The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and
Trang 1many particular solutions The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation
by means of the initial and boundary conditions
Throughout this section, we consider linear equations in the n-dimensional Euclidean
space Rn or in an open domain V
Rn (exclusive of the boundary) with a sufficiently
smooth boundary S = ∂V
14.2.1-1 Parabolic equations Initial and boundary conditions
In general, a linear second-order partial differential equation of the parabolic type with
nindependent variables can be written as
∂w
∂t – Lx,t [w] = Φ(x, t), (14.2.1.1) where
Lx,t [w]≡ n
i,j=1
a ij (x, t) ∂
2w
∂x i ∂x j +
n
i=1
b i (x, t) ∂x ∂w
i + c(x, t)w, (14.2.1.2)
x ={x1, , x n},
n
i,j=1
a ij (x, t)ξ i ξ j ≥σ
n
i=1
ξ2
i σ>0
Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent
on time t.
Equation (14.2.1.1) is called homogeneous ifΦ(x, t)≡ 0
Cauchy problem (t≥ 0, xRn ) Find a function w that satisfies equation (14.2.1.1) for
t>0and the initial condition
w = f (x) at t=0 (14.2.1.3)
Boundary value problem* (t ≥ 0, x V ) Find a function w that satisfies equa-tion (14.2.1.1) for t >0, the initial condition (14.2.1.3), and the boundary condition
Γx,t [w] = g(x, t) at xS (t >0) (14.2.1.4)
In general, Γx,t is a first-order linear differential operator in the space variables x with
coefficient dependent on x and t The basic types of boundary conditions are described in
Subsection 14.2.2
The initial condition (14.2.1.3) is called homogeneous if f (x) ≡ 0 The boundary
condition (14.2.1.4) is called homogeneous if g(x, t)≡ 0
14.2.1-2 Hyperbolic equations Initial and boundary conditions
Consider a second-order linear partial differential equation of the hyperbolic type with n
independent variables of the general form
∂2w
∂t2 + ϕ(x, t)
∂w
∂t – Lx,t [w] = Φ(x, t), (14.2.1.5)
* Boundary value problems for parabolic and hyperbolic equations are sometimes called mixed or
initial-boundary value problems.
Trang 2592 LINEARPARTIALDIFFERENTIALEQUATIONS
where the linear differential operator Lx,t is defined by (14.2.1.2) Hyperbolic equations
govern unsteady wave processes, which depend on time t.
Equation (14.2.1.5) is said to be homogeneous ifΦ(x, t)≡ 0
Cauchy problem (t≥ 0, xRn ) Find a function w that satisfies equation (14.2.1.5) for
t>0and the initial conditions
w = f0(x) at t=0,
∂ t w = f1(x) at t=0 (14.2.1.6)
Boundary value problem (t≥0, xV ) Find a function w that satisfies equation (14.2.1.5) for t >0, the initial conditions (14.2.1.6), and boundary condition (14.2.1.4)
The initial conditions (14.2.1.6) are called homogeneous if f0(x)≡ 0and f1(x)≡ 0
Generalized Cauchy problem In the generalized Cauchy problem for a hyperbolic
equation with two independent variables, values of the unknown function and its first
derivatives are prescribed on a curve in the (x, t) plane Alternatively, values of the
unknown function and its derivative along the normal to this curve may be prescribed For more details, see Paragraph 14.8.4-4
Goursat problem On the characteristics of a hyperbolic equation with two independent
variables, the values of the unknown function w are prescribed; for details, see
Para-graph 14.8.4-5)
14.2.1-3 Elliptic equations Boundary conditions
In general, a second-order linear partial differential equation of elliptic type with n
inde-pendent variables can be written as
where
Lx[w]≡ n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j +
n
i=1
b i(x)∂x ∂w
i + c(x)w, (14.2.1.8)
n
i,j=1
a ij (x)ξ i ξ j ≥σ
n
i=1
ξ2
i σ >0
Elliptic equations govern steady-state thermal, diffusion, and other phenomena independent
of time t.
Equation (14.2.1.7) is said to be homogeneous ifΦ(x)≡ 0
Boundary value problem Find a function w that satisfies equation (14.2.1.7) and the
boundary condition
Γx[w] = g(x) at xS. (14.2.1.9)
In general,Γx is a first-order linear differential operator in the space variables x The basic
types of boundary conditions are described below in Subsection 14.2.2
The boundary condition (14.2.1.9) is called homogeneous if g(x) ≡ 0 The boundary value problem (14.2.1.7)–(14.2.1.9) is said to be homogeneous ifΦ≡ 0and g≡ 0
Trang 3TABLE 14.3 Boundary conditions for various boundary value problems specified by
parabolic and hyperbolic equations in two independent variables (x1 ≤x≤x2)
Type of problem Boundary condition at x = x1 Boundary condition at x = x2
First boundary value problem w = g1(t) w = g2(t)
Second boundary value problem ∂ x w = g1(t) ∂ x w = g2(t)
Third boundary value problem ∂ x w + β1w = g1(t) (β1 < 0) ∂ x w + β2w = g2(t) (β2 > 0) Mixed boundary value problem w = g1(t) ∂ x w = g2(t)
Mixed boundary value problem ∂ x w = g1(t) w = g2(t)
14.2.2 First, Second, Third, and Mixed Boundary Value Problems
For any (parabolic, hyperbolic, and elliptic) second-order partial differential equations, it is conventional to distinguish four basic types of boundary value problems, depending on the form of the boundary conditions (14.2.1.4) [see also the analogous condition (14.2.1.9)]
For simplicity, here we confine ourselves to the case where the coefficients a ij of equations (14.2.1.1) and (14.2.1.5) have the special form
a ij (x, t) = a(x, t)δ ij, δ ij =
1 if i = j,
0 if i≠j.
This situation is rather frequent in applications; such coefficients are used to describe various phenomena (processes) in isotropic media
The function w(x, t) takes prescribed values at the boundary S of the domain:
w(x, t) = g1(x, t) for xS. (14.2.2.1)
Second boundary value problem The derivative along the (outward) normal is
pre-scribed at the boundary S of the domain:
∂w
∂N = g2(x, t) for xS. (14.2.2.2)
In heat transfer problems, where w is temperature, the left-hand side of the boundary condition (14.2.2.2) is proportional to the heat flux per unit area of the surface S.
Third boundary value problem A linear relationship between the unknown function
and its normal derivative is prescribed at the boundary S of the domain:
∂w
∂N + k(x, t)w = g3(x, t) for xS. (14.2.2.3)
Usually, it is assumed that k(x, t) = const In mass transfer problems, where w is
concen-tration, the boundary condition (14.2.2.3) with g3≡ 0describes a surface chemical reaction
of the first order
Mixed boundary value problems Conditions of various types, listed above, are set at
different portions of the boundary S.
If g1 ≡ 0, g2 ≡ 0, or g3 ≡ 0, the respective boundary conditions (14.2.2.1), (14.2.2.2), (14.2.2.3) are said to be homogeneous
Boundary conditions for various boundary value problems for parabolic and hyperbolic
equations in two independent variables x and t are displayed in Table 14.3 The equation
coefficients are assumed to be continuous, with the coefficients of the highest derivatives
being nonzero in the range x1≤x≤x2considered
Remark. For elliptic equations, the first boundary value problem is often called the Dirichlet problem, and the second boundary value problem is called the Neumann problem.
Trang 4594 LINEARPARTIALDIFFERENTIALEQUATIONS
14.3 Properties and Exact Solutions of Linear Equations
14.3.1 Homogeneous Linear Equations and Their Particular
Solutions
14.3.1-1 Preliminary remarks
For brevity, in this paragraph a homogeneous linear partial differential equation will be written as
For second-order linear parabolic and hyperbolic equations, the linear differential opera-torL[w] is defined by the left-hand side of equations (14.2.1.1) and (14.2.1.5), respectively.
It is assumed that equation (14.3.1.1) is an arbitrary homogeneous linear partial differential
equation of any order in the variables t, x1, , x nwith sufficiently smooth coefficients
A linear operatorL possesses the properties
L[w1+ w2] =L[w1] +L[w2],
L[Aw] = AL[w], A= const
An arbitrary homogeneous linear equation (14.3.1.1) has a trivial solution, w≡ 0
A function w is called a classical solution of equation (14.3.1.1) if w, when substituted into (14.3.1.1), turns the equation into an identity and if all partial derivatives of w that
occur in (14.3.1.1) are continuous; the notion of a classical solution is directly linked to the range of the independent variables In what follows, we usually write “solution” instead of
“classical solution” for brevity
14.3.1-2 Usage of particular solutions for the construction of other solutions
Below are some properties of particular solutions of homogeneous linear equations
1◦ Let w1= w1(x, t), w2= w2(x, t), , w k = w k (x, t) be any particular solutions of the
homogeneous equation (14.3.1.1) Then the linear combination
w = A1w1+ A2w2+· · · + A k w k (14.3.1.2)
with arbitrary constants A1, A2, , A kis also a solution of equation (14.3.1.1); in physics,
this property is known as the principle of linear superposition.
