Fourth-Order One-Dimensional Nonstationary
9.2. Fourth-Order One-Dimensional Nonstationary
Equations
9.2.1. Equations of the Form ✽ ✾ + ❀ 2 ✽ ✾ = ❂ ( ❁ , ✿ )
9.2.1-1. Particular solutions of the homogeneous equation ( ❃ ≡ 0):
( ✝ , ✞ )= ✒ ❄ sin( ✍ ✝ )+ ❅ cos( ✍ ✝ )+ ❆ sinh( ✍ ✝ )+ ❇ cosh( ✍ ✝ ) ✗ exp(− ✍ 4 ✟ 2 ✞ ), where ❄ , ❅ , ❆ , ❇ , and ✍ are arbitrary constants.
9.2.1-2. Domain: 0 ≤ ✝ ≤ ❈ . Solution in terms of the Green’s function.
1 ☎ . We consider problems on an interval 0 ≤ ✝ ≤ ❈ with the general initial condition
= ✙ ( ✝ ) at ✞ =0
and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as
2 ☎ . Paragraphs 9.2.1-3 through 9.2.1-10 present the Green’s functions for various types of boundary conditions. The Green’s functions can be evaluated from the formula ■ ■
where the ❑ and ( ✝ ) are determined by solving the self-adjoint eigenvalue problem for the fourth-order ordinary differential equation ❑
subject to appropriate boundary conditions; the prime denotes differentiation with respect to ❑ ❑ ✝ . The norms of eigenfunctions can be calculated by the formula
Relations (1) and (2) are written under the assumption that ❑
= 0 is not an eigenvalue.
9.2.1-3. The function and its first derivative are prescribed at the boundaries:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ .
Green’s function:
( ✝ ■ )= ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ; the ❑ ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = 1. The numerical values
of the roots can be calculated from the formulas given in Paragraph 9.2.3-2.
9.2.1-4. The function and its second derivative are prescribed at the boundaries:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ .
Green’s function:
sin( ✍ ✝ ) sin( ✍ ❊ ) exp(− ✍ 4 ✟ 2 ❈ ✞ ),
9.2.1-5. The first and third derivatives are prescribed at the boundaries:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
cos( ✍ ✝ ) cos( ✍ ❊ ) exp(− ✍ ✟ ❈ ✞ ),
9.2.1-6. The second and third derivatives are prescribed at the boundaries:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )+cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )+sin( ✍ ✝ ) ✗ ; the ❑ ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = 1. The numerical values
of the roots can be calculated from the formulas given in Paragraph 9.2.3-2.
9.2.1-7. Mixed conditions are prescribed at the boundaries (case 1):
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation tan( ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.1-8. Mixed conditions are prescribed at the boundaries (case 2):
= ▼ =0 at
= 0, ✆ = =0 at ✝ = ❈ .
Green’s function: ■
( ✝ )= ■ ✒ sinh( ✍ ❈ )+sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )+cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = −1.
9.2.1-9. Mixed conditions are prescribed at the boundaries (case 3):
= ✆ =0 at = 0, = =0 at ✝ = ❈ . Green’s function:
9.2.1-10. Mixed conditions are prescribed at the boundaries (case 4):
=0 at ✝ = ❈ . Green’s function:
) = sin( ✍ ❈ ) sinh( ✍ ✝ ) + sinh( ✍ ❈ ) sin( ✍ ✝ ); the ✍ are positive roots of the transcendental equation tan( ❑ ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.2. Equations of the Form 2 ✽ ✾
This equation is encountered in studying transverse vibration of elastic rods. ❁ 9.2.2-1. Particular solutions:
( ✝ , ✞ )= ✒❖❄ sin( ✍ ✝ )+ ❅ cos( ✍ ✝ )+ ❆ sinh( ✍ ✝ )+ ❇ cos( ✍ ✝ ) ✗ sin( ✍ 2 ✟ ✞ ),
( ✝ , ✞ )= ✒❖❄ sin( ✍ ✝ )+ ❅ cos( ✍ ✝ )+ ❆ sinh( ✍ ✝ )+ ❇ cos( ✍ ✝ ) ✗ cos( ✍ 2 ✟ ✞ ), where ❄ , ❅ , ❆ , ❇ , ❄ 1 , ❅ 1 , ❆ 1 , ❇ 1 , and ✍ are arbitrary constants.
9.2.2-2. Domain: − ✘ < ✝ < ✘ . Cauchy problem. Initial conditions are prescribed:
( ✆ = ✟ ✢✣✢ ( ✝ ) at ✞ = 0. Boussinesq solution:
Reference ◆ : I. Sneddon (1951).
9.2.2-3. Domain: 0 ≤ ✝ < ✘ . Free vibration of a semiinfinite rod. The following conditions are prescribed:
=0 ✆ at ✞ = 0, ✜ =0
(initial conditions), = ✆ ✙ ( ✞ ) at ✝ = 0, =0 at ✝ =0
at
(boundary conditions). Boussinesq solution:
Reference : I. Sneddon (1951).
9.2.2-4. Domain: 0 ≤ ✝ ≤ ❈ . Boundary value problems. For solutions of various boundary value problems, see Subsection 9.2.3 for ❃ ≡ 0.
9.2.3. Equations of the Form ✽ ✾
This equation is encountered in studying forced (transverse) vibration of elastic rods. ❁
9.2.3-1. Domain: 0 ≤ ✝ ≤ ❈ . Solution in terms of the Green’s function.
1 ☎ . We consider boundary value problems on an interval 0 ≤ ✹ ✝ ≤ ❈ with the general initial condition
= ( ✝ ) at ✞ =0 and various homogeneous boundary conditions. The solution can be represented in terms of the
( ✝ ) at ✞ = 0,
Green’s function as
2 ☎ . Paragraphs 9.2.3-2 through 9.2.3-9 present the Green’s functions for various types of boundary
conditions. The Green’s functions can be evaluated from the formula ■
where the ✍ and ( ✝ ) are determined by solving the self-adjoint eigenvalue problem for the ❑
fourth-order ordinary differential equation ❑
subject to appropriate boundary conditions; the prime denotes differentiation with respect to ❑ ❑ ✝ . The norms of eigenfunctions can be calculated by Krylov’s formula [see Krylov (1949)]: ■
Relations (1) and (2) are written under the assumption that ❑ ✍ = 0 is not an eigenvalue.
9.2.3-2. Both ends of the rod are clamped. Boundary conditions are prescribed:
= ✆ =0 at = 0, = =0 at ✝ = ❈ .
Green’s function:
( ✝ ■ )= ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation cosh( ■ ✍ ❈ ) cos( ✍ ❈ ) = 1. The numerical values of the roots can be calculated from the formulas ■
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
9.2.3-3. Both ends of the rod are hinged. Boundary conditions are prescribed:
= ✆ =0 at = 0, = =0 at ✝ = ❈ .
Green’s function:
2 sin( ✍ ✝ ) sin( ✍ ❊
References : A. N. Krylov (1949), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
9.2.3-4. Both ends of the rod are free. Boundary conditions are prescribed:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ . Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )+cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )+sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = 1. For the numerical values of the roots, see Paragraph 9.2.3-2. The first two terms in the expression of the Green’s function correspond to the zero eigenvalue
= 0, to which two orthogonal eigenfunctions ▲ (1)
0 2 ▲ ❈ ▲ 0 = 1 and 0 =2 − correspond with =
and (2) 2
Reference : A. N. Krylov (1949).
9.2.3-5. One end of the rod is clamped and the other is hinged. Boundary conditions are prescribed:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ .
Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation tan( ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.3-6. One end of the rod is clamped and the other is free. Boundary conditions are prescribed:
= ▼ =0 at
= 0, ✆ = =0 at ✝ = ❈ .
Green’s function: ■
( ✝ ■ )= ✒ sinh( ✍ ❈ )+sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )+cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ; the ❑ ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = −1.
