\documentstyle[12pt,amsfonts,amssymb]{book}
\raggedbottom
\textheight=225mm
\textwidth=140mm
\topmargin=0cm
\oddsidemargin=0cm
\evensidemargin=0cm

\begin{document}
\def\theequation{\arabic{equation}}
\setcounter{equation}{0}
 
\begin{center} 
BOUNDARY VALUE PROBLEMS IN (1.0) APPROXIMATION OF A 
MATHEMATICAL 
MODEL OF BARS\footnote{This research was supported by a grant of Georgian government for universities} 
 
\vspace*{0.2cm} 
{\it G. Jaiani} 
 
\vspace*{0.2cm} 
{\it I.Vekua Institute of Applied Mathematics}\\ 
{\it I.Javakhishvili Tbilisi State University} 
\end{center} 
 
\vspace*{0.1cm} 
\par Generalizing an idea of I. Vekua [1] of constructing a version of the 
theory of plates and shells,  
 in the elastic bar model suggested in [2,3] fields of displacements, strains and stresses 
of the three-dimensional theory of linear elasticity 
have been expanded into orthogonal double Fourier-Legendre series  
with respect to the variables along thickness, and width 
of the bar with a variable rectangular cross-section. 
\par 
Now let us consider symmetric $(2{\bar h}_i=0,\;i=2,3)$ bar in (1.0) approximation  
of the above model. The corresponding full system of equations has the following form: 
\begin{equation} 
(\lambda+2\mu)(h_2h_3{\mathop v\limits^{0,0}}_{1,1}),_1+3 
\lambda(h_2h_3{\mathop v\limits^{1,0}}_{3}),_1+{\mathop {X^0_1}\limits^{0,0}} 
=0, 
\end{equation} 
\begin{equation} 
\mu (h_2h^3_3{\mathop v\limits^{1,0}}_{3,1})_{,1}-3(\lambda+2\mu)h_2h_3 
{\mathop v\limits^{1,0}}_3-\lambda h_2h_3{\mathop v\limits^{0,0}}_{1,1}+ 
h_3{\mathop {X^0_3}\limits^{1,0}}=0, 
\end{equation} 
\begin{equation} 
\mu (h_2h_3{\mathop v\limits^{0,0}}_{2,1})_{,1}+{\mathop {X^0_2}\limits^{0,0}}=0, 
\end{equation} 
\begin{equation} 
\mu (h_2h^3_3{\mathop v\limits^{1,0}}_{2,1})_{,1}-3\mu h_2h_3{\mathop 
v\limits^{1,0}}_2+ 
h_3{\mathop {X^0_2}\limits^{1,0}}=0, 
\end{equation} 
\begin{equation} 
\mu (h_2h_3{\mathop v\limits^{0,0}}_{3,1})_{,1}+3\mu (h_2h_3{\mathop 
v\limits^{1,0}}_1)_{,1}+ 
{\mathop {X^0_3}\limits^{0,0}}=0, 
\end{equation} 
\begin{equation} 
(\lambda+2\mu)(h_2h_3^3{\mathop v\limits^{1,0}}_{1,1})_{,1}-\mu h_2h_3 
{\mathop v\limits^{0,0}}_{3,1}-3\mu h_2h_3{\mathop v\limits^{1,0}}_{1}+ 
h_3{\mathop {X^0_1}\limits^{1,0}}=0, 
\end{equation} 
where 
$$\mathop {v_{k}} \limits ^{\hspace{-0.7ex}r,s}(x_1):=\frac{\mathop {u_k}\limits ^{\hspace{-
0.7ex}{r,s}}(x_1)} 
{h_2^{r+1}(x_1)h_3^{s+1}(x_1)} 
$$ 
$\lambda$ and $\mu$ are Lam\'{e}'s constants, index 1 after comma means  
differentiation with respect to $x_1$, 
$$ 
\mathop {X_j^0}\limits^{\hspace{-0.7ex}n_3,n_2}:=\displaystyle
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2} 
\limits^{\hspace{-1ex}(+)}}\left[ \sqrt{1+ 
\left(\;\mathop {h_{3,1}}\limits^{\hspace{-1.8ex}(+)}\right)^2}
Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(+)}} 
(x_1,x_2,\mathop {h_3}\limits^{\hspace{-0.7ex}(+)})+ 
(-1)^{n_3} 
\sqrt{1+\left(\;\mathop {h_{3,1}}\limits^{\hspace{-1.8ex}(-)} 
\right)^2}\times\right. 
