The procedure of constructing solutions to nonstationary boundary value problems is further different for parabolic and hyperbolic equations; see Subsections 14.4.2 and 14.4.3 below for
Trang 1where ε ij (λ) =
s i(2ϕj
x + k i 2ϕ jx=x
i For system (14.4.1.10) to have nontrivial solutions, its determinant must be zero; we have
ε11(λ)ε22(λ) – ε12(λ)ε21(λ) =0 (14.4.1.11)
Solving the transcendental equation (14.4.1.11) for λ, one obtains the eigenvalues λ = λ n,
where n =1, 2, For these values of λ, equation (14.4.1.6) has nontrivial solutions,
ϕ n (x) = ε12(λ n)2ϕ1(x, λ n ) – ε11(λ n)2ϕ2(x, λ n), (14.4.1.12)
which are called eigenfunctions (these functions are defined up to a constant multiplier).
To facilitate the further analysis, we represent equation (14.4.1.6) in the form
[p(x)ϕ x] x + [λρ(x) – q(x)]ϕ =0, (14.4.1.13) where
p (x) = exp
b (x)
a (x) dx
, q(x) = – c (x)
a (x)exp
b (x)
a (x) dx
, ρ(x) = 1
a (x) exp
b (x)
a (x) dx
(14.4.1.14)
It follows from the adopted assumptions (see the end of Paragraph 14.4.1-1) that p(x),
p
x (x), q(x), and ρ(x) are continuous functions, with p(x) >0and ρ(x) >0
The eigenvalue problem (14.4.1.13), (14.4.1.8) is known to possess the following prop-erties:
1 All eigenvalues λ1, λ2, are real, and λ n → ∞ as n → ∞; consequently, the
number of negative eigenvalues is finite
2 The system of eigenfunctions ϕ1(x), ϕ2(x), is orthogonal on the interval
x1≤x≤x2with weight ρ(x), i.e.,
x2
x1
ρ (x)ϕ n (x)ϕ m (x) dx =0 for n≠m (14.4.1.15)
3 If
q (x)≥ 0, s1k1≤ 0, s2k2≥ 0, (14.4.1.16)
there are no negative eigenvalues If q ≡ 0 and k1 = k2 = 0, the least eigenvalue is
λ1 =0and the corresponding eigenfunction is ϕ1 = const Otherwise, all eigenvalues are positive, provided that conditions (14.4.1.16) are satisfied; the first inequality in (14.4.1.16)
is satisfied if c(x)≤ 0
Subsection 12.2.5 presents some estimates for the eigenvalues λ n and
eigenfunc-tions ϕ n (x).
The procedure of constructing solutions to nonstationary boundary value problems is further different for parabolic and hyperbolic equations; see Subsections 14.4.2 and 14.4.3 below for results (elliptic equations are treated in Subsection 14.4.4)
14.4.2 Problems for Parabolic Equations: Final Stage of Solution
14.4.2-1 Series solutions of boundary value problems for parabolic equations
Consider the problem for the parabolic equation
∂w
∂t = a(x) ∂
2w
∂x2 + b(x)
∂w
∂x +
c (x) + γ(t)
Trang 2(this equation is obtained from (14.4.1.1) in the case α(t)≡ 0and β(t) =1) with homogeneous linear boundary conditions (14.4.1.2) and initial condition (14.4.1.3)
First, one searches for particular solutions to equation (14.4.2.1) in the product form
(14.4.1.5), where the function ϕ(x) is obtained by solving an eigenvalue problem for the
ordinary differential equation (14.4.1.6) with the boundary conditions (14.4.1.8) The
solution of equation (14.4.1.7) with α(t)≡ 0and β(t) =1corresponding to the eigenvalues
λ = λ n and satisfying the normalizing conditions ψ n(0) =1has the form
ψ n (t) = exp
–λ n t+
t
0 γ (ξ) dξ
(14.4.2.2)
Then the solution of the nonstationary boundary value problem (14.4.2.1), (14.4.1.2), (14.4.1.3) is sought in the series form
w (x, t) =
∞
n=1
A n ϕ n (x) ψ n (t), (14.4.2.3)
where the A n are arbitrary constants and the functions w n (x, t) = ϕ n (x) ψ n (t) are particular
solutions (14.4.2.1) satisfying the boundary conditions (14.4.1.2) By the principle of linear superposition, series (14.4.2.3) is also a solution of the original partial differential equation that satisfies the boundary conditions
To determine the coefficients A n, we substitute series (14.4.2.3) into the initial condi-tion (14.4.1.3), thus obtaining
∞
n=1
A n ϕ n (x) = f0(x).
Multiplying this equation by ρ(x)ϕ n (x), where the weight function ρ(x) is defined in (14.4.1.14), then integrating the resulting relation with respect to x over the interval
x1≤x≤x2, and taking into account the properties (14.4.1.15), we find
ϕ n 2
x2
x1
ρ (x)ϕ n (x)f0(x) dx, ϕ n 2=
x2
x1
ρ (x)ϕ2n (x) dx. (14.4.2.4)
Relations (14.4.2.3), (14.4.2.2), (14.4.2.4), and (14.4.1.12) give a formal solution of the nonstationary boundary value problem (14.4.2.1), (14.4.1.2), (14.4.1.3)
Example Consider the first (Dirichlet) boundary value problem on the interval0 ≤x≤ lfor the heat equation
∂w
∂t = ∂
2w
with the general initial condition (14.4.1.3) and the homogeneous boundary conditions
w= 0 at x= 0, w= 0 at x = l. (14.4.2.6)
The function ψ(t) in the particular solution (14.4.1.5) is found from (14.4.2.2), where γ(t) =0:
ψ n (t) = exp(–λ n t) (14.4.2.7)
The functions ϕ n (x) are determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8) with a(x) =1,
b (x) = c(x) =0, s 1= s2= 0, k 1= k2= 1, x 1 = 0, and x 2= l:
ϕ xx + λϕ =0; ϕ= 0 at x= 0, ϕ= 0 at x = l.
Trang 3So we obtain the eigenfunctions and eigenvalues:
ϕ n (x) = sin
nπx l
, λ n=
nπ l
2
, n= 1, 2, (14.4.2.8) The solution to problem (14.4.2.5)–(14.4.2.6), (14.4.1.3) is given by formulas (14.4.2.3), (14.4.2.4) Taking into account thatϕ n 2 = l/2, we obtain
w (x, t) =
∞
n=1
A nsin
nπx l
exp
–n
2π2t
l
, A n= 2
l
l
0
f0(ξ) sin
nπξ l
dξ (14.4.2.9)
If the function f0(x) is twice continuously differentiable and the compatibility conditions (see
Para-graph 14.4.2-2) are satisfied, then series (14.4.2.9) is convergent and admits termwise differentiation, once with
respect to t and twice with respect to x In this case, formula (14.4.2.9) gives the classical smooth solution
of problem (14.4.2.5)–(14.4.2.6), (14.4.1.3) [If f0(x) is not as smooth as indicated or if the compatibility
conditions are not met, then series (14.4.2.9) may converge to a discontinuous function, thus giving only a generalized solution.]
Remark For the solution of linear nonhomogeneous parabolic equations with nonhomogeneous boundary conditions, see Section 14.7.
14.4.2-2 Conditions of compatibility of initial and boundary conditions
Suppose the function w has a continuous derivative with respect to t and two continuous derivatives with respect to x and is a solution of problem (14.4.2.1), (14.4.1.2), (14.4.1.3).
Then the boundary conditions (14.4.1.2) and the initial condition (14.4.1.3) must be con-sistent; namely, the following compatibility conditions must hold:
[s1f
0+ k1f0]x=x1 =0, [s2f
0+ k2f0]x=x2 =0 (14.4.2.10)
If s1=0or s2=0, then the additional compatibility conditions
[a(x)f0 + b(x)f0]x=x1 =0 if s1=0,
[a(x)f0 + b(x)f0]x=x2 =0 if s2=0 (14.4.2.11)
must also hold; the primes denote the derivatives with respect to x.