9.2.3-7. One end of the rod is hinged and the other is free. Boundary conditions are prescribed:
=0 at ✝ = ❈ . Green’s function:
( ✝ ) = sin( ✍ ❈ ) sinh( ✍ ✝ ) + sinh( ✍ ❈ ) sin( ✍ ✝ ); the ✍ are positive roots of the transcendental equation tan( ❑ ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.3-8. The first and third derivatives are prescribed at the ends:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ . Green’s function:
2 cos( ✍ ✝ ) cos( ✍ ❊ ) sin( ✍ 2 ✟ ✞ ),
9.2.3-9. Mixed boundary conditions are prescribed at the ends:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
2 sin( ✍ ✝ ) sin( ✍ ❊ ) sin( ✍ ✟ ✞ ),
9.2.4. Equations of the Form ✽ ✾ + ❀ 2 ✽ ✾
9.2.4-1. Particular solutions of the homogeneous equation ( ❃ ≡ 0):
, ✞ )=( ❄ ✝ 3 + ❅ ✝ 2 + ❆ ✝ + ❇ ) sin ✎❚✞ ✌ ✏ ,
, ✞ )=( ❄ ✝ 3 + ❅ ✝ 2 + ❆ ✝ + ❇ ) cos ✎❚✞ ✌ ✏ ,
, ✞ )= ✒ ❄ sin( ✍ ✝ )+ ❅ cos( ✍ ✝ )+ ❆ sinh( ✍ ✝ )+ ❇ cos( ✍ ✝ ) ✗ sin ✎ ✞ ✟ 2 ✍ 4 + ✌ P ✏ ,
( ✝ , ✞ )= ✒ ❄ sin( ✍ ✝ )+ ❅ cos( ✍ ✝ )+ ❆ sinh( ✍ ✝ )+ ❇ cos( ✍ ✝ ) ✗ cos ✎ ✞ ✟ 2 ✍ 4 + ✌ ✏ , where ❄ , ❅ , ❆ , ❇ , and ✍ are arbitrary constants.
9.2.4-2. Domain: 0 ≤ ✝ ≤ ❈ . Solution in terms of the Green’s function.
1 ☎ . We consider boundary value problems on an interval 0 ≤ ✝ ≤ ❈ with the general initial condition
= ( ✝ ) at ✞ =0 and various homogeneous boundary conditions. The solution can be represented in terms of the
( ✝ ) at ✞ = 0,
Green’s function as
2 ☎ . Paragraphs 9.2.4-3 through 9.2.4-10 present the Green’s functions for various types of boundary conditions. The Green’s functions can be evaluated from the formula ■ ■ ■
where the ✍ and ( ✝
) are determined by solving the self-adjoint eigenvalue problem for the fourth- ❑ order ordinary differential equation ✢✣✢✣✢✣✢ ✍ 4 ❑ − = 0 subject to appropriate boundary conditions. The
norms of eigenfunctions can be calculated by formula (2) from Paragraph 9.2.3-1. ❑ ❑
9.2.4-3. The function and its first derivative are prescribed at the ends:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
( ✝ ■ )= ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ; the ❑ ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = 1.
9.2.4-4. The function and its second derivative are prescribed at the ends:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
sin( ✍ ✝ ) sin( ✍ ❊
9.2.4-5. The first and third derivatives are prescribed at the ends:
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
cos( ✍ ✝ ) cos( ✍ ❊ )
9.2.4-6. The second and third derivatives are prescribed at the ends:
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ . Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )+cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )+sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = 1. For the numerical values of the roots, see Paragraph 9.2.3-2.
9.2.4-7. Mixed boundary conditions are prescribed at the ends (case 1):
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )−sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )−cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ; the ❑ ✍ are positive roots of the transcendental equation tan( ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.4-8. Mixed boundary conditions are prescribed at the ends (case 2):
= ✆ =0 at = 0, = =0 at ✝ = ❈ .
Green’s function:
( ✝ )= ■ ✒ sinh( ✍ ❈ )+sin( ✍ ❈ ) ✗ ✒ cosh( ✍ ✝ )−cos( ✍ ✝ ) ✗ − ✒ cosh( ✍ ❈ )+cos( ✍ ❈ ) ✗ ✒ sinh( ✍ ✝ )−sin( ✍ ✝ ) ✗ ;
the ✍ are positive roots of the transcendental equation cosh( ✍ ❈ ) cos( ✍ ❈ ) = −1.
9.2.4-9. Mixed boundary conditions are prescribed at the ends (case 3):
= ✆ =0 at ✝ = 0, = =0 at ✝ = ❈ .
