Issue 38

N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01 4 u x u x v x x v x v x u x x u u a v v a 2 2 ( -1) 2 3 " ( ) ( ) ' ( ) '( ) 1 1 1 ( 1) 2 1 " ( ) ( ) ' ( ) ( ) 1 1 1 (0) 0, ( ) 0 (0) 0, ( ) 0                                                      (9) Here the new unknown function is inputted         x v x x v x , 0 , ' ' , 0     . As it is seen from the boundary condition (6),     u x x y , 0 '      , so the condition (6) is satisfied automatically. With the aim to reduce the problem to the vector boundary problem one must input the vectors and the matrixes       u x y x v x              ,       x f x x 3 ' 1 1 1                          , P 1 0 1 1 0 1                      , Q 1 0 1 1 0 1                 . Then the equations in the vector form will be written as the vector equation     L y x f x 2     , where L 2 is a differential operator of the second order         L y x Iy x Qy x Py x 2 2 " 2 '              , I is an identity matrix. The integral transformations also should be applied to the boundary conditions, with the aim to formulate the boundary functionals in the transformations’ domain. As a result the vector boundary problem is constructed         L y x f x y y a 2 0 0, 0           (10) T HE SOLVING OF THE VECTOR BOUNDARY VALUE PROBLEM he solution of the vector boundary problem (10) will be searched as the superposition of a homogenous vector equation’s general solution   y x 0   and a particular solution of the inhomogeneous one   y x 1         y x y x y x 0 1         These solutions were constructed with the help of the matrix differential calculation apparatus earlier [18].         c c y x Y x Y x y x c c 1 3 1 1 2 2 4                    where   i Y x i , 0,1  are the matrix system of the fundamental matrix solutions [18]:             x x x x x x e e Y x Y x x x x x 1 2 1 1 1 1 , 2 2 1 1 1 1                                                                  where constants i c i , 1, 4  are founded from the boundary conditions [18]. T