A Numerical Model for Steel Continuous Casting Problem in a Time-variable Domain

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.

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.

The influence of the temperature dependence of the thermal conductivity and heat transfer coefficients, as well as of the emissivity of a dielectric film and of an insulating base on formation of a stationary temperature field of the dielectric–base system exposed to irradiation by a continuous ion beam in air, has been studied theoretically. Analytical expressions have been obtained for the temperature fields of a sample and base, as well as a system of interrelated nonlinear algebraic equations for characteristic temperatures. With the aid of numerical solution of a system of nonlinear algebraic equations, the dependences of the temperature of the irradiated surface of the sample–base system and of the rear side of this system on the incident ion beam parameters have been found. It has been established that these dependences are nonlinear. It has been found that heating of a sample, including its irradiated surface, decreases substantially with increase in the thermal conductivity of the base.

The melting process with moving boundary for mixture consisting of two fluids

Modified bounded homotopies to enable a narrow bounding zone

Systems of nonlinear algebraic equations have a special place in solving modeling and calculation problems in various areas of construction Sciences. These are tasks for calculating building structures, calculating distributed loads, mass and heat transfer tasks, calculating engineering networks, and many other computational and optimization tasks related to the construction industry. As is known, there are a sufficient number of fundamentally different methods for solving systems of nonlinear algebraic equations, which is related to the specifics of the systems, their degree of nonlinearity and conditionality. Non-linear ill-conditioned systems are particularly difficult; an erroneous choice of the method for solving such a system can lead to significantly distorted or simply incorrect results, which is absolutely unacceptable when performing the calculated stages of solving construction problems. In this paper, we propose to use a modification of the well – known Levenberg – Marquardt method developed by the authors, based on the regularization of the Jacobi matrix used in the classical Newton method, to solve complex, ill-conditioned systems of nonlinear equations. The modified method allows solving poorly defined systems and can significantly reduce the amount of calculations needed to ensure the required accuracy by reducing the number of iterative procedures. The paper presents a detailed description of the algorithm, given the solution of the model problem and, as an example the task of modeling and optimization of the port berthing facilities – mooring wall of massive masonry with various add-ons. When composing mathematical equations, the method of limit States was used, which is generally accepted for calculating such port hydraulic structures. The choice of the optimum scheme was carried out by minimizing the cost of structure during limit conditions the reliability of structures when using existing construction materials. The most significant geometric dimensions of the structure were selected using independent parameters to be optimized.

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

A multiple scale time domain collocation method for solving non-linear dynamical system

A numerical approach is defined to get the solution of the regularized long wave (RLW) equation. The hyperbolic- trigonometric tension (HTT) B-spline is adapted as a trial function to the Galerkin method. Space integration of the RLW equation is managed trough HTT B-spline Galerkin method over the finite elements. Resulting system of ordinary differential equations is integrated using the Crank Nicolson technique. Thus RLW equation is fully integrated into system of nonlinear algebraic equations. Solution of this system gives the approximate solution of the RLW equation over the given problem domain.

We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

In this paper, a residual-norm based algorithm (RNBA) is applied to solve determinate/indeterminate systems of nonlinear algebraic equations. The RNBA is derived from a manifold and defined in terms of a squared residual norm and a fictitious time variable, from which a robust iterative algorithm with either fixing or adjusting parameters can be obtained. Besides, some convergent indexes, such as manifold factor A0 and ratio of residual errors S, are defined to indicate the manifold attracting effect. Through the convergent indexes, the convergent mechanism of the RNBA displays a Hopf bifurcation when approximating the true solutions. Several numerical examples, including the root finding in two-, three-variable and in elliptic-type partial differential equations (PDEs), are examined. Comparisons of numerical results and exact solutions show that the proposed algorithm has good computational efficiency and accuracy simultaneously.

The process of building mathematical models of static objects and systems, which includes the solution of two problems: structural and parametric identification, is considered. At the same time, the task of identifying the structure of the model is more difficult and primary. The problem of structural identification of interval models of the characteristic of a static object is the problem of multiple solving of problems of parametric identification of this model, and therefore, from a computational point of view, it is NP complex. The procedure for finding the optimal structure of the model is considered as a directed selection of a defined set of structures in such a way as to minimize the number of iterations of the formation of interval systems of nonlinear algebraic equations. The article formulates the problem of structural identification of interval models of static objects, as a problem of repeatedly searching for solutions of interval systems of nonlinear algebraic equations, in the form of optimization problems with a nonlinear objective function and nonlinear constraints. For the first time, a method of structural identification of interval models of the characteristics of static objects based on the analysis of interval data is proposed and substantiated, which, unlike the existing ones, is based on self-organization and self-adaptation procedures of computing procedures by analogy with artificial bee colony (АBC), which gives the ability to implement model structure identification procedures with lower computational complexity and obtain interval models with simpler structures compared to known methods. The proposed method was tested on the example of building an interval model of the characteristics of a small hydroelectric plant for the purpose of research and ensuring the maximum efficiency of the use of hydropower resources, which demonstrated the effectiveness of using computational procedures based on the artificial bee colony. Accordingly, the proposed method makes it possible to obtain simple, from the point of view of complexity, interval models of complex static objects with a given guaranteed accuracy and with a lower computational complexity of identifying these models. Such features of the method ensure the effective development of the mathematical apparatus, which is used both in decision-making processes and in the processes of preparing decisions in intellectualized data-oriented systems.

Model of the nitrogen transformation cycle

Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics