14.13.1.3 Table 14.14 gives examples of such transformations for linear boundary value problems with one space variable for parabolic and hyperbolic equations.. Transformations That Lead
Trang 114.12.1-2 Hyperbolic equations with two independent variables.
Consider the problem for the homogeneous linear hyperbolic equation
∂2w
∂t2 + ϕ(x)
∂w
∂t = a(x) ∂
2w
∂x2 + b(x)
∂w
∂x + c(x)w (14.12.1.9) with the homogeneous initial conditions
w=0 at t =0,
∂ t w=0 at t =0, (14.12.1.10) and the boundary conditions (14.12.1.3) and (14.12.1.4)
The solution of problem (14.12.1.9), (14.12.1.10), (14.12.1.3), (14.12.1.4) with the
non-stationary boundary condition (14.12.1.3) at x = x1can be expressed by formula (14.12.1.5)
in terms of the solution u(x, t) of the auxiliary problem for equation (14.12.1.9) with the initial conditions (14.12.1.10) and boundary condition (14.12.1.4), for u instead of w, and the simpler stationary boundary condition (14.12.1.6) at x = x1
In this case, the remark made in Paragraph 14.12.1-1 remains valid
14.12.1-3 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,
∂ k w
∂t k =
n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j +
n
i=1
b i(x)∂x ∂w
i + c(x)w, (14.12.1.11)
where k =1, 2and x ={x1, , x n}
Let V be some bounded domain inRn with a sufficiently smooth surface S = ∂V The
solution of the boundary value problem for equation (14.12.1.11) in V with the homogeneous initial conditions (14.12.1.2) if k = 1or (14.12.1.10) if k = 2, and the nonhomogeneous linear boundary condition
Γx[w] = g(t) for xS, (14.12.1.12)
is given by
w (x, t) = ∂
∂t
t
0 u (x, t – τ ) g(τ ) dτ =
t
0
∂u
∂t (x, t – τ ) g(τ ) dτ Here, u(x, t) is the solution of the auxiliary problem for equation (14.12.1.11) with the same
initial conditions, (14.12.1.2) or (14.12.1.10), for u instead of w, and the simpler stationary
boundary condition
Γx[u] =1 for xS Note that (14.12.1.12) can represent a boundary condition of the first, second, or third kind; the coefficients of the operatorΓxare assumed to be independent of t.
Trang 2648 LINEARPARTIALDIFFERENTIALEQUATIONS
14.12.2 Problems for Nonhomogeneous Linear Equations
14.12.2-1 Parabolic equations
The solution of the nonhomogeneous linear equation
∂w
∂t =
n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j +
n
i=1
b i(x)∂x ∂w
i + c(x)w + Φ(x, t)
with the homogeneous initial condition (14.12.1.2) and the homogeneous boundary condi-tion
Γx[w] =0 for xS (14.12.2.1)
can be represented in the form (Duhamel’s second principle)
w (x, t) =
t
0 U (x, t – τ , τ ) dτ (14.12.2.2)
Here, U (x, t, τ ) is the solution of the auxiliary problem for the homogeneous equation
∂U
∂t =
n
i,j=1
a ij(x) ∂
2U
∂x i ∂x j +
n
i=1
b i(x)∂x ∂U
i + c(x)U
with the boundary condition (14.12.2.1), in which w must be substituted by U , and the
nonhomogeneous initial condition
U =Φ(x, τ) at t =0,
where τ is a parameter.
Note that (14.12.2.1) can represent a boundary condition of the first, second, or third kind; the coefficients of the operatorΓxare assumed to be independent of t.
14.12.2-2 Hyperbolic equations
The solution of the nonhomogeneous linear equation
∂2w
∂t2 + ϕ(x)
∂w
∂t =
n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j +
n
i=1
b i(x)∂x ∂w
i + c(x)w + Φ(x, t)
with the homogeneous initial conditions (14.12.1.10) and homogeneous boundary
con-dition (14.12.2.1) can be expressed by formula (14.12.2.2) in terms of the solution U =
U (x, t, τ ) of the auxiliary problem for the homogeneous equation
∂2U
∂t2 + ϕ(x)
∂U
∂t =
n
i,j=1
a ij(x) ∂
2U
∂x i ∂x j +
n
i=1
b i(x)∂x ∂U
i + c(x)U
with the homogeneous initial and boundary conditions, (14.12.1.2) and (14.12.2.1), where
w must be replaced by U , and the nonhomogeneous initial condition
∂ t U =Φ(x, τ) at t =0,
where τ is a parameter.
