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).