A Shooting Method for Fully Implicit Index-2 Differential Algebraic Equations

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.

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.

Modified bounded homotopies to enable a narrow bounding zone

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 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.

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.

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.

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

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.

Chapter 5 - Life Is Really Nonlinear

Modern energy and electronic systems comprise a large number of nonlinear elements operating in dynamic modes. Accurate modeling of such systems requires proper consideration of nonlinear current-voltage characteristics (CVC) and transient processes, which is especially important when designing rectifiers, inverters, control systems and other power electronics devices. Existing electrical circuit simulators do not always provide users with the necessary flexibility, scalability, or compatibility with enterprise safety standards, and may have legal restrictions. Custom effective modeling methods for such systems allow for creating specialized software solutions to analyze dynamic modes of electrical circuits, free from the limitations of commercial simulators and tailored to specific engineering and scientific tasks. The purpose of the work is to develop a method for numerical modeling of dynamic modes of electrical circuits comprising semiconductor diodes or other elements with nonlinear CVC, described by nodal equations. The scientific novelty lies in the development of a homotopy approach to overcoming high condition number of the Jacobian matrix when solving nonlinear differential-algebraic equations (DAEs) of electrical circuits, based on deformation of CVC with adaptation of the deformation parameter depending on the residual norm and condition number of the matrix; in the development of a method for extracting linearly independent differential equations from DAEs of electrical circuits without preliminary analysis of their topology; in the development of a universal stamp of a nonlinear element, based on linearization of the functional in the vicinity of the current approximation, allowing for integration of diode models with various CVCs into nodal equations. Materials and methods. Theoretical electrical engineering methods were used in the work, including the modified nodal potential method. The proposed numerical modeling method involves integration of elements with nonlinear CVC into nodal equations; extraction of differential and algebraic parts from DAEs, based on singular value decomposition of the matrix standing before the derivative vector; transformation of DAEs into a nonlinear system of algebraic equations using backward differentiation formulas (BDF) with variable time step. Initial points for BDF were determined by the diagonally implicit Runge–Kutta method of second order accuracy. Numerical solution of the obtained nonlinear equations was performed by the damped Newton–Raphson method. To reduce the condition number of the Jacobian matrix in transient modes, when the spread of differential conductivities reaches 12 orders and higher, a homotopy approach was proposed, consisting of gradual deformation of the diode CVC from a smoothed to the original curve during convergence, while maintaining a given value of the condition number. Results. To demonstrate the proposed solutions, computer simulation of a bridge rectifier operating on an active-inductive load with two types of diode CVC was performed: piecewise-linear and smooth, corresponding to the Shockley equation with series resistance. The deformation parameter and damping coefficient were adaptively changed depending on the residual norm of the functional and the condition number of the Jacobian matrix. Comparison of simulation results with different methods of specifying diode CVC showed that differences appear predominantly in transient processes of switching diode operation modes. It has been found that to ensure convergence of numerical solution in diode switching modes, characterized by high condition number of the Jacobian matrix, the homotopy approach is more effective than diagonal regularization. The proposed method for numerical modeling of dynamic modes of electrical circuits with nonlinear elements has a natural algorithmic structure, allowing for simple software implementation. Conclusions. 1. The most universal diode stamp, obtained on the basis of linearization of the functional derived from the CVC equation in the vicinity of the current approximation, has been identified. 2. A method for extracting linearly independent differential equations from DAEs of electrical circuits without preliminary analysis of circuit topology has been proposed. 3. A method for calculating the Jacobian matrix for solving nonlinear DAE has been proposed. 4. To ensure convergence of numerical solution with high condition number of the Jacobian matrix, it is preferable to apply the homotopy approach.

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

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.

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.

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