14.4.3 Problems for Hyperbolic Equations: Final Stage of Solution
14.4.3-1 Series solution of boundary value problems for hyperbolic equations
For hyperbolic equations, the solution of the boundary value problem (14.4.1.1)–(14.4.1.4)
is sought in the series form
w (x, t) =
∞
n=1
ϕ n (x)
A n ψ n1(t) + B n ψ n2(t)
(14.4.3.1)
Here, A n and B n are arbitrary constants The functions ψ n1(t) and ψ n2(t) are particular solutions of the linear equation (14.4.1.7) for ψ (with λ = λ n) that satisfy the conditions
ψ n1(0) =1, ψ
n1(0) =0; ψ n2(0) =0, ψ
n2(0) =1 (14.4.3.2)
The functions ϕ n (x) and λ nare determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8)
Trang 4Substituting solution (14.4.3.1) into the initial conditions (14.4.1.3)–(14.4.1.4) yields
∞
n=1
A n ϕ n (x) = f0(x),
∞
n=1
B n ϕ n (x) = f1(x).
Multiplying these equations by ρ(x)ϕ n (x), where the weight function ρ(x) is defined
in (14.4.1.14), then integrating the resulting relations with respect to x on the interval
x1 ≤x≤x2, and taking into account the properties (14.4.1.15), we obtain the coefficients
of series (14.4.3.1) in the form
ϕ n 2
x2
x1
ρ (x)ϕ n (x)f0(x) dx, B n= 1
ϕ n 2
x2
x1
ρ (x)ϕ n (x)f1(x) dx (14.4.3.3)
The quantityϕ n is defined in (14.4.2.4).
Relations (14.4.3.1), (14.4.1.12), and (14.4.3.3) give a formal solution of the
nonsta-tionary boundary value problem (14.4.1.1)–(14.4.1.4) for α(t) >0
Example Consider a mixed boundary value problem on the interval0 ≤x≤lfor the wave equation
∂2w
∂t2 = ∂
2w
with the general initial conditions (14.4.1.3)–(14.4.1.4) and the homogeneous boundary conditions
w= 0 at x= 0, ∂ x w= 0 at x = l. (14.4.3.5)
The functions ψ n1(t) and ψ n2(t) are determined by the linear equation [see (14.4.1.7) with α(t) =1,
β (t) = γ(t) =0, and λ = λn]
ψ tt + λψ =0 with the initial conditions (14.4.3.2) We find
ψ n1(t) = cos λ n t
, ψ n2(t) = √1
λ nsin λ n t
The functions ϕ n (x) are determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8) with a(x) =1,
b (x) = c(x) =0, s 1= k2= 0, s 2= k1= 1, x 1 = 0, and x 2= l:
ϕ xx + λϕ =0; ϕ= 0 at x =0, ϕ x= 0 at x = l.
So we obtain the eigenfunctions and eigenvalues:
ϕ n (x) = sin(μ n x), μ n=√
λ n = π(2n– 1)
2l , n= 1, 2, (14.4.3.7) The solution to problem (14.4.3.4)–(14.4.3.5), (14.4.1.3)–(14.4.1.4) is given by formulas (14.4.3.1) and (14.4.3.3) Taking into account thatϕ n 2= l/2, we have
w (x, t) =
∞
n=1
A n cos(μ n t ) + B n sin(μ n t)
sin(μ n x), μ n= π(2n– 1)
2l ,
A n= 2
l
l
0
f0(x) sin(μ n x ) dx, B n= 2
lμ n
l
0
f1(x) sin(μ n x ) dx.
(14.4.3.8)
If f0(x) and f1(x) have three and two continuous derivatives, respectively, and the compatibility conditions
are met (see Paragraph 14.4.3-2), then series (14.4.3.8) is convergent and admits double termwise differentiation.
In this case, formula (14.4.3.8) gives the classical smooth solution of problem (14.4.3.4)–(14.4.3.5), (14.4.1.3)– (14.4.1.4).
Remark For the solution of linear nonhomogeneous hyperbolic equations with nonhomogeneous bound-ary conditions, see Section 14.8.
Trang 514.4.3-2 Conditions of compatibility of initial and boundary conditions.
Suppose w is a twice continuously differentiable solution of problem (14.4.1.1)–(14.4.1.4).
Then conditions (14.4.2.10) and (14.4.2.11) must hold In addition, the following conditions
of compatibility of the boundary conditions (14.4.1.2) and initial condition (14.4.1.4) must
be satisfied:
[s1f
1+ k1f1]x=x1 =0, [s2f
1+ k2f1]x=x2 =0
14.4.4 Solution of Boundary Value Problems for Elliptic Equations
14.4.4-1 Solution of special problem for elliptic equations
Now consider a boundary value problem for the elliptic equation
a (x) ∂2w
∂x2 + α(y)
∂2w
∂y2 + b(x)
∂w
∂x + β(y) ∂w
∂y +
c (x) + γ(y)
w=0 (14.4.4.1)
with homogeneous boundary conditions (14.4.1.2) in x and the following mixed (homoge-neous and nonhomoge(homoge-neous) boundary conditions in y:
σ1∂ y w + ν1w=0 at y = y1,
σ2∂ y w + ν2w = f (x) at y = y2 (14.4.4.2)
We assume that the coefficients of equation (14.4.4.1) and boundary conditions (14.4.1.2) and (14.4.4.2) meet the following requirements:
a (x), b(x), c(x) α(y), β(y), and γ(t) are continuous functions,
a (x) >0, α(y) >0, |s1|+|k1|>0, |s2|+|k2|>0, |σ1|+|ν1|>0, |σ2|+|ν2|>0 The approach is based on searching for particular solutions of equation (14.4.4.1) in the product form
w (x, y) = ϕ(x) ψ(y). (14.4.4.3)
As before, we first arrive at the eigenvalue problem (14.4.1.6), (14.4.1.8) for the function
ϕ = ϕ(x); the solution procedure is detailed in Paragraph 14.4.1-3 Further on, we assume the λ n and ϕ n (x) have been found The functions ψ n = ψ n (y) are determined by solving
the linear ordinary differential equation
α (y)ψ yy + β(y)ψ y + [γ(y) – λ n ]ψ =0 (14.4.4.4) subject to the homogeneous boundary condition
σ1∂ y ψ + ν1ψ=0 at y = y1, (14.4.4.5)
which is a consequence of the first condition (14.4.4.2) The functions ψ nare determined
up to a constant factor
Taking advantage of the principle of linear superposition, we seek the solution to the boundary value problem (14.4.4.1), (14.4.4.2), (14.4.1.2) in the series form
w (x, y) =
∞
n=1
A n ϕ n (x)ψ n (y), (14.4.4.6)
Trang 6where A n are arbitrary constants By construction, series (14.4.4.6) will satisfy equa-tion (14.4.4.1) with the boundary condiequa-tions (14.4.1.2) and the first boundary condiequa-tion
(14.4.4.2) In order to find the series coefficients A n, substitute (14.4.4.6) into the second boundary condition (14.4.4.2) to obtain
∞
n=1
A n B n ϕ n (x) = f (x), B n = σ2dψ n
dy
y=y2
+ ν2ψ n (y2). (14.4.4.7)
Further, we follow the same procedure as in Paragraph 14.4.2-1 Specifically, multiplying
(14.4.4.7) by ρ(x)ϕ n (x), then integrating the resulting relation with respect to x over the interval x1≤x≤x2, and taking into account the properties (14.4.1.15), we obtain
B n ϕ n 2
x2
x1
ρ (x)ϕ n (x)f (x) dx, ϕ n 2 =
x2
x1
ρ (x)ϕ2n (x) dx, (14.4.4.8)
where the weight function ρ(x) is defined in (14.4.1.14).
Example Consider the first (Dirichlet) boundary value problem for the Laplace equation
∂2w
∂x2 +∂
2w
subject to the boundary conditions
w= 0 at x= 0, w= 0 at x = l1 ;
w= 0 at y= 0, w = f (x) at y = l2 (14.4.4.10)
in a rectangular domain 0 ≤x≤l , 0 ≤y≤l .
