Ordinary Differential Algebraic Equations Research Articles

The Algebro-Geometric Method for Solving Algebraic Differential Equations — A Survey

This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations (ADEs). An algebraic differential equation is a polynomial relation between a function, some of its partial derivatives, and the variables in which the function is defined. Regarding all these quantities as unrelated variables, the polynomial relation leads to an algebraic relation defining a hypersurface on which the solution is to be found. A solution in a certain class of functions, such as rational or algebraic functions, determines a parametrization of the hypersurface in this class. So in the algebro-geometric method the author first decides whether a given ADE can be parametrized with functions from a given class; and in the second step the author tries to transform a parametrization into one respecting also the differential conditions. This approach is relatively well understood for rational and algebraic solutions of single algebraic ordinary differential equations (AODEs). First steps are taken in a generalization to systems and to partial differential equations.

Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

In this paper, we introduce certain $n$-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth are at most two. Moreover, for the second order factorizable algebraic ODEs, all the meromorphic solutions of them (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

Computation of all rational solutions of first-order algebraic ODEs

In this paper, we discuss three different approaches to attack the problem of determining all rational solutions for a first-order algebraic ordinary differential equation (AODE). We first give a sufficient condition for first-order AODEs to have the property that poles of rational solutions can only occur at the zeros of the leading coefficient. A combinatorial approach is presented to determine all rational solutions, if there are any, of the family of first-order AODEs satisfying this condition. Algebraic considerations based on algebraic function theory yield another algorithm for quasi-linear first-order AODEs. And finally ideas from algebraic geometry combine these results to an algorithm for finding all rational solutions of parametrizable first-order AODEs.

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

We study the rational solutions of systems of first-order algebraic partial differential equations and relate them to those of an associated autonomous system. We also describe how rational general solutions of these systems are related, and provide an algorithm in some particular case concerning the dimension of the associated algebraic variety. Our results can be considered as a generalization of the approach by L. X. C. Ngô and F. Winkler on algebraic ordinary differential equations of order one, adapted to systems of first-order algebraic partial differential equations.

The Maillet–Malgrange type theorem for generalized power series

There is proposed the Maillet–Malgrange type theorem for a generalized power series (having complex power exponents) formally satisfying an algebraic ordinary differential equation. The theorem describes the growth of the series coefficients.

Deciding the existence of rational general solutions for first-order algebraic ODEs

In this paper, we consider the class of first-order algebraic ordinary differential equations (AODEs), and study their rational general solutions. A rational general solution contains an arbitrary constant. We give a decision algorithm for finding a rational general solution, in which the arbitrary constant appears rationally, of the whole class of first-order AODEs. As a byproduct, this leads to an algorithm for determining a rational general solution of a class of first-order AODE which covers almost all first-order AODEs from Kamke's collection. The method is based intrinsically on the consideration of the AODE from a geometric point of view. In particular, parametrizations of algebraic curves play an important role for a transformation of a parametrizable first-order AODE to a quasi-linear differential equation.

КЛАССЫ ФОРМАЛЬНЫХ РЕШЕНИЙ КОНЕЧНОГО ПОРЯДКА АЛГЕБРАИЧЕСКОГО ОБЫКНОВЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ

In this paper we select general classes of finite order formal solutions of an algebraic (polynomial) ordinary differential equation (ODE), that can be calculated by the methods of the plane power geometry based on the method of determining leading terms of the equation by Newton-Bruno polygon. Beside that in this paper we prove the theorem that if a formal solution of the selected class exists than the first approximation (the truncation) of this solution is the (formal) solution of the first approximation of the initial equation (that is called the truncated equation). Calculated formal solutions by means of these methods relate to much more general classes of the formal solutions that are called grid-based series and transseries in the foreign papers. Grid-based series and transseries are fairly new objects and in spite of the large number of publications they are slightly studied. They appear among formal solutions of the differential equations including equations that are important in physics. Other general methods of the calculation of such series do not exist yet. Therefore it is important to select the classes of the formal solutions that can be calculated algorithmically by the methods of the plane power geometry.

On the convergence of generalized power series satisfying an algebraic ODE

Here we propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.

Hayman’s classical conjecture on some nonlinear second-order algebraic ODEs

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second-order nonlinear algebraic ordinary differential equations (ODEs). We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the classical conjecture proposed by Hayman in 1996.

Birational transformations preserving rational solutions of algebraic ordinary differential equations

We characterize the set of all rational transformations with the property of preserving the existence of rational solutions of algebraic ordinary differential equations (AODEs). This set is a group under composition and, by its action, partitions the set of AODEs into equivalence classes for which the existence of rational solutions is an invariant property. Moreover, we describe how the rational solutions, if any, of two different AODEs in the same class are related.

Does the Riemann zeta function satisfy a differential equation?

In Hilbert's 1900 address at the International Congress of Mathematicians, it was stated that the Riemann zeta function is the solution of no algebraic ordinary differential equation on its region of analyticity. It is natural, then, to inquire as to whether ζ(z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that ζ(z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[ζ−1]=1/(z−1), is given. We also show that this infinite order differential operator T may be inverted, and through inversion of T we obtain a series representation for ζ(z) which coincides exactly with the Euler–MacLauren summation formula for ζ(z). Relations to certain known results and specific values of ζ(z) are discussed.

Transient numerical modeling of catalytic channels using a quasi-steady gas phase

This paper presents a transient model of internal catalytic combustion suitable for isolated channels and monolith reactors. Due to time-scales in the problem, the model considers a quasi-steady gas phase with a transient solid. The gas is described by axially varying bulk temperature and species. The gas includes lateral diffusion via transfer coefficients and the specification of a gas-phase species concentration at the wall; axial diffusion is neglected. The solid phase is a thermally thin shell with axially varying temperature, surface species, and surface species concentrations. The solid includes axial heat conduction and external heat loss by convection and radiation. The combustion process utilizes detailed gas and surface reaction models. The gas-phase model becomes a system of stiff ordinary differential equations with respect to axial position; the upstream (inlet) boundary conditions are specified and the axially varying solid properties are parameters in integration. The solid phase discretizes into a system of stiff ordinary differential-algebraic equations with respect to time. The time evolution of the system comes from alternating integrations of the quasi-steady gas phase and transient solid.The model is compared to two experimental cases using CO fuel: (1) steady-state conversion in an isothermal platinum tube and (2) transient propagation of a catalytic reaction inside a small platinum tube and includes external tube temperature measurements. This work presents sensitivity studies on important parameters including internal transfer coefficients, catalytic surface site density, external heat-loss, and others. Under mass-transfer limited conditions, global transfer coefficients are adequate to predict fuel conversion. Near light-off, the model predictions improve for the first case after adjusting the surface kinetics such that the net rate of CO adsorption increases compared to O2. For the second case, predictions of transient propagation speeds are good for equivalence ratios near unity and greater but require adjustment of external heat loss or kinetics to match under lean conditions.

Three-dimensional extended Kantorovich solution for Levy-type rectangular laminated plates with edge effects

An accurate three dimensional elasticity solution for rectangular laminated plates subjected to Levy-type boundary conditions is presented using the recently developed mixed-field multi-term extended Kantorovich method (EKM) in conjunction with Fourier series. Such a solution accurately characterizes the edge effects in finite laminated plates under general bending, which can be used as a benchmark for assessing two-dimensional (2D) laminate theories for predicting of edge effects. The variables are expanded in Fourier series along the direction intersecting the simply-supported edges, and the 3D EKM is applied for the inplane and thickness directions. Using the Reissner-type variational principle, the governing partial differential equations are converted into two sets of 9n algebraic-ordinary differential equations for each Fourier term. The boundary conditions are satisfied exactly at all points on the boundaries. It is shown that similar to the case of cylindrical bending of infinite panels, the present solution for the general bending of finite plate also converges within two or three terms in the trial function. The numerical results are presented for composite and sandwich plates with four different boundary conditions. The effect of aspect ratio and width to length ratio on the boundary layer effect in the in-plane and interlaminar stresses are investigated.

A review of the application of SimuLink S-functions to multi domain modelling and building simulation

Multi domain modelling provides great opportunities for possible synergy between the building simulation domain and other scientific and technological domains. Although domains may have quite different models, they often use common mathematical representations, based on differential algebraic equations (DAEs) and/or ordinary differential equations (ODEs). This paper reviews the use of S-Functions in SimuLink for DAEs and ODEs modelling regarding building simulation and its potential for multi domain applications. It is concluded that ODEs are directly implementable using S-Functions in SimuLink. DAEs are indirectly implementable by a manual process of integrating Dymola/Modelica models. Examples from the literature confirm the great opportunities for the combined building thermal, geothermal, electrical and grid performance simulation.

Rational general solutions of higher order algebraic odes

This paper generalizes the method of Ngo and Winkler (2010, 2011) for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation (AODE) to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.

Optimal control of coupled multiphysics problems: Guidelines for real‐life applications demonstrated for a complex fuel cell model

AbstractIn practice one often is confronted with the simulation and optimization of dynamical processes of multiphysics systems depending on time and space. This paper shall serve as a guidebook for practioneers who wants to proceed from simulation to optimization. We are specially concerned in this paper with optimal control problems involving partial differential equations. After providing an insight into the theory of optimal control problems with partial differential equations, the focus of the paper lies on their numerical treatment to obtain approximate optimal solutions, or more precisely approximate candidate optimal solutions. The two basic approaches, first optimize then discretize (FOTD) and first discretize then optimize (FDTO) are first described in an abstract setting and then evaluated by means of the fuel cell example of this paper which can serve as a prototype problem for multiphysics processes. The example describes the dynamical behaviour inside a certain type of high temperature fuel cell where gas flows, heat distribution, and potential fields as well as chemical reactions are modelled by hyperbolic and parabolic partial differential as well as ordinary differential algebraic equations. The underlying problem is an optimal control problem where up to 28 instationary partial differential algebraic equations in one, respectively two spatial dimensions are involved. In the conclusion pros and cons of the aforementioned approaches are discussed. Hereby, not only the numerical efficiency but also the investigation of human resources are balanced against each other (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the numerical and computational aspects of non-smoothnesses that occur in railway vehicle dynamics

The paper contains a report of the experiences with numerical analyses of railway vehicle dynamical systems, which all are nonlinear, non-smooth and stiff high-dimensional systems. Some results are shown, but the emphasis is on the numerical methods of solution and lessons learned. But for two examples the dynamical problems are formulated as systems of ordinary differential-algebraic equations due to the geometric constraints. The non-smoothnesses have been neglected, smoothened or entered into the dynamical systems as switching boundaries with relations, which govern the continuation of the solutions across these boundaries. We compare the resulting solutions that are found with the three different strategies of handling the non-smoothnesses. Several integrators – both explicit and implicit ones – have been tested and their performances are evaluated and compared with respect to accuracy, and computation time.

Multiterm Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Plates

In an article recently published in this journal, the powerful single-term extended Kantorovich method (EKM) originally proposed by Kerr in 1968 for two-dimensional (2D) elasticity problems was further extended by the authors to the three-dimensional (3D) elasticity solution for laminated plates. The single-term solution, however, failed to predict accurately the stress field near the boundaries; thus limiting its applicability. In this work, the method is generalized to the multiterm solution. The solution is developed using the Reissner-type mixed variational principle that ensures the same order of accuracy for displacements and stresses. An n-term solution generates a set of 8n algebraic-ordinary differential equations in the in-plane direction and a similar set in the thickness direction for each lamina, which are solved in close form. The problem of large eigenvalues associated with higher order terms is addressed. In addition to the composite laminates considered in the previous article, results are also presented for sandwich laminates, for which the inaccuracy in the single-term solution is even more prominent. It is shown that considering just one or two additional terms in the solution (n = 2 or 3) leads to a very accurate prediction and drastic improvement over the single-term solution (n = 1) for all entities including the stress field near the boundaries. This work will facilitate development of near-exact solutions of many important unresolved problems involving 3D elasticity, such as the free edge stresses in laminated structures under bending, tension and torsion.

Rational general solutions of first-order algebraic ODEs

No abstract available.

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

In this paper, a Chebyshev spectral collocation domain decomposition (DD) semi-discretization by using a grid mapping, derived by Kosloff and Tal-Ezer in space is applied to the numerical solution of the generalized Burger’s–Huxley (GBH) equation. To reduce roundoff error in computing derivatives we use the above mentioned grid mapping. In this work, we compose the Chebyshev spectral collocation domain decomposition and Kosloff and Tal-Ezer grid mapping, elaborately. Firstly, the theory of application of the Chebyshev spectral collocation method with grid mapping and DD on the GBH equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use a fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. Application of this modified method to the GBH equation show that this method (M-DD) is faster and more accurate than the standard Chebyshev spectral collocation DD (S-DD) method.