$$
$$
\displaystyle 
\left.\times Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} 
(x_1,x_2,\mathop {h_3}\limits^{\hspace{-0.7ex}(-)})\right]P_{n_2}(a_2x_2-b_2)dx_2
+\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3} 
\limits^{\hspace{-1ex}(+)}}\left[
\sqrt{1+\left(\;\mathop {h_{2,1}}\limits^{\hspace{-1.8ex}(+)}\right)^2}\; 
 Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(+)}} 
(x_1,\mathop {h_2}\limits^{\hspace{-0.7ex}(+)},x_3)+ \right.
$$ 
\par
\normalsize
$$ \left.
\displaystyle+(-1)^{n_3} 
\sqrt{1+\left(\;\mathop {h_{2,1}}\limits^{\hspace{-1.8ex}(-)} 
\right)^2}\; 
 Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} 
(x_1,\mathop {h_2}\limits^{\hspace{-0.7ex}(-)},x_3)\right]
P_{n_3}(a_3x_3-b_3)dx_3+ 
\mathop {X_j}\limits^{\hspace{-0.7ex}n_3,n_2}, \;\;i=1,2,3, 
$$ 
$Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(+)}},\;Q_j^{\mathop {h_3} 
\limits^{\hspace{-1ex}(-)}},\; Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(+)}}, 
\;Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(-)}}$  
are components of surface forces acting on surfaces  
$\mathop {h_3}\limits^{\hspace{-0.7ex}(+)},\;  
\mathop {h_3}\limits^{\hspace{-0.7ex}(-)},\; 
\mathop {h_2}\limits^{\hspace{-0.7ex}(+)},\; 
\mathop {h_2}\limits^{\hspace{-0.7ex}(-)}$ respectively; the bar occupies the  
following domain 
$$ 
B:=\{(x_1,x_2,x_3):\;\;0<x_1<l, \;\;\mathop {h_i}\limits^{\hspace{-0.7ex}(-)} 
(x_1)<x_i<\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}(x_1),\;\; 
i=2,3,\;\;l={\rm const}\}; 
$$ 
$$ 
2h_i(x_1):=\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}(x_1)-
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}(x_1)\ge 0, \;\;
2{\bar h}_i(x_1):=\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}(x_1)+
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}(x_1)\ge 0,\;\; i=2,3;\;\; h_i\in C^1; 
$$ 
$$ 
\displaystyle 
\mathop {X_j}\limits^{\hspace{-0.7ex}n_3,n_2}(x_1):= 
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2} 
\limits^{\hspace{-1ex}(+)}}\;  
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3} 
\limits^{\hspace{-1ex}(+)}} 
 X_j(x_1,x_2,x_3) P_{n_2}(a_2x_2-b_2)P_{n_3}(a_3x_3-b_3)dx_2dx_3,\;\;j=1,2,3, 
$$ 
$X_j$ are volume forces; $P_n(\cdot)$ are Legendre Polynomials; 
$$ 
a_i(x_1):=\displaystyle 
\frac{1}{h_i(x_1)},\;\;\;\; 
b_i(x_1):=\displaystyle \frac{{\bar h}_i(x_1)}{h_i(x_1)},\;\;\;\;i=2,3; 
$$ 
$$ 
\displaystyle 
\mathop {u_j}\limits^{\hspace{-0.7ex}n_3,n_2}(x_1):= 
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2} 
\limits^{\hspace{-1ex}(+)}} \; 
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3} 
\limits^{\hspace{-1ex}(+)}}  u_j(x_1,x_2,x_3) 
P_{n_2}(a_2x_2-b_2)P_{n_3}(a_3x_3-b_3)dx_2dx_3,\;\;j=1,2,3, 
$$ 
$u_j(x_1,x_2,x_3)$ are components of the displacement vector. 
\par 
Boundary value problems for equation (3) have been solved in [2]. Therefore 
for this equation
e.g. 
the Dirichlet problem is correct if only ${\mathop I\limits^{0,0}}_0<\infty$, 
${\mathop I\limits^{0,0}}_\ell<\infty$. For (4) the Dirichlet problem is 
correct if only ${\mathop I\limits^{1,0}}_0<\infty$, ${\mathop 
I\limits^{1,0}}_\ell<\infty$ 
(see also [4],[5]), where 
$$ 
{\mathop {I_0}\limits^{N_3,N_2}}:=\int\limits_{o}^{\varepsilon} 
h_2^{-2N_2-1}(\tau)h_3^{-2N_3-1}(\tau)d\tau,\;\;\varepsilon=const>0, 
$$ 
$$ 
{\mathop {I_\ell}\limits^{N_3,N_2}}:=\int\limits_{\ell-\varepsilon}^{\ell} 
h_2^{-2N_2-1}(\tau)h_3^{-2N_3-1}(\tau)d\tau,\;\;\varepsilon=const>0.
$$ 
Hence the approximate value of 
$$ 
u_2(x_1,x_2,x_3)\cong \frac{1}{4}{\mathop v\limits^{0,0}}_2(x_1)+ 
\frac{3x_3}{4}{\mathop v\limits^{1,0}}_2(x_1) 
$$ 
can be preassigned on the bar ends if only ${\mathop I\limits^{1,0}}_0<+\infty$, 
${\mathop I\limits^{1,0}}_\ell<+\infty$. 
\par Systems (5), (6) and (1), (2) can be correspondigly reduced to the following systems 
$$ 
3\mu{\mathop v\limits^{1,0}}_{1}=-\mu{\mathop 
v\limits^{0,0}}_{3,1} 
-h_2^{-1}(x_1)h_3^{-1}(x_1)\left({\mathop\int\limits_{x_1^0}^{x_1}}{\mathop 
{X^0_3}\limits^{0,0}}dx_1-C_1\right),\;\; 
C_1=const,\;x_1^0\in ]0,\ell[, 
$$ 
$$ 
(\lambda+2\mu)(h_2h_3^3{\mathop v\limits^{0,0}}_{3,11})_{,11}= 
3{\mathop {X^0_3}\limits^{0,0}}+3(h_3{\mathop {X^0_1}\limits^{1,0}})_{,1}-
\frac{\lambda+2\mu}{\mu}\left(h_2h_3^3\left(h_2^{-1}h_3^{-1} 
{\mathop\int\limits_{x_1^0}^{x_1}}{\mathop {X^0_3}\limits^{0,0}}dx_1- 
\right.\right.
$$
$$
\left.\left.
-h_2^{-1}h_3^{-1}C_1\right)_{,1} \right)_{,11} ;
$$ 
and 
$$ 
3\lambda {\mathop v\limits^{1,0}}_3=-(\lambda+2\mu){\mathop v\limits^{0,0}}_{1,1}- 
h_2^{-1}(x_1)h_3^{-1}(x_1)\left({\mathop\int\limits_{x_1^0}^{x_1}} 
{\mathop {X^0_1}\limits^{0,0}}(t)dt 
 -C_2\right),\;\;C_2=const, 
$$ 
$$ 
(\lambda+2\mu)\mu(h_2h_3^3{\mathop v\limits^{0,0}}_{1,11})_{,11}-
12(\lambda+\mu)\mu 
(h_2h_3{\mathop v\limits^{0,0}}_{1,1})_{,1}= 
$$ 
$$ 
=-\mu\left(h_2h_3^3\left[\left(  
h_2^{-1}h_3^{-1}{\mathop\int\limits_{x_1^0}^{x_1}}{\mathop 
{X^0_1}\limits^{0,0}}(t)dt 
\right)_{,1}-
C_2\left(h_2^{-1}h_3^{-1}\right)_{,1}\right]\right)_{,11} 
+12\frac{(\lambda+\mu)\mu}{\lambda+2\mu}{\mathop {X^0_1}\limits^{0,0}}+ 
$$ 
$$ 
+\frac{3\lambda^2}{\lambda+2\mu}{\mathop {X^0_1}\limits^{0,0}} 
+3\lambda(h_3{\mathop {X^0_3}\limits^{1,0}})_{,1}. 