Particular solutions to equation (14.4.4.9) are sought in the form (14.4.4.3) We have the following
eigenvalue problem for ϕ(x):
ϕ xx + λϕ =0; ϕ= 0 at x= 0, ϕ= 0 at x = l1
On solving this problem, we find the eigenfunctions with respective eigenvalues
ϕ n (x) = sin(μ n x), μ n=√
λ n = πn
l , n= 1, 2, (14.4.4.11)
The functions ψ n = ψ n (y) are determined by solving the following problem for a linear ordinary differential
equation with homogeneous boundary conditions:
ψ yy – λ n ψ= 0; ψ= 0 at y= 0 (14.4.4.12)
It is a special case of problem (14.4.4.4)–(14.4.4.5) with α(y) =1, β(y) = γ(y) = 0, σ 1 = 0, and ν 1 = 1 The nontrivial solutions of problem (14.4.4.12) are expressed as
ψ n (y) = sinh(μ n ), μ n=√
λ n = πn
l , n= 1, 2, (14.4.4.13)
Using formulas (14.4.4.6), (14.4.4.8), (14.4.4.11), (14.4.4.13) and taking into account the relations B n=
ψ n (l2) = sinh(μ n l ), ρ(x) =1, and ϕ n 2 = l/2, we find the solution of the original problem (14.4.4.9)–
(14.4.4.10) in the form
w (x, y) =
∞
n=1
A n sin(μ n x ) sinh(μ n ), A n= 2
l sinh(μ n l )
l1
0 f (x) sin(μ n x ) dx, μ n= πn
l .
Trang 7TABLE 14.5 Description of auxiliary problems for equation (14.4.4.1) and problems for associated
functions ϕ(x) and ψ(y) that determine particular solutions of the form (14.4.4.3).
The abbreviation HBC below stands for a “homogeneous boundary condition”
Auxiliary
problem
Functions vanishing
in the boundary conditions (14.4.4.14)
Eigenvalue problem with homogeneous boundary conditions
Another problem with one homogeneous boundary
condition (for λ nfound) Problem 1 f2(y) = f3(x) = f4(x) =0,
function f1(y) prescribed functions ψto be determinedn (y) and values λ n
functions ϕ n (x) satisfy an HBC at x = x2
Problem 2 f1(y) = f3(x) = f4(x) =0,
function f2(y) prescribed
functions ψ n (y) and values λ n
to be determined
functions ϕ(x) satisfy an HBC at x = x1
Problem 3 f1(y) = f2(y) = f4(x) =0,
function f3(x) prescribed
functions ϕ n (x) and values λ n
to be determined
functions ψ(y) satisfy an HBC at y = y2
Problem 4 f1(y) = f2(y) = f3(x) =0,
function f4(x) prescribed functions ϕto be determinedn (x) and values λ n
functions ψ(y) satisfy an HBC at y = y1
14.4.4-2 Generalization to the case of nonhomogeneous boundary conditions
Now consider the linear boundary value problem for the elliptic equation (14.4.4.1) with general nonhomogeneous boundary conditions
s1∂ x w + k1w = f1(y) at x = x1, s2∂ x w + k2w = f2(y) at x = x2,
σ1∂ y w + ν1w = f3(x) at y = y1, σ2∂ y w + ν2w = f4(x) at y = y2 (14.4.4.14) The solution to this problem is the sum of solutions to four simpler auxiliary problems for equation (14.4.4.1), each corresponding to three homogeneous and one nonhomogeneous boundary conditions in (14.4.4.14); see Table 14.5 Each auxiliary problem is solved using the procedure given in Paragraph 14.4.4-1, beginning with the search for solutions
in the form of the product of functions with different arguments (14.4.4.3), determined
by equations (14.4.1.6) and (14.4.4.4) The separation parameter λ is determined by the
solution of a eigenvalue problem with homogeneous boundary conditions; see Table 14.5 The solution to each of the auxiliary problems is sought in the series form (14.4.4.6)
Remark For the solution of linear nonhomogeneous elliptic equations subject to nonhomogeneous bound-ary conditions, see Section 14.9.
14.5 Integral Transforms Method
Various integral transforms are widely used to solve linear problems of mathematical physics The Laplace transform and the Fourier transform are in most common use (its and other integral transforms are considered in Chapter 11 in detail)
14.5.1 Laplace Transform and Its Application in Mathematical
Physics
14.5.1-1 Laplace and inverse Laplace transforms Laplace transforms for derivatives
The Laplace transform of an arbitrary (complex-valued) function f (t) of a real variable t (t≥ 0) is defined by
2
f (p) =L5f (t)6
, where L5f (t)6
≡ ∞
0 e
–pt f (t) dt, (14.5.1.1)
where p = s + iσ is a complex variable, i2= –1