Duhamel’s Principles in Nonstationary Problems

0.10. Duhamel’s Principles in Nonstationary Problems

0.10.1. Problems for Homogeneous Linear Equations

0.10.1-1. Parabolic equations with two independent variables. Consider the problem for the homogeneous linear equation of parabolic type

(1) with the homogeneous initial condition

(2) and the boundary conditions

(4) By appropriately choosing the values of the coefficients 1 , 2 , 1 , and 2 in (3) and (4), one can

obtain the first, second, third, and mixed boundary value problems for equation (1). The solution of problem (1)–(4) with the nonstationary boundary condition (3) at ✚ = ✚ 1 can be expressed by the formula (Duhamel’s first principle)

in terms of the solution ( ✚ , ✕ ) of the auxiliary problem for equation (1) with the initial and boundary conditions (2) and (4), for ✮ instead of ✬ Ï , and the following simpler stationary boundary condition

A similar formula also holds for the homogeneous boundary condition at ✮

= ✚ 1 and

a nonhomogeneous nonstationary boundary condition at ✚ = ✚ 2 .

0.10.1-2. Hyperbolic equations with two independent variables. Consider the problem for the homogeneous linear hyperbolic equation

(7) with the homogeneous initial conditions

and the boundary conditions (3) and (4). The solution of problem (7), (8), (3), (4) with the nonstationary boundary condition (3) at

= ✚ 1 can be expressed by formula (5) in terms of the solution ( ✚ , ✕ ) of the auxiliary problem for equation (7) with the initial conditions (8) and boundary condition (4), for ✮ instead of Ï , and the

simpler stationary boundary condition (6) at ✚ = ✚ 1 .

In this case, the remark made in Paragraph 0.10.1-1 remains valid.

0.10.1-3. Second-order equations with several independent variables. Duhamel’s first principle can also be used to solve homogeneous linear equations of the parabolic

or hyperbolic type with many space variables,

where = 1, 2 and x = { ✚ 1 , ✛✜✛✜✛ , ✚ ✢ }.

be some bounded domain in ✴ with a sufficiently smooth surface ✵ = ✳ . The solution of the boundary value problem for equation (9) in ✳ with the homogeneous initial conditions (2) if = 1 or (8) if = 2, and the nonhomogeneous linear boundary condition

Let ☛

(10) is given by

Here, (x, ✕ ) is the solution of the auxiliary problem for equation (9) with the same initial conditions, (2) or (8), for ✮ instead of Ï , and the simpler stationary boundary condition

x [ ]=1 for x ✥ ✵ .

Note that (10) can represent a boundary condition of the first, second, or third kind; the ✮ coefficients of the operator ✶ x are assumed to be independent of

0.10.2. Problems for Nonhomogeneous Linear Equations

0.10.2-1. Parabolic equations. The solution of the nonhomogeneous linear equation

with the homogeneous initial condition (2) and the homogeneous boundary condition

(11) can be represented in the form (Duhamel’s second principle)

Here, (x, ✕ , ✖ ) is the solution of the auxiliary problem for the homogeneous equation

with the boundary condition (11), in which Ï must be substituted by , and the nonhomogeneous initial condition

= ✙ (x, ✖ ) at ✕ = 0,

where ✷

is a parameter. Note that (11) can represent a boundary condition of the first, second, or third kind; the ✶

coefficients of the operator x are assumed to be independent of ✕ .

0.10.2-2. Hyperbolic equations. The solution of the nonhomogeneous linear equation

with the homogeneous initial conditions (8) and homogeneous boundary condition (11) can be expressed by formula (12) in terms of the solution = (x, ✕ , ✖ ) of the auxiliary problem for the homogeneous equation

with the homogeneous initial and boundary conditions, (2) and (11), where Ï must be replaced by , and the nonhomogeneous initial condition

= ✙ (x, ✖ ) at ✕ = 0,

where ✖ is a parameter. Note that (11) can represent a boundary condition of the first, second, or third kind.

References for Section 0.10: E. Butkov (1968), S. J. Farlow (1982), E. Zauderer (1989), R. Courant and D. Hilbert (1989), D. Zwillinger (1998).