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.

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:

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]:

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].

2- Group Vision

4- General Objectives:

6- Suggested Researches by the group:

8- The Expected Outcome: Proposed group will be able to: