Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.     

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations. Received: May 26, 2000; revised version: September 6, 2000     

Explicit determination of isovector fields of equivalence groups for balance equations of arbitrary order—Part II

The explicit solutions of previously given determining equations for isovector components (infinitesimal generators) associated with Lie groups of equivalence transformations for the most general family of balance equations of arbitrary order involving arbitrary but finite number of independent and dependent variables are provided. Equivalence transformations which are considered in this work map the solutions of partial differential equations containing arbitrary functions or parameters to the solutions of the equations of the same structure but with different functions or parameters. The general solutions exposed here are employed to determine the isovector fields corresponding to a sort of Korteweg–de Vries type equation involving third derivative with respect to spatial variable.     

TOTAL ORTHOGONALIZATION OF REDUNDANTS IN THE FORCE METHOD

If the force method is fully optimized by introducing two appropriate co-ordinate transformations on the internal generalized force unknowns, the self-stress states (or redundant force systems) are determined so as to be an orthonormal set which is also orthogonal to the particular solution. The magnitudes of the redundant forces then become zero for any loading. This eliminates all operations with the redundants. In other words, the particular solution which is found is the solution to the problem. The result is a large decrease in the amount of computation necessary in the force method by eliminating the need to form up and solve the matrices associated with the compatibility equations. Although a different approach, this optimized form of the force method is shown to result in the same numerical procedures as those for the natural factor formulation of the displacement method. The same transformations developed to orthogonalize completely the self-stress states may also be applied directly to the compatibility and equilibrium equations as an alternative procedure. This approach to the solution of the structural equations is designated as the force transformation method.     

Non-Gaussian models for stochastic mechanics

Memoryless transformations of Gaussian processes and transformations with memory of the Brownian and Lévy processes are used to represent general non-Gaussian processes. The transformations with memory are solutions of stochastic differential equations driven by Gaussian and Lévy white noises. The processes obtained by these transformations are referred to as non-Gaussian models. Methods are developed for calibrating these models to records or partial probabilistic characteristics of non-Gaussian processes. The solution of the model calibration problem is not unique. There are different non-Gaussian models that are equivalent in the sense that they are consistent with the available information on a non-Gaussian process. The response analysis of linear and non-linear oscillators subjected to equivalent non-Gaussian models shows that some response statistics are sensitive to the particular equivalent non-Gaussian model used to represent the input. This observation is relevant for applications because the choice of a particular non-Gaussian input model can result in inaccurate predictions of system performance.     

Analysis of the equations of motion of linearized controlled structures

The linearized equations of motion of controlled structures possess coefficient matrices that lack the familiar properties of symmetry and definiteness. A method is developed for the efficient analysis of linearized controlled structures. This constructive method utilizes equivalence transformations in Lagrangian coordinates and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, this method can offer substantial reduction in computational effort and ample physical insight. However, it is often necessary to draw upon some type of decoupling approximation for fast solution. Many numerical techniques involve discretized equations resembling those of linearized controlled structures. These numerical techniques can also be greatly streamlined if the method of equivalence transformations is incorporated. This research has been supported in part by the Alexander von Humboldt Foundation and by the National Science Foundation under Grant No. MSS-8657619. Opinions, findings, and conclusions expressed in this paper are those of the author and do not necessarily reflect the views of the sponsors.     

On the solution of hydrodynamical equations via Lie groups

Lie groups of homothetic transformations in the Euclidean space R₂ have been employed to determine and investigate certain classes of solutions of hydrodynamical equations of a perfect fluid. In particular, the conditions for a solution to be regular with respect to the one-parameter group of transformations have been determined. Furthermore, it has been shown that if the regularity conditions mentioned above are satisfied then the problem of obtaining regular solutions reduces to that of solving a system of equations not involving λ in the canonical coordinates (λ, μ) of the subgroup. Some special classes of flows have also been investigated.     

On some mixed finite element methods for plane membrane problems

An analysis of some quadrilateral dual mixed finite element methods for plane membrane problems is presented. The methods are based on a variational formulation which a priori does not involve symmetric stresses. After having presented the governing equations of the problem under discussion in the linear framework, a detailed analysis of a method already proposed in Cazzani and Atluri (1993) is performed. In particular, a result setting the equivalence of the method and another one involving symmetric stresses is established. Two other methods, this time not equivalent to any symmetric stress method, are presented and for them an analysis is outlined. Finally, some numerical tests showing the method performances are provided.     

