of nuclear reactors, the boundary layer control in aerodynamics, crystal growth, food processing and cooling towers. Research on boundary layer flow with heat and mass transfer has been conducted by several researchers in various aspects [13-18]. The boundary layer equations are highly nonlinear and it is not possible to solve analytically. Semi-analytical and numerical methors are used to solve these boundary layer equations, for example, shooting method with Runge-Kutta algorithm, finte difference method, homotopy analysis method, perturbation methods, transform methods.

    Computational Fluid Dynamics

    The Computational Fluid Dynamics (CFD) uses applied mathematics and numerical simulation using powerful computers to model fluid flow situations. As a developing science, CFD has been received extensive attention throughout the international community since the advent of the digital computer. There has been considerable growth in the development and application of CFD to all aspects of fluid dynamics from last three decades. As a result, CFD has become an integral part of the engineering design and analysis environment of many companies because of its ability to predict the performance of new designs or processes [19-20].

    In design and development, CFD programs are considered to be standard numerical tools which predict not only fluid flow behaviour, but also the transfer of heat, transfer of mass (such as in perspiration or dissolution), phase change (such as in freezing, melting or boiling), chemical reaction (such as combustion or rusting), mechanical movement (such as an impeller turning, pistons, fans or rudders), stress or deformation of related solid structures (such as a mast bending in the wind) and in environment and architecture [21-22]. An analysis of heat transfer is of great interest for technical researchers, engineers, developers, designers and manufacturers because many processes such as petroleum reservoirs, nuclear waste disposal, chemical reactor catalytic and others utilized heat transfer. CFD Simulation is used for optimization of HVAC designs, validation of diverse design parameters for example, the location, the number of exhausts and diffusers, flow rate and temperature of supplied air for meeting the design criteria. It helps in designing of the smoke system, cooling applications, ventilation systems, etc. [23]. Now-a-days control volume and finite element methods are widely used in CFD calculations [24-25].

    CFD has been used in several areas in our daily life. Some of them are [26]:

  • Construction & Environment
  • Chemical & Petrochemical Processes
  • Defence & Security
  • Food & Beverage
  • Electronic Equipment design & cooling
  • Renewable Energy (geothermal, wind power, hydropower)
  • Marine engineering & Ship Building
  • Sports
  • The CFD modelling is a very chanlanging task for the above mensioned applications. The Navier-Stokes equations are used to model for flow situation as well as the energy equation is used to thermal transport [23-26].

    Mathematical Biology

    Mathematical biology is a branch of mathematics that deals with the applications of mathematics to understand biological phenomena [30, 31]. It is a multidisciplinary research-field that combined mathematics with biology as the name suggests. Mathematical biology employs theoretical analysis, mathematical models and live abstractions to investigate the principles that govern the interaction, structure, development and behaviour of the systems. Most biological processes are characterized by complex, non-linear and super-complex mechanisms, as it is increasingly recognized that the result of such interactions can only be understood through a combination of mathematical, logical, physical / chemical, molecular and computational models. Mathematical biology covers research fields such as epidemics, evolutionary biology (ecology), complex systems biology (cellular and molecular biology), neuroscience (neural network), theoretical biophysics (deterministic and stochastic processes and spatial modelling), etc.

    Mathematical biology aims at the mathematical representation and the modelling of biological processes, using techniques and applied mathematics tools. It has both theoretical and practical applications in biological, biomedical and biotechnological research. Describing systems quantitatively means that their behaviour can be simulated better and, therefore, it is possible to predict properties that may not be evident to the experimenter. This requires precise mathematical models subjected to mathematical analysis.

    2- Group Vision

  • Make the group an internationally known research group by providing new environment & deeper research and development partnership between Saudi and international researchers to advance research of strategic importance to the Kingdom, in line with Vision 2030.
  • Provide research environment of global level.
  • Coordination between group members and experts.
  • 3- Mission of the proposed research group

  • To introduce modern approximation techniques; to explain how, why, and when they can be expected to work; and to provide a foundation for further study of numerical methods and scientific computing.
  • Enhance student’s research experties to complete their projects by enrolling in either MSc or PhD programs in Mathematics or in Scientific Computing by providing the potential supervisor/active researcher to become valueable member of society.
  • To bridge the gap between the Mathematics and its users from other fields (engineering, geology, petroleum, Banking, Insurance, cyber crime, medical sciences and industry).
  • Hunting talent through post doctoral progam.
  • Periodically seminars and colloquium series.
  • 4- General Objectives:

  • Ministry of Education in Kingdom of Saudi Arabia is seeking to develop general and basic skills of all students to enable them, to face modern life requirements, in addition to specialized skills for each profession that covers all professional fields for young generation.
  • Start of teacher training program for the fresh graduates to serve in the reputed schools and colleges for the Kindom of Saudi Arabia.
  • Research training and report/article writing skills for graduate and postgraduate students.
  • Emphasizing next step to include joint proposals development and submission to the international collaboration fund for prospective funding opportunities
  • Organizing seminars, workshops and conferences on specific topics.
  • Increase number of publications in top level (Q1) journals.
  • 5- Specific Objectives:

  • To provide new iterative techniques better than the existing ones in terms of residual error, COC, CPU time, Efficiency, etc.
  • To provide the theoretical radius of convergence of iterative methods with the help of Banach space.
  • To optimize the position and size of the heat source and the sink.
  • To enhance the heat transfer in enclosures.
  • Explore the impact of different Newtonian and non-Newtonian fluid models in specific situations.
  • 6- Suggested Researches by the group:

  • Dumitru Baleanu
  • Anjan Biswas
  • Masood Khan
  • Joydev Chattopadhyay
  • Juan Ramón Torregrosa
  • 7- The Importance of the Work of Proposed Research Group to the University and the Society:

  • With the help of teacher training program, we can provide expert faculty to serve in the reputed schools and colleges for the Kindom of Saudi Arabia. This will help to produce expert doctors, practical engineers and scientists with real applications.
  • Providing a model for improving education in Kindom of Saudi Arabia for other fields of science and engineering through teacher training, certification course, and research participation programs.
  • This group will also enhance the knowledge of existing teachers from schools and colleges by providing a good foundation for students to study science and engineering.
  • Strengthening overall mathematics education in the Kingdom if these programs are expanded into all areas of science, e.g., Chemistry, Physics, Zoology, Botony and different desciplins of Engineering at all levels.
  • This group will formulate a committee of experts for developing/modifying effective mathematics curriculum to improve level of knowledge in Kingdom. This provide good series of text books and e-learning materials.
  • Group experts will support the young scholars in writing manuscripts and reports.
  • This program will increase number of good publications to enhance the glory of KAU.
  • Creating and open environment, in which Saudi and international researchers can discuss their area of research interest and identifying areas of overlapping their mutual interest.
  • 8- The Expected Outcome: Proposed group will be able to:

  • Provide good teachers for schools and colleges.
  • Produce good researchers of international level.
  • Find solutions for "unsolved real-life problems" through new invented numerical techniques.
  • Implementing new algorithms to solve/improve the industrial problems.
  • Make secure data communation through coding theory application.
  • Provide a universal plateform to Saudi researchers.
  • Increase the ranking of KAU by supporting through number of publications in ISI listed journal of good fame.
  • Provide training for the development of upto date mathematics curriculum and series of standard books for schools and colleges according to their needs through dedicated academicians.
  • Provide plateform to the Saudi and international researchers for strengthening their ties and to discuss new ideas to meet the challenges.

    Last Update
    10/19/2023 9:05:35 AM