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In the present paper the system of differential equations for the nonlinear theory of non-shallow shells is obtained. Some basic boundary value problems are solved.

Consider the system of differential equations $$ \frac{{dy}}{{dt}} = f\left( {t,y,\frac{{dy}}{{dt}}} \right), $$ (1) or y 0307; = f(t,y,y 0307;), where t ∈ R + = [0,+∞); y = (y 1,y 2…,y n ), f = (f 1,, f 2,…, f n ) are vectors in the n — dimensional Euclidean space R n . Let f:R + × Ω → R n ; Ω is a domain defined as follows: Ώ:{(y,z):y ∈ D ⊂ R n ;z ∈ R n :||z

In this chapter we consider methods of approximate solution of two-dimensional boundary value problems for the system of differential equations, which arose also as a necessary step for solving the initial problem (1.1)–(1.5).

Consider a system of differential equations $$ \frac{{dx(t)}}{{dt}} = f(t,x(t),x(t - \Delta )), $$ (2.1) (1.1)where x=(x 1,x 2...,x n is an n-dimensional vector f(t, x, y,),...,f n(f(t, x, y,)) is an n-dimensional vector function periodic in t with period 2π and Δ is a constant.

The system of differential equations for the dissipative free state, in matrix form, is that of relation (10.1) with the right-hand side set to zero.

We shall be looking at the following system of differential equations: $$\frac{{dx}}{{dt}} = X\left( x \right)$$ (1.1) the right hand side of which is defined, continuous and satisfies a Lipschitz condition in a domain D of n-dimensional Euclidean space. Under these assumptions, for every x0∈D there exists a unique solution x(t, x0) of system (1.1) such that x(0, x0)=x0. This solution is defined and belongs to the domain D for a maximal interval (T-,T+) in ℝ where \(- \infty \leqslant {{T}^{ - }} < 0 < {{T}^{ + }} \leqslant + \infty ,{{T}^{ - }}\left( {{{x}_{0}}} \right),{{T}^{ + }} = {{T}^{ + }}\left( {{{x}_{0}}} \right)\). Since the system of equations (1.1) is autonomous, the function x(t, x0) satisfies the group condition with respect to t: $$ x\left( {t + \tau ,{x_0}} \right) = x\left( {t,x\left( {\tau ,{x_0}} \right)} \right) $$ for any x0∈D and any t,τ and t + τ in (T-,T+).

By viewing material states as superpositions of perfect lattice and dislocated substates, we can adopt the approach of continuum mechanics to arrive at a system of differential equations describing the interaction effects between stress, dislocations and deformation. The various arguments that we utilize are broadly related to ideas earlier advanced in theories of continuous distribution of dislocations and in models of dislocation dynamics. Following a brief account of the general theoretical structure in the introduction, we discuss in the first part of this paper the ability of the present method to recover previous phenomenological models of material behavior, such as macroscopic theories of plasticity. In the second part of the paper, we illustrate the ability of the present method to predict dislocation phenomena associated with the formation of plastic zones, deformation bands and dislocation patterns.

Thin-walled structures play an important role in structural engineering. These structures — due to the thinness of the walls — are very sensitive to external excitation which can be caused either by harmonically fluctuating loading or by impact. The aim of the paper is to determine optimality criteria for vibrating thin-walled structures. In deriving these conditions, one must be very careful since thin-walled structures exhibit a different mechanical behaviour to solid sections. These special effects are the influence of shear deformations and of the deformation of the cross-section. Moreover, for special cross-sections there is a coupling of bending and torsion. In order to consider all these effects, a system of differential equations is introduced. Then, optimization problems are discussed and optimality conditions are derived, including some known and several new results.

A hybrid model is established for the spread of helminthic infections in a community. The qualitative behavior of a resulting system of differential equations is given epidemiological interpretations. A threshold effect is shown to exist for isolated communities. The threshold effect disappears if the community is exposed to even a small amount of continuous external infection.

It has been shown in Chapter 3 that linear time-invariant vibrating systems can be described by the system (3.10) of f second order differential equations $$ \begin{array}{l} M\ddot y(t) + (D + G) \dot y (t) + (K + N) + (t) = h(t),\\ \begin{array}{*{20}{c}} {}&y \end{array}({t_0}) = {y_0},\begin{array}{*{20}{c}} {}&{\dot y} \end{array}({t_0}) = {{\dot y}_0} \end{array} $$ (4.1) or, more generally, by the system (3.43) of n first order diferrential equations $$ \dot x(t) = Ax(t) + b(t),\begin{array}{*{20}{c}} {}&x \end{array}({t_0}) = {x_0}. $$ (4.2) The general solution $$ y(t) = y(t; {y_0},{\dot y_0},{t_0};h(t)),\begin{array}{*{20}{c}} {}&t \end{array} \ge {t_0}, $$ (4.3) respectively $$ x(t) = x(t;{x_0},{t_0};b(t)),\begin{array}{*{20}{c}} {}&{t \ge {t_0}} \end{array}, $$ (4.4) of these systems of differential equations yields information about the motion of the vibrating system when it is perturbed from its equilibrium position, or if it is subject to contnuing excitation. The general solution is in principle determined by the fundamental matrix of the system of differential equations. Therefore, this chapter will be devoted to discussing the properties of the fundamental matrix and the possibilities of its calculation.

This chapter opens the concluding part of the book, which is devoted to suboptimal control laws for dynamic objects. Similar problems were considered in preceding chapters; however, only recurrence relations or difference equations were used as a mathematical description of the objects’ behavior over a finite time set. (An exception is Section 4.5, where a denumerable set of moments was considered.) Further to this, we shall take a system of differential equations as a model that is more applicable to engineering.

On a complex plane, we consider the system of differential equations $$ \begin{array}{*{20}{c}} {{\xi _x} - {\eta _y} = a\xi + b\eta }\\ {{\xi _x} - {\eta _y} = {a_1}\xi + {b_1}\eta .} \end{array} $$ (19.1) This system is written in the complex form $$ \begin{array}{*{20}{c}} {\bar \partial u = Au + B\bar u,}&{u = \xi + i\eta ,}\\ {\bar \partial = \frac{1}{2}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right),}&{x = x + iy.} \end{array} $$ (19.2) One calls the system (19.1), (19.2) the Carleman system or the Bers-Vekua system. We prefer to call this system the system of Carleman-Bers-Vekua (CBV). Solutions to this system are called generalized analytic functions or pseudo-analytic functions. The generalized analytic function theory was founded in the 1950s by Bers [1], [a] and Vekua [35], [36].

A Backlund map is a transformation of the dependent variables in a system of differential equations, whereby the first derivatives of the new variables are given in terms of the new variables themselves as well as of the old variables and their derivatives. In this section, only the first derivatives of the old variables enter in. The general case, where higher derivatives are admitted, is discussed in the next section.

The system of differential equations which describes the motion of continuum media of gas, liquid, Reiner-Rievling-type liquid, etc., is considered.

Until now, analytic solutions for the problem of line formation in magnetic fields have been given only under special assumptions concerning either the magnetic field or the atmospheric model. Conditions are given for which the system of differential equations can be solved analytically. For the general case a method is given to solve the system in approximation and analytically. LTE is generally assumed which means that we use the system given by Unno (1956) or that given by Beckers (1969) including the Faraday rotation.