$$ 
The advantages of these systems consist in the following: the second equations 
are equations with respect to a only one unknown function and after their solution 
(see [6]) from the first equations correspondinly ${\mathop v\limits^{1,0}}_1$ and 
${\mathop v\limits^{1,0}}_3$ can be readily calculated. 
\par 
{\bf Remark 1.} Let us consider $(N_3,N_2)$ approximation of a model of bars given
in [2,3]. Further the notation of above works will be used.
\par
Let the width and the thickness of the bar be given by $2h_i(x_1)=h_0^ix_1^{\alpha_i},$
$i=2,3\;(h^i_0={\rm const}>0,\;\;\alpha_i={\rm const}\ge 0),$ respectively.
\par
{\bf Problem 1.} {\it Find bounded weighted double moments 
${\mathop{v_j}\limits^{n_3,n_2}}\in C^3(]0,\ell[)\cap C(]0,\ell]),$ 
$j=1,2,3,$ $n_i=\overline{0,N_i},$ $i=2,3,$ of displacement vector components
$u_j(x_1,x_2,x_3)$ under boundary conditions:
\begin{equation}
\begin{array}{r}
{\mathop{v_j}\limits^{n_3,n_2}}(0)={\mathop{\varphi_j^0}\limits^{n_3,n_2}},\;\;
j=1,2,3,\;n_i=\overline{0,N_i},\;i=2,3,\\{\rm if}\; 
(2N_2+1)\alpha_2+(2N_3+1)\alpha_3<1;
\end{array}
\end{equation}
$$
{\mathop{v_j}\limits^{n_3,n_2}}(\ell)={\mathop{\varphi_j^\ell}\limits^{n_3,n_2}},\;\;
j=1,2,3,\;n_i=\overline{0,N_i},\;i=2,3,
$$
provided that the lateral boundaries $(x_i={\mathop{h_i}\limits^{(\pm)}}(x_1),$
$i=2,3,$ $0<x_1<\ell)$ of the bar are arbitrarily loaded. $\mathop{\varphi_j^0}\limits^{n_3,n_2},$
$\mathop{\varphi_j^\ell}\limits^{n_3,n_2},$ and loadings 
$\mathop{f_j^i}\limits^{(\pm)}$ are given.}
\par
{\bf Statement.} Problem 1 is correct in the sense of Hadamard.
\par
If we consider the limit $N_i\rightarrow +\infty,$ $i=2,3,$ then from (7) we have
$$
\alpha_i<\frac{1}{2N_i+1},\;\;{\rm and}\;\;
{\mathop{\rm lim}\limits_{N_i\rightarrow +\infty}}\alpha_i=0,\;\;i=2,3.
$$
Thus, Problem 1 tends to the following three-dimensional model
\par
{\bf Problem 2.} {\it Find 
$$u_i(x_1,x_2,x_3)\in C^3(V)\cap\left\{
\begin{array}{l}
 C(\overline{V})\;{\rm if}\;\alpha_i=0,\;i=2,3,\\
 C(\overline{V}\backslash\{0,x_2,x_3\})\;{\rm if}\;\alpha_2+\alpha_3>0,\end{array}\right.