Similarity solutions of unsteady boundary layer equations of a special third grade fluid

Two-dimensional, unsteady, laminar boundary layer equations of a special model of non-Newtonian fluids are considered. The fluid can be considered as a special type of power-law fluid. The problem investigated is the flow over a moving surface, with suction of injection. Two different type of ordinary differential equations system are found using the transformations. Using scaling and translation transformations, equations and boundary conditions are transformed into a partial differential system with two variables. Using translation and a more general transformation, the boundary value problem is transformed into an ordinary differential equations system. Finally, we numerically solve two different ordinary differential equations, separately.     

Symmetry properties for a generalised thin film equation

Symmetry properties are presented for a fourth-order parabolic equation written in conservation form. It was introduced in the literature as a generalisation of the fourth-order thin film equation. We derive equivalence transformations, Lie symmetries, potential symmetries, non-classical symmetries and potential non-classical symmetries. A chain of such equations is introduced. We conclude by presenting similar results for the third-order equation of this chain.     

Traveling wave solutions of nonlinear partial differential equations

Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Bäcklund Transformation.     

Geometric method of reconstructing systems from experimental data

A method of reconstructing model dynamic evolution equations reduced to the central invariant manifold is described, which is based on an analysis of experimental data from controlled objects. The proposed method takes into account the group transformations of phase trajectories, which retain the topological equivalence of local regions.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

What does integrability of finite-gap or soliton potentials mean?

In the example of the Schr?dinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit Psi-function, trace formula and the Jacobi problem-provide a complete solution. The Theta-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.     

Thermodynamics and properties of relaxation functions of materials with memory

Consideration is given to thermodynamic restrictions imposed on relaxation functions within the framework of the Coleman theory for materials with memory. It is shown that for the Second Law of Thermodynamics to be held, some integral transformations of relaxation functions should neccessarily be sign defined. This, in particular, implies the requirement of the dissipative property of the relaxation functions which results in some properties of the relaxation functions which restrict their behavior. These properties contain, as particular cases, the restrictions already known and strengthen them. Moreover, the obtained results, in contrast to some earlier ones, are valid for the case of general nonlinear constitutive equations, where relaxation functions describe their linear part.     

On the isovectors of a class of nonlinear diffusion equations

A class of nonlinear diffusion equations is expressed in terms of an ideal of exterior differential forms. The components of the associated isovector field are constructed by using its transport property under Lie's derivative. The solution of the corresponding orbital equations generate an invariant group of transformations which reduce the nonlinear diffusion equation to an ordinary differential equation. For the problem of heat propagation, in an initially cool infinite media, due to a plane source, an exact solution is derived.     

A systematic approach to the search for variational principles for plane steady flows

Using Vainberg's theorem of nonlinear potential operators, alternate potential principles associated with the differential equations governing the gasdynamics of plane steady irrotational diabatic flow and isoenergetic rotational adiabatic flow are formulated and their equivalence with Bateman's principle is established. Further, the advantage usefulness of treating a single nonlinear equation for the existence and hence formulation of a functional over the equivalent system for the same problem is brought into sharper focus.     

Radial basis functions as approximate particular solutions: review of recent progress

Solution of Poisson type differential equations can be achieved by finding an approximate particular solution to the forcing term followed by a boundary element method or more simply by using a boundary collocation method. The approximate particular solution is often found by using radial basis functions approximations to the forcing function. The advantage of radial basis functions is they involve a single independent variable regardless of the dimension of the problem. They prove particularly attractive when the domain cannot be expressed as product domains of lower dimensions. This paper provides a review of some of the recent progress in this field (in connection with the solution of Poisson type differential equations) together with some new results. Particular emphasis will be on the solution of non-linear and time dependent differential equations.     

Potential systems for PDEs having several conservation laws

A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with a multi-dimensional space of local conservation laws, instead of using only basis conservation laws, their arbitrary linear combinations are used that are inequivalent with respect to the equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa; in this sense the number of independent linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible that a great number of inequivalent potential systems are missed. Examples of all these possibilities are given.     

On convergence of Newton's method for load flow problems

The Kantorovich theorem is virtually the only known sufficiency condition for convergence of Newton's method on a set of non-linear equations. The theorem gives very conservative bounds and is difficult to apply. In this paper the power system load flow problem is solved by Newton's method, and a sufficiency condition for the convergence of the iterations on this particular problem is given. The criterion relates to a condition number defined on the Jacobian matrix associated with the non-linear equations. Computational feasibility is considered and the merits of its use are discussed.