PSI - Issue 28

O. Pozhylenkov et al. / Procedia Structural Integrity 28 (2020) 458–463 Author name / Structural Integrity Procedia 00 (2019) 000–000

462

5

(11)

Next step is to find fundamental solution basis matrices, which were derived in such form: Ψ � � � � � � � � � � � � � � � � � , � � 1,2 (12) Applying the boundary conditions � �Ψ � � �� � �,� , �,� � � �, � � � �, � � � , one obtained the linear algebraic system from which matrices � � , � � 1,2, � � 1,2 were found. With the help of the Green`s matrix the solution of the non-homogeneous vector problem was constructed: � � � � � � � � � � � � Ψ � � � ∗ � � Ψ � � � ∗ � (13) Green`s matrix has been found in the form: � � � � Ψ � � �Ψ � � �, � � � Ψ � � �Ψ � � �, � � (14) Solution of the stated problem in the transforms domain: (15) Where �,� � � , �,� � � elements of the Green`s matrix, � � � �� � �� � ,�� ∗ sin� � � ∗ ��1 � � � � . With the help of the inversion Fourier`s transform the solution of stated problem was found: � � � 2 � � � � ∗ sin� � � � ��� � � � 2 � � � � ∗ cos� � � � ��� , � � �� � 1 2 � , � � 1, � (16) Applying boundary condition � � � � � � � to the solution (16) integral equation has been constructed: 2 � � � � � � � � � � � � � � 2 �����2 � � � � � � � � � �� cos� � �� � � � ��� � � � 2 ���2 � �Ψ �� � � � � �� � �� � cos� � � � � � � � ��� (17) To find the solution of the integral equation orthogonal polynomial`s method was applied. It leads the solution of the singular integral equation to the infinite algebraic system, which can be solved applying reduction method that has been shown at Popov G., (1982) . ⎩⎪ ⎨ ⎪ ⎧ � � � � � �,� � � ∗ � � � � � � � � � �,� � � ∗ � � � � � Ψ � � � ∗ � � Ψ � � � ∗ � � Ψ � � � ∗ � � Ψ � � � ∗ �