$$
bounded in the latter case, under boundary conditions }
$$
u_j(0,x_2,x_3)=\varphi_j^0(x_2,x_3),\;\;j=1,2,3,\;\;{\rm if}\;\;
\alpha_i=0,\;\;i=2,3;
$$
$$
u_j(\ell,x_2,x_3)=\varphi_j^\ell(x_2,x_3),\;\;j=1,2,3,
$$
$$
X_{ij}(x_1,x_2,x_3)\left\vert_{_{x_i={\mathop{h_i}\limits^{(\pm)}}(x_1)} }=
{\mathop{f_j^i}\limits^{(\pm)}}(x_1,x_{5-i}),\;\;j=1,2,3,\;i=2,3.\right.
$$
\par 
{\bf Remark 2.} In the forthcoming paper the similar mathematical model of bars
with the variable cross-section of general form will be constructed.
\par 
{\bf Remark 3.} In [7] to deduce the system of one-dimensional equations of ceramic 
bar of uniform cross-section from the three-dimensional equations of  
thermopiezoelasticity, the method of power series expansion is used for the  
variables along the bar thickness and width. In contrast to this, by using the method 
of the present paper, one-dimensional equations of a ceramic bar of a variable  
cross-section can be deduced. 
\par 
{\bf Remark 4.} If we waive the assumption that $\ell$ is much greater 
than width and thickness  
which is necessary in case of bars then $B$ should be understood 
as an essentially three-dimensional elastic body. Generalizing the  
idea of this paper 
we could expand the displacement vector components into triple Fourier-Legendre 
series. 
This approach leads in the general case to the infinite system of algebrical  
equations. In case of a prismatic body $(a_j=const,\;j=1,2,3),$ $a_1=\ell^{-1}$ 
$(N_3,N_2,N_1)$ 
approximation is equivalent to the polynomial approximation of solution of  
three-dimensional boundary value problems of the linear theory of elasticity.
In dynamical case $(0,0,0)$ approximation characterizes rigid transfer of the 
body under action of volume forces; (1.1.1) approximation coincides with the
homogeneous deformation of the body.  This approach is convinient especially in point 
of view of numerical solution of the three-dimensional problems. 
 
\vspace*{0.3cm} 
\footnotesize 
\begin{center} 
{\bf R e f e r e n c e s} 
 
\vspace*{0.2cm} 
\end{center} 
\par 
[1] {Vekua I.N., Shell Theory: General Methods of Construction. Pitman 
Advanced Publishing Program, Boston-London-Melburne, 1985,  1-287.} 
\par
[2] Jaiani G.V., On a Model of a Bar with Variable Thickness, BULLETIN of TICMI, 
2, 1998, 36-40.  (electronic version: http://www.viam.hepi. 
edu.ge/others/TICMI).
\par
[3] Jaiani G.V., On a Mathematical Model of a Bar with Variable Rectangular 
Cross-section, Universitaet Potsdam, Institut fuer Mathematik, Preprint 98/21, 
1998, 1-24.
\par
[4] Kiguradze I.T.,  Shekhter B.L., Singular Boundary Value Problems for Ordinary
Differential Equations of Second Order. Itogi Nauki i Tekhniki, Seria, Sovremennje
problemi Matematiki, Noveishje Dostizhenia, Moskva, 30, 1987, 105-202.
\par
[5] Jaiani G.V., The First Boundary Value Problem of Cusped Prismatic Shell 
Theory in Zero Approximation of Vekua Theory, Proceedings of I.Vekua 
Institute of Applied Mathematics, 29, 1988, 5-38, (Russian, 
Georgian and English summaries).
\par 
[6] Jaiani G.V., Bending of an Orthotropic Cusped Plate, Universitaet Potsdam, 
Institut fuer Mathematik, Preprint 98/23, 1998, 1-30.
\par
[7] Askar Altay G., D\"{o}kmeci M.C., Numerical Algorithms for Dynamics of Thermopiezoceramic
Bars. Journal of Applied Mathematics and Mechanics (ZAMM) 77, 6, 1997, 429-445.
 

\end{document}