Iterative Methods for Nonlinear Systems

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

The solution of a system of nonlinear algebraic or transcendental equations is probably one of the most difficult problems in numerical analysis. In general, it is almost impossible to find the solution, unless an approximate value is already known. Linear equations occur more frequently in numerical calculations, simply because it is difficult and sometimes impossible to solve the corresponding nonlinear problem. Most of the equations governing the physical world are nonlinear, but very often for simplicity and tractability linear models are considered. Solution of boundary value problems for differential equations also lead to a nonlinear equation or a system of nonlinear equations. Solution of a system of nonlinear equations also arises in numerical analysis, while deriving certain formulae. As we have seen in Chapter 6, the Gaussian quadrature formulae can be obtained by solving a system of nonlinear equations.

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

The problem of numerical solution of systems of nonlinear algebraic equations with even powers is studied in this paper. Nondifferentiable (nonsmooth) unconstrained minimization problems are associated with the system under consideration, with objective functions, based on discrete l 1 - and l ∞ - norm, respectively, by using these two norms as proximity criteria, and the considered system is solved by minimizing the sum of absolute values of residuals, in the case of l 1 - norm, or by minimizing the maximal residual, in the case of l ∞ - norm, respectively. Also, a “differentiable” unconstrained minimization problem (least squares problem) is associated with the considered system of nonlinear equations, with objective function, based on the discrete l 2 -norm, and the considered system is solved by minimizing the sum of squares of the residuals. Subgradients of the objective functions of unconstrained minimization problems are calculated, and a subgradient method for solving the associated problems is presented. The iterative gradient method for solving the corresponding “differentiable” discrete least squares problem, based on l 2 - norm, is also outlined, and gradients of the objective function of the corresponding “differentiable” unconstrained minimization problems are calculated.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

98/03268 Reopening of pulverized coal injection in blast furnace after a pause

99/00738 On the combustion of heat pump with solar steam turbine

99/00035 ESR study of the macromolecular structure of copper-porphyrin-labeled coal

A theoretical analysis of steady and unsteady magnetohydrodynamic (MHD) flow of blood through a stenosed artery with variable viscosity and slip conditions is presented in this work. Slip conditions and variable viscosity dependent on hematocrit for both steady and unsteady blood flow models are taken into consideration in this study. The governing equations have been derived by treating blood as incompressible magnetohydrodynamic (MHD) Newtonian fluid. Galerkin Weighted Residual method, Newton Raphson method and fourth order Runge-Kutta method are used to solve the non-linear differential equations for velocity profiles of the blood flow, flow rate, shear stress and flow resistance for different values of hematocrit parameter, magnetic field parameter and slip velocity. The Galerkin Weighted Residual method is applied to transform the partial differential equations describing the flow into systems of non-linear equations and system of non-linear first order differential equations for the time independent and time dependent MHD flow models respectively. Newton Raphson’s method with the aid of maple software is used to solve the system of non-linear equations while the fourth order Runge-Kutta method is used to solve the system of non-linear first order differential equations. The research analyzed how the flow velocity, flow rate, shear stress and resistance to fluid flow respond to changes in some key parameters such as hematocrit parameter, magnetic field parameter and slip velocity. The results as presented graphically reveal that flow velocity and flow rate decrease while shear stress and resistance to flow increase when the hematocrit parameter increases. Also, flow velocity, flow rate and shear stress increase while resistance to flow decrease when the slip velocity increases. Finally, the increase in magnetic field parameter led to a decrease in flow velocity, flow rate and shear stress but increased the flow resistance. The flow velocity of the unsteady blood flow model is higher when compared to steady blood flow model.

This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

A mathematical model and numerical method for simulation of the continuous casting process in a variable in time domain are presented. The variable geometry of the slab is caused by the change in time of the width of the mould. The mathematical model of the process is a Stefan problem with prescribed convection and non-linear Robin boundary condition. Considered differential equation is approximated by a finite difference scheme, which is constructed in several steps. First, a semi-discrete problem is constructed using the method of characteristics with respect to the time variable. Then, at each time level, the current elliptic problem in a curvilinear domain is replaced by a problem in the parallelepiped domain using the fictitious domain method. Finally, the boundary-value problem in the parallelepiped domain is approximated by a finite-difference scheme. The constructed fully discrete problem in algebraic form is a system of nonlinear equations containing a diagonal monotone operator and a linear part with a symmetric and positive definite $$M$$ -matrix. To solve the resulting system of nonlinear algebraic equations, well-known iterative solution methods can be applied.

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.

This paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.

Self-consistent amplitude profile of ducted VLF transmitter signal due to resonant interaction with energetic electrons in the magnetosphere

Existence of solutions for a singular system of nonlinear fractional differential equations