Differential — Algebraic Equations

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.

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.

Transient simulation becomes a bottleneck for modern IC designs due to large numbers of transistors, interconnects and tight design margins. For modified nodal analysis (MNA) formulation, we could have differential algebraic equations (DAEs) which consist ordinary differential equations (ODEs) and algebraic equations. Study of solving DAEs with conventional multi-step integration methods has been a research topic in the last few decades. We adopt matrix exponential based integration method for circuit transient analysis, its stability and accuracy with DAEs remain an open problem. We identify that potential stability issues in the calculation of matrix exponential and vector product (MEVP) with rational Krylov method are originated from the singular system matrix in DAEs. We then devise a robust algorithm to implicitly regularize the system matrix while maintaining its sparsity. With the new approach, $\varphi$ functions are applied for MEVP to improve the accuracy of results. Moreover our framework no longer suffers from the limitation on step sizes thus a large leap step is adopted to skip many simulation steps in between. Features of the algorithm are validated on large-scale power delivery networks which achieve high efficiency and accuracy.

Rational general solutions of systems of first-order algebraic partial differential equations

In his algebraic theory of differential equations, J. F. Rittf has developed a decomposition theory for systems of algebraic differential equations by introducing the idea of irreducible systems and proving that every system is equivalent to one and essentially only one finite set of irreducible systems.The analogous theorem for algebraic difference equations was given by Ritt and Doob.fThe purpose of the present paper is to derive a decomposition theorem for algebraic mixed difference equations; i.e., equations which contain algebraically one or more unknown functions y(x), their "transforms" y(x), y(x+l), y(x+2), ---, and the derivatives of those transforms.

The numerical solution of a system of nonlinear algebraic or transcendental equations is examined within the framework of the parameter continuation method. An earlier result of the author according to which the best parameters should be sought in the tangent space of the solution set of this system is now refined to show that the directions of the eigenvectors of a certain linear self-adjoint operator should be used for finding these parameters. These directions correspond to the extremal values of the quadratic form associated with the above operator. The parametric approximation of curves and surfaces is considered.

This paper proposes the implementation of dynamic power systems as a system of differential algebraic equations (DAEs) and to solve it by means of DAE solvers, because it is easier and more natural to write the equations as a system of DAEs than to write them as a system of ordinary differential equations (ODEs). Problem-solving environments have recently included routines in their packages to solve DAEs. One of these routines has been used to simulate a power system composed of a nonlinear, three-phase, three-legged transformer, a load and all their possible connections. The magnetic transformer relations and the connections give some of the algebraic equations of the DAE system.

The simulation of a high-temperature gas-cooled reactor pebble-bed module (HTR-PM) plant is discussed. This lumped parameter model has the form of a set differential algebraic equations (DAEs) that include stiff equations to model point neutron kinetics. The nested approach is the most common method to solve DAE, but this approach is very expensive and time-consuming due to inner iterations. This paper deals with an alternative approach in which a simultaneous solution method is used. The DAEs are discretized over a time horizon using collocation on finite elements, and Radau collocation points are applied. The resulting nonlinear algebraic equations can be solved by existing solvers. The discrete algorithm is discussed in detail; both accuracy and stability issues are considered. Finally, the simulation results are presented to validate the efficiency and accuracy of the simultaneous approach that takes much less time than the nested one.

In this paper, the regularization problem is investigated for nonlinear differential-algebraic equation (DAE) systems with unknown parameters, which appear linearly in both differential and algebraic equations. It is shown that the feasibility of the proposed algorithms guarantees the existence of a feedback controller so that the resulting closed-loop systems admit equivalent ordinary differential equation (ODE) systems with lower triangular forms. As an application case of DAE systems, a constrained manipulator with flexible joints is studied to illustrate the proposed methodology.

ICS: a package for computing involutive characteristic set of algebraic partial differential equation systems

In this paper, we solve higher index differential algebraic equations (DAEs) by transforming them into integral algebraic equations (IAEs). We apply collocation methods on continuous piece-wise polynomials space to solve the obtained higher index IAEs. The efficiency of the given method is improved by using a recursive formula for computing the integral part. Finally, we apply the obtained algorithm to solve a trajectory-prescribed path control problem and a model of simple pendulum. The numerical experiments show efficiency of the given techniques.

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

A shooting method for two-point boundary value problems (BVPs) for fully implicit index-1 and index-2 differential algebraic equations (DAEs) is presented. A combination of the shooting equations with a method of the calculation of consistent initial values leads to a system of nonlinear algebraic equations with a nonsingular Jacobian matrix. Examples are given.

Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability

Direct Sturm–Liouville problem for surface Love waves propagating in layered viscoelastic waveguides