Note that (14.12.2.1) can represent a boundary condition of the first, second, or third kind
Trang 314.13 Transformations Simplifying Initial and Boundary
Conditions
14.13.1 Transformations That Lead to Homogeneous Boundary
Conditions
A linear problem with arbitrary nonhomogeneous boundary conditions,
Γ(k)
x,t [w] = g k (x, t) for xS k, (14.13.1.1) can be reduced to a linear problem with homogeneous boundary conditions To this end, one should perform the change of variable
w (x, t) = ψ(x, t) + u(x, t), (14.13.1.2)
where u is a new unknown function and ψ is any function that satisfies the nonhomogeneous
boundary conditions (14.13.1.1),
Γ(k)
x,t [ψ] = g k (x, t) for xS k. (14.13.1.3) Table 14.14 gives examples of such transformations for linear boundary value problems with one space variable for parabolic and hyperbolic equations In the third boundary value
problem, it is assumed that k1 <0and k2>0
TABLE 14.14
Simple transformations of the form w(x, t) = ψ(x, t) + u(x, t) that lead to
homogeneous boundary conditions in problems with one space variables (0 ≤x≤l)
No Problems Boundary conditions Function ψ(x, t)
1 First boundaryvalue problem w = g1(t) at x =0
w = g2(t) at x = l ψ (x, t) = g1(t) +
x l
g2(t) – g1(t)
2 Second boundaryvalue problem ∂ x w = g1(t) at x =0
∂ x w = g2(t) at x = l ψ (x, t) = xg1(t) +
x2
2l
g2(t) – g1(t)
3 Third boundaryvalue problem ∂ x w + k1w = g1(t) at x =0
∂ x w + k2w = g2(t) at x = l ψ (x, t) =
(k2x– 1– k2 )g1(t) + (1 – k1x )g2(t)
k2– k1– k1k2
4 Mixed boundaryvalue problem w = g1(t) at x =0
∂ x w = g2(t) at x = l ψ (x, t) = g1(t) + xg2(t)
5 Mixed boundaryvalue problem ∂ x w = g1(t) at x =0
w = g2(t) at x = l ψ (x, t) = (x – l)g1(t) + g2(t)
Note that the selection of the function ψ is of a purely algebraic nature and is not connected with the equation in question; there are infinitely many suitable functions ψ that
satisfy condition (14.13.1.3) Transformations of the form (14.13.1.2) can often be used at the first stage of solving boundary value problems
Trang 4650 LINEARPARTIALDIFFERENTIALEQUATIONS
14.13.2 Transformations That Lead to Homogeneous Initial and
Boundary Conditions
A linear problem with nonhomogeneous initial and boundary conditions can be reduced
to a linear problem with homogeneous initial and boundary conditions To this end, one
should introduce a new dependent variable u by formula (14.13.1.2), where the function ψ
must satisfy nonhomogeneous initial and boundary conditions
Below we specify some simple functions ψ that can be used in transformation (14.13.1.2)
to obtain boundary value problems with homogeneous initial and boundary conditions To
be specific, we consider a parabolic equation with one space variable and the general initial condition
w = f (x) at t =0 (14.13.2.1)
1 First boundary value problem: the initial condition is (14.13.2.1) and the boundary
conditions are given in row 1 of Table 14.14 Suppose that the initial and boundary
condi-tions are compatible, i.e., f (0) = g1(0) and f (l) = g2(0) Then, in transformation (14.13.1.2), one can take
ψ (x, t) = f (x) + g1(t) – g1(0) +x
l
g2(t) – g1(t) + g1(0) – g2(0)
2 Second boundary value problem: the initial condition is (14.13.2.1) and the boundary
conditions are given in row 2 of Table 14.14 Suppose that the initial and boundary
condi-tions are compatible, i.e., f (0) = g1(0) and f (l) = g2(0) Then, in transformation (14.13.1.2), one can set
ψ (x, t) = f (x) + x
g1(t) – g1(0)
+ x
2
2l
g2(t) – g1(t) + g1(0) – g2(0)
3 Third boundary value problem: the initial condition is (14.13.2.1) and the boundary
conditions are given in row 3 of Table 14.14 If the initial and boundary conditions are compatible, then, in transformation (14.13