ARC-length method for differential equations

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Symmetry and ordinary differential constraints

The representative generalized symmetries of any ordinary differential equation are described in terms of its invariants. This identifies the evolution equations compatible with a given constraint. The restriction of the flow of a compatible equation to the solution space of the constraint is generated by the corresponding internal symmetry. This reduces the evolution equation to a finite dimensional system of first-order ordinary differential equations. The Euler–Lagrange equation of any conserved density of a given evolution equation yields such a reduction. Other examples include the generalized method of separation of variables, the characterization of separable evolution equations, and the characterization of equations with complete families of wave solutions. A Newton equation is compatible with an ordinary differential constraint if and only if the constraint is affine, with force field symmetry, in which case the equation reduces to a finite-dimensional dynamical system. Newton equations with complete families of characteristic solutions reduce to central force problems on solution spaces of linear constraints.     

The differential geometry of bending

Summary The deformation associated with uniform bending has been treated using the techniques of differential geometry. The Burgers circuit as well as various related tensor quantities such as torsion and dislocation density have all been determinated with respect to these various states of deformation.

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Sommario La deformazione associata con la flessione uniforme viene trattata usando le tecniche della geometria differenziale. Il circuito di Burgers e altre quantità tensoriali e la densità di dislocazione, vengono determinate nei vari stati di deformazione.

     

Barkhausen noise in differential transformers

A treatment is presented of the Barkhausen noise in differential transformers in analogy to our noise calculations on fluxgate magnetometers. A numerical estimate is given which for practical situations leads to an accuracy limit of the order of 10^?10 m.     

Periodic viable trajectories of differential inclusions

In this paper, the periodic viable trajectories of differential inclusions are discussed. Firstly, a simplified property of differential inclusions is given. Then, an existence theorem of periodic viable trajectories of differential inclusions in a finite dimensional space is proved. With the above results and Galerkin’s approximation, an existence theorem of periodic viable trajectories of partial differential inclusions in a Hilbert space is proved.     

A generalized differential effective medium theory

A generalization of the Differential Effective Medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration φ of the added phases in the composite. As φ→ 1, the Effective Medium Approximation (EMA) is obtained. For φ < 1, the result depends upon the backbone and the mixture path that is followed. The approach to EMA for φ ? 1 is analysed and a generalization of Archie's law for conductor-insulator composites is described. The conductivity mimics EMA above the percolation threshold and DEM as the conducting phase vanishes.     

Averaging of impulsive fuzzy differential equations

For fuzzy differential equations with pulses at fixed times, we substantiate the schemes of partial and complete averaging. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 530–540, October–December, 2008.     

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation $\text{dx}\text{dt}\text{= A(t)x + f(t,x)}$ by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.     

Unified expressions of all differential variational principles

A mathematical expression of the quantitative causal principle is given, using the expression this letter shows the unified expressions of D’Alembert–Lagrange, virtual work, Jourdian, Gauss and general D’Alembert–Lagrange principles of differential style, first finds the intrinsic relations among these variational principles, the conservation quantities of the above principles are first found, finally the Noether conservation charges of the all differential variational principles in the systems with the symmetry of Lie group D_m are first obtained.     

The exponential asymptotic solution of differential equation

In this paper, the exponential asymptotic solution (E.A.S.) of differential equation is discussed. Firstly, E.A.S. of the second-order differential equation is studied and the orthogonal conditions of the uniformly valid E.A.S. are found out. Next, E.A.S. in matched asymptotic method is discussed. Finally, some examples are given.     

Non-linear oscillations of a third-order differential equation

An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed.