ABSTRACT
I ) + N
),
where
=
1 TCU
) ñ
=
1R
) K ( ó ) E ó ,
= arctan Q
) E ó = F2 + ( Q 2
+ > 22) ;
the Q
are positive roots of the transcendental equation cot( Q F
) = Q 2
Q ( > 1 + > 2) . 3 W . For the solution of the third boundary value problem with nonhomogeneous boundary conditions, see Paragraph 4.1.2-6 with X ( ó , I ) ≡ 0. YZ
4.1.1-8. Domain: 0 ≤ ó ≤ F
. Mixed boundary value problem. 1 W . Longitudinal vibration of an elastic rod with one end rigidly fixed and the other free. The following conditions are prescribed:[
=
( ó ) at I = 0 (initial condition), \ J
= K ( ó ) at I = 0 (initial condition),[ = 0 at ó = 0 (boundary condition),
= 0 at ó = F
(boundary condition). Solution:
( ó , I ) = L M
I ) + N
= ]
(2 ð + 1) 2 F
,
=
0 ñ ( ó ) sin( Q
=
2R
0 K ( ó ) sin( Q
2 W . For the solution of the mixed boundary value problem with nonhomogeneous boundary conditions, see Paragraph 4.1.2-7 with X ( ó , I ) ≡ 0. YZ
4.1.1-9. Goursat problem. The boundary conditions are prescribed to the equation characteristics:[
=
( ó ) for ó − R
I = 0 (0 ≤ ó ≤ ^ ),[ = K ( ó ) for ó +
I = 0 (0 ≤ ó ≤ _ ), where
(0) = K (0). Solution: [
( ó , I ) = ñ G
ó +
2 H − ñ (0). The solution propagation domain is bounded by four lines:
ó −
I = 0, ó +
I = 0, ó −
I = 2 _ , ó +
I = 2 ^ . YZ
4.1.2-1. Domain: − f < ó < f . Cauchy problem. Initial conditions are prescribed: [
= g ( ó ) at I = 0, \ J
= K ( ó ) at I = 0. Solution:
( ó , I ) = 12 [ g ( ó − R
I ) + g ( ó + R
I )] + 12 R h
K ( k ) l k + 12 R
4.1.2-2. Domain: 0 ≤ ó < f . First boundary value problem. The following conditions are prescribed:[
= g ( ó ) at I = 0 (initial condition), \ J
= K ( ó ) at I = 0 (initial condition),[ = o ( I ) at ó = 0 (boundary condition).
