Numerical iterative methods for nonlinear problems.

The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms

An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model

An efficient numerical iterative method is constructed for the static deflection of an infinite beam on a nonlinear elastic foundation. The proposed iterative scheme consists of quasilinear method (QLM) and Green’s function technique. The QLM translates the nonlinear ordinary differential equation into iterative linear ordinary differential equation. The successive iterations of quasilinear form of ordinary differential equation (ODE) show the quadratic convergence if an initial guess is chosen in the neighbourhood of true solution. The Green’s function technique converts the differential operator into an integral operator and the integral operator is approximated by discrete summation which finally gives us an iterative formula for the resulting set of algebraic equations.The numerical validity and efficiency are proved by simulating some nonlinear problems.Peer ReviewedPostprint (published version

An Efficient Computation of Effective Ground Range Using an Oblate Earth Model

An effcient method is presented to calculate the ground range of a ballistic missile trajectory on a nonrotating Earth. The spherical Earth model does not provide good approximation of distance between two locations on the surface of Earth. We used oblate spheroid Earth model because it provides better approximations. The effective ground range of a ballistic missile is an arc-length of a planner elliptic (or circle) curve which passes through the launch and target points on the surface of Earth model. A general formulation is presented to calculate the arc-length of an elliptic (or circle) curve which is the intersection of oblate Earth model and a plane. Explicit formulas are developed to calculate the coordinates of center of the ellipse as well as major and minor axes which are necessary ingredients for the calculation of effective ground range

A preconditioned iterative method for solving systems of nonlinear equations having unknown multiplicity

A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions.Peer ReviewedPostprint (published version

A Numerical Simulation for Darcy-Forchheimer Flow of Nanofluid by a Rotating Disk With Partial Slip Effects

This study examines Darcy-Forchheimer 3D nanoliquid flow caused by a rotating disk with heat generation/absorption. The impacts of Brownian motion and thermophoretic are considered. Velocity, concentration, and thermal slips at the surface of the rotating disk are considered. The change from the non-linear partial differential framework to the non-linear ordinary differential framework is accomplished by utilizing appropriate variables. A shooting technique is utilized to develop a numerical solution of the resulting framework. Graphs have been sketched to examine how the concentration and temperature fields are affected by several pertinent flow parameters. Skin friction and local Sherwood and Nusselt numbers are additionally plotted and analyzed. Furthermore, the concentration and temperature fields are enhanced for larger values of the thermophoresis parameter

A fast and efficient Newton-type iterative scheme to find the sign of a matrix

This work proposes a new scheme under the umbrella of iteration methods to compute the sign of an invertible matrix. To this target, a review of the exiting solvers of the same type is given and then a new scheme is derived based on a multi-step Newton-type nonlinear equation solver. It is shown that the new method and its reciprocal converge globally with wider convergence radii in contrast to their competitors of the same order from the general Padé schemes. After investigation on the theoretical parts, numerical experiments based on complex matrices of various sizes are furnished to reveal the superiority of the proposed solver in terms of elapsed CPU time

Heavy metals in selected vegetables from markets of Faisalabad, Pakistan

Two hundred ten samples of selected vegetables (okra, pumpkin, tomato, potato, eggplant, spinach, and cabbage) from Faisalabad, Pakistan, were analyzed for the analysis of heavy metals: cadmium (Cd), lead (Pb), arsenic (As), and mercury (Hg). Inductively coupled plasma optical emission spectrometry was used for the analysis of heavy metals. The mean levels of Cd, Pb, As, and Hg were 0.24, 2.23, 0.58, and 7.98 mg/kg, respectively. The samples with Cd (27%), Pb (50%), and Hg (63%) exceeded the maximum residual levels set by the European Commission. The mean levels of heavy metals found in the current study are high and may pose significant health concerns for consumers. Furthermore, considerable attention should be paid to implement comprehensive monitoring and regulations