A VARIATIONAL PRINCIPLE FOR COMPLEX BOUNDARY VALUE PROBLEMS

This paper provides a variational formalism for boundary value problems which arise in certain feilds of research such as that of electricity, where the a,sociated boundary conditions contain complex periodic conditions. A functional is provided which embodies the boundary conditions of the problem and hence the expansion (trial) functions need not satisfy any of them.

Let R be a given complex domain with boundary F. Following the work of Delves and Hall [I] we split the boundary into four non-overlapping segments F i 2 3 4 and assume i that periodicity conditions are imposed on the segments F3 and F such that for some fixed , F {Z + I e F3 }.In this case we have the relations" 4 ( + (3 "-94 "3 () and (1.1)I I()ds I I(+) ds r r3 where 3 and 4 are the unit outward normals to F and F respectively and fds is a line integral along the boundary with positive direction taken counterclockwise.

THE PROBLEM
Let the problem whose solution is sought be of the following form" -V2u + d(x)u g(!), &c 2 (2 .l.a) with the prescribed boundary conditions- Vu.(!) =-e Vu.n(x + A), !E F3 where F2 and/or F3 may be void.
In modelling the stator of a turbogenerator where the rotor rotates at angular fre- quency and is effectively a bar magnet generating a rotating magnetic field, periodic boundary conditions of the form" i@ u(i) e u(+i) arise for the first harmonic component" and the normal gradient condition has: -i@ Vu.n(x) -e Vu.n(x +) where @ is the sector angle.These tvo conditions are exactly the last two conditionso (2.l.b).

3.
A FUNCTIONAL 'EMBODYING THE BOUNDARY CONDITIONS.
In this section we produce a functional which is stationary at the solution of (2.1) for a class of functions which do not satisfy any of the boundary conditions sinc thee conditJo-s are incornorated via suitable terms in the functional J given as" Next, it will be shown that if we expand the trial function V about the true solution u, of (2.1)-V u + Ew, where E is a scalar and w is an arbitrary variation, then J(V) is stationary.

Define
G(E) J(u + Ew), then  where we have written the line integral of (3.3) as the sum of four llne integrals along the boundarles into which F has been decomposed.The integrals over Ft and F2 of 3. 4) cancel the corresponding integrals over Ft and F2 in (3.2) taking into consideration the boundary conditions in (2.l.b).Also from (2.l.b), it is obvious that the first of the two llne integrals over F3 in (3.2) is equal to zero.What is left is to show that the last integral in (3.2)(hereafter referred to as LI) cancels the line integrals over Fa and F in (3.4).But Vu(x + a) (n__3.9. 4 ).n__ 3 ds w(_x + a) [-e iO Vu(x) .n]ds I w(_x +a) [Vu(_x +a) (n3.4) .n_.These line integrals over F3 and F cancel the corresponding ones in (3.4).Hence the functional J is stationary at the solution u. 4.
MATRIX SET-UP.To descibe the matrix set-up stage, we consider for convenience and simplicity the solution of the following one dimensional problem: together with the boundary conditions: where z is regarded as a parameter that takes any complex value.
We seek an approximate solution fN(zx) to f(zx) of the form" N fN(zx) [ an(Z) hn(X), -l_<x_< (4.2) n=l Then the problem represents a one-dimensional form of (2.1); and the functional J given in (3.1) reduces to" 2 J(V) [(V') + BV 2 2GV] dx 2[a V(-1)]V'(-1) + 218 V(1)]V'(1) 3) The coefficients a (z) are defined by the stationary point of J (at the solution where V n f) that is, by the equations" La A + B + S ]a G + H (4.4.a) where A, ., and S (rc _,7 mtrices" a}u ,J are -vectors, with components" h' h' A dx, B hiB(zx)h dx, G i hiG(zx) dx, Sj hi(-l)h'j(-]) + hj(-l)i(-I hi(1)j(1) hj(1)h(1), When using global expansion functions, it is desirable for stability reasons to use orthogonal polynomials see Mikhlin [2]).Accordingly, in (4.2) we take h_2 h_l x h n -x 2) Tn(X n=0,1, where r=N-3 and T (x) is the nth Chebyshev polynomial of the first kind.The reason for n this choice of basis is the need to handle the derivative terms in the matrix A wthout introducing artificial singularities.To calculate the elements in (4.4.b), we expand the functions B(zx) and G(zx) by Chebyshev series and use Fast Fourier Techniques to approx- imate the expansion coefficients.Thence we relate the elements Aij Bij and Gi of (4.4.b) to the coefficients of these expansions.This together with a numerical example will be considered in a subsequent paper.
While we do not attempt an error analysis here,the rapidity of convergence in calc- ulating the matrix equation (4.4) has been considered formally by Delves and Mead [3], Freeman et al [4] and Delves and Bian [5].In these papers it is shown that a complete characterisation of the convergence of the calculation can be given in terms of an assumed structure of the matix L in (4.4) and the convergence of the Fourier coefficients of the right hand function G(zx) in (4.l.a).Both a priori and a posterior in error estimates are provided by Delves [6] where a very similar treatment to the one given in this section is used for Frdholm integral equations and from which we take ignoring the a priori estimate since it contains an unknown constant ): A posteriori estimate: -I which is a standard bound; s min(p,q) where p and q depend on the differentiability of B(zx) and G(zx).The procedure iven in this section can easily be extended to two dime- sions in a straightforward manner and details are omitted.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: [(gl u)Vw wVu].ds