Green’s function:
sin( ✍ ✝ ) sin(
9.2.4-10. Mixed boundary conditions are prescribed at the ends (case 4):
= 0, ✆ = =0 at ✝ = ❈ . Green’s function:
( ✝ ) = sin( ✍ ❈ ) sinh( ✍ ✝ ) + sinh( ✍ ❈ ) sin( ✍ ✝ ); the ✍ are positive roots of the transcendental equation tan( ❑ ✍ ❈ ) − tanh( ✍ ❈ ) = 0.
9.2.5. Other Equations
9.2.5-1. Equations containing the first derivative with respect to ✞ .
The change of variable ✜ (
, ✞ )= ❳ − ( ✝ , ✞ ) leads to the equation
which is discussed in Subsection 9.2.1.
This is a special case of equation 9.6.4.2 with ❩
This is a special case of equation 9.6.4.1 with ❩
= 4. The transformation
( ❵ , ❛ )= ( ✩ , ✥ ) exp ✴❝❜ ❞ ( ❛ ) ❡ ❛✷❢ , ✩ = ❵ ❣ ( ❛ )+ ❜ ( ❛ ) ❣ ( ❛ ) ❡ ❛ , ❤ = ❜ ❫ ( ❛ ) ❣ 4 ( ❛ ) ❡ ❛ ,
where ❣ ( ❛ ) = exp ✐❝❜ ❥ ( ❛ ) ❡ ❛✷❢ , leads to the constant coefficient equation
which is discussed in Subsection 9.2.1.
This is a special case of equation 9.6.4.4 with q = 1 and ✼ = 4.
9.2.5-2. Equations containing the second derivative with respect to r .
( ✈ , r ) ≡ 0 this equation governs transverse vibration of an elastic rod in a resisting medium with velocity-proportional resistance coefficient. ❧
The change of variable ✇ ( ✈ , r ) = exp ① − 1 2 q r✑② ( ✈ , r ) leads to the equation
+ 2 ③ ❧ ❦ − 1 4 q 2 = exp ① 1 2 q r✑② r ✉ 2 ✈ 4 ( ✈ , r ),
which is discussed in Subsection 9.2.4.
This is a special case of equation 9.6.4.2 with q = 2 and ✼ = 4.
This is a special case of equation 9.6.4.4 with q = 2 and ✼ = 4.
. General solution (two representations): ❧
, r ) is an arbitrary function satisfying the heat equation − = 0; q = 1, 2.
2 ⑥ . Fundamental solution:
( ✈ , r )= ❶
exp ④ − r ⑤ .
3 ❷ ⑥ . Domain: − ❸ < ✈ < ❸ . Cauchy problem. Initial conditions are prescribed:
Reference : G. E. Shilov (1965).
1 ⑥ . Fundamental solution: ⑩ ⑩
( ✈ , r )=
ln ❶ ✈ 2 + r 2 − ✈ arctan r − ( r + ✈ ) ln | r + ✈ |
− ( r − ✈ ) ln | r − ✈ |+ | r + ✈ |+ | r − ✈ | ❼ .
2 ⑥ . Domain: − ❸ < ✈ < ❸ . Cauchy problem. Initial conditions are prescribed:
= ❥ ( ✈ ) at r = 0. Solution:
Reference : G. E. Shilov (1965).
4 ♥ = 0. s ♥ s ♥ ♣ ♥ ♣
General solution (three representations):
( ✈ , r )= ❥ 1 ( r − ✈ )+ ❥ 2 ( r + ✈ )+ r ❾✣❿ 1 ( r − ✈ )+ ❿ 2 ( r + ✈ ) ➀ ,
( ✈ , r )= ❥ 1 ( r − ✈ )+ ❥ 2 ( r + ✈ )+ ✈ ❾ ❿ 1 ( r − ✈ )+ ❿ 2 ( r + ✈ ) ➀ ,
( ✈ , r )= ❥ 1 ( r − ✈ )+ ❥ 2 ( r + ✈ )+( r + ✈ ) ❿ 1 ( r − ✈ )+( r − ✈ ) ❿ 2 ( r + ✈ ), where ❥ 1 ( ➁ ), ❥ 2 ( ➂ ), ❿ 1 ( ➁ ), and ❿ 2 ( ❺✓✫ ➂ ) are arbitrary functions.
Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).