Suppose {w k} is an infinite sequence of solutions of equation (14.3.1.1) Then the series∞
k=1w k , irrespective of its convergence, is called a formal solution of (14.3.1.1) If the
solutions w kare classical, the series is uniformly convergent, and the sum of the series has all the necessary particular derivatives, then the sum of the series is a classical solution of equation (14.3.1.1)
2◦ Let the coefficients of the linear differential operator L be independent of time t If
equation (14.3.1.1) has a particular solution 2w = 2w(x, t), then the partial derivatives of 2 w
with respect to time,*
∂ w2
∂t , ∂
22w
∂t2 , .,
∂ k 2w
∂t k , .,
are also solutions of equation (14.3.1.1)
* Here and in what follows, it is assumed that the particular solutionw2 is differentiable sufficiently many
times with respect to t and x1, , x n(or the parameters).
Trang 53◦ Let the coefficients of the linear differential operator L be independent of the space variables x1, , x n If equation (14.3.1.1) has a particular solutionw2 = 2w(x, t), then the
partial derivatives of2w with respect to the space coordinates
∂ w2
∂x1,
∂ w2
∂x2,
∂ w2
∂x3, .,
∂22w
∂x2 1
2w2
∂x1∂x2, .,
∂ k+m w2
∂x k
2∂x m3
, .
are also solutions of equation (14.3.1.1)
If the coefficients of L are independent of only one space coordinate, say x1, and equation (14.3.1.1) has a particular solutionw2 = 2w(x, t), then the partial derivatives
∂ w2
∂x1,
∂2w2
∂x2 1
, ., ∂ k 2w
∂x k
1
, .
are also solutions of equation (14.3.1.1)
4◦ Let the coefficients of the linear differential operator L be constant and let
equa-tion (14.3.1.1) have a particular soluequa-tionw2 = 2w(x, t) Then any particular derivatives of 2 w
with respect to time and the space coordinates (inclusive mixed derivatives)
∂ w2
∂t , ∂ w2
∂x1, .,
∂2w2
∂x2 2
22w
∂t∂x1, .,
∂ k w2
∂x k
3
, .
are solutions of equation (14.3.1.1)
5◦ Suppose equation (14.3.1.1) has a particular solution dependent on a parameter μ,
2w = 2w(x, t; μ), and the coefficients of the linear differential operator L are independent of μ
(but can depend on time and the space coordinates) Then, by differentiating 2w with respect
to μ, one obtains other solutions of equation (14.3.1.1),
∂ w2
∂μ, ∂
22w
∂μ2, .,
∂ k 2w
∂μ k, .
Example 1 The linear equation
∂w
∂t = a ∂
2w
∂x2 + bw
has a particular solution
w (x, t) = exp[μx + (aμ2+ b)t], where μ is an arbitrary constant Differentiating this equation with respect to μ yields another solution
w (x, t) = (x +2aμt ) exp[μx + (aμ2+ b)t].
Let some constants μ1, , μ k belong to the range of the parameter μ Then the sum
w = A1w(x, t; μ2 1) +· · · + A k w(x, t; μ2 k), (14.3.1.3)
where A1, , A k are arbitrary constants, is also a solution of the homogeneous linear equation (14.3.1.1) The number of terms in sum (14.3.1.3) can be both finite and infinite
6◦ Another effective way of constructing solutions involves the following The particular
solution w(x, t; μ), which depends on the parameter μ (as before, it is assumed that the2
coefficients of the linear differential operatorL are independent of μ), is first multiplied by
an arbitrary function ϕ(μ) Then the resulting expression is integrated with respect to μ over some interval [α, β] Thus, one obtains a new function,
β
α w(x, t; μ)ϕ(μ) dμ,2
which is also a solution of the original homogeneous linear equation
The properties listed in Items 1◦–6◦ enable one to use known particular solutions
to construct other particular solutions of homogeneous linear equations of mathematical physics
Trang 6596 LINEARPARTIALDIFFERENTIALEQUATIONS
TABLE 14.4 Homogeneous linear partial differential equations that admit multiplicative separable solutions
No Form of equation (14.3.1.1) Form of particular solutions
1 Equation coefficients areconstant w (x, t) = A exp(λt + β1x1+· · · + β n x n),
λ , β1, , β nare related by an algebraic equation
2 Equation coefficients areindependent of time t w (x, t) = e λt ψ(x),
λis an arbitrary constant, x ={x1, , x n}
3 Equation coefficients are independentof the coordinates x
1, , x n
w (x, t) = exp(β1x1+· · · + β n x n )ψ(t),
β1, , β nare arbitrary constants
4 Equation coefficients are independentof the coordinates x
1, , x k
w (x, t) = exp(β1x1+· · ·+β k x k )ψ(t, x k+1 , , x n),
β1, , β kare arbitrary constants 5
L t [w] + Lx[w] =0,
operator L t depends on only t,
operator Lx depends on only x
w (x, t) = ϕ(t)ψ(x),
ϕ (t) satisfies the equation L t [ϕ] + λϕ =0,
ψ (x) satisfies the equation Lx[ψ] – λψ =0
6
L t [w] + L1[w] + · · · + L n [w] =0,
operator L t depends on only t,
operator L k depends on only x k
w (x, t) = ϕ(t)ψ1(x1) ψ n (x n),
ϕ (t) satisfies the equation L t [ϕ] + λϕ =0,
ψ k (x k ) satisfies the equation L k [ψ k ] + β k ψ k= 0,
λ + β1+· · · + β n= 0
7
f0(x1)L t [w] +n
k=1 f k (x1)L k [w] =0,
operator L t depends on only t,
operator L k depends on only x k
w (x, t) = ϕ(t)ψ1(x1) ψ n (x n),
L t [ϕ] + λϕ =0,
L k [ψ k ] + β k ψ k= 0, k = 2, , n,
f1(x1)L1[ψ1 ] – *
λf0(x1 ) + n
k=2 β k f k (x1 ) +
ψ1 = 0
8
∂w
∂t + L1 ,t [w] + · · · + L n,t [w] =0,
where L k,t [w] = mk
s=0 f ks (x k , t) ∂
s w
∂x s k
w (x, t) = ψ1(x1, t)ψ2(x2, t) ψ n (x n , t),
∂ψ k
∂t + L k,t [ψ k ] = λ k (t)ψ k , k =1, , n,
λ1(t) + λ2(t) + · · · + λ n (t) =0
14.3.1-3 Multiplicative and additive separable solutions
1◦ Many homogeneous linear partial differential equations have solutions that can be
represented as the product of functions depending on different arguments Such solutions
are referred to as multiplicative separable solutions; very commonly these solutions are briefly, but less accurately, called just separable solutions.
Table 14.4 presents the most commonly encountered types of homogeneous linear dif-ferential equations with many independent variables that admit exact separable solutions Linear combinations of particular solutions that correspond to different values of the
separa-tion parameters, λ, β1, , β n, are also solutions of the equations in question For brevity, the word “operator” is used to denote “linear differential operator.”
For a constant coefficient equation (see the first row in Table 14.4), the separation parameters must satisfy the algebraic equation
D(λ, β1, , β n) =0, (14.3.1.4) which results from substituting the solution into equation (14.3.1.1) In physical
applica-tions, equation (14.3.1.4) is usually referred to as a dispersion equation Any n of the n +1 separation parameters in (14.3.1.4) can be treated as arbitrary
Trang 7Example 2 Consider the linear equation
∂2w
∂t2 + k ∂w
∂t = a2∂
2w
∂x2 + b ∂w
∂x + cw.
A particular solution is sought in the form
w = A exp(βx + λt).
This results in the dispersion equation λ2+ kλ = a2β2+ bβ + c, where one of the two parameters β or λ can
be treated as arbitrary.
For more complex multiplicative separable solutions to this equation, see Subsection 14.4.1.
Note that constant coefficient equations also admit more sophisticated solutions; see the second and third rows, the last column
The eighth row of Table 14.4 presents the case of incomplete separation of variables where the solution is separated with respect to the space variables x1, , x n, but is not
separated with respect to time t.
Remark 1. For stationary equations that do not depend on t, one should set λ =0, Lt [w]≡ 0, and ϕ(t) ≡ 1
in rows 1, 6, and 7 of Table 14.4.
Remark 2 Multiplicative separable solutions play an important role in the theory of linear partial differ-ential equations; they are used for finding solutions to stationary and nonstationary boundary value problems; see Sections 14.4 and 14.7–14.9.
2◦ Linear partial differential equations of the form
L t [w] + Lx[w] = f (x) + g(t),
where L t is a linear differential operator that depends on only t and Lxis a linear differential
operator that depends on only x, have solutions that can be represented as the sum of
functions depending on different arguments
w = u(x) + v(t).
Such solutions are referred to as additive separable solutions.
Example 3 The equation from Example 2 admits an exact additive separable solution w = u(x) + v(t)
with u(x) and v(t) described by the linear constant-coefficient ordinary differential equations
a2u xx + bu x + cu = C,
v tt + kv t – cv = C, where C is an arbitrary constant, which are easy to integrate A more general partial differential equation with variable coefficients a = a(x), b = b(x), k = k(t), and c = const also admits an additive separable solution.
14.3.1-4 Solutions in the form of infinite series in t.
1◦ The equation
∂w
∂t = M [w], where M is an arbitrary linear differential operator of the second (or any) order that only
depends on the space variables, has the formal series solution
w(x, t) = f (x) +
∞
k=1
t k
k! M
k [f (x)], M k [f ] = M
M k–1[f ]
,
where f (x) is an arbitrary infinitely differentiable function This solution satisfies the initial condition w(x,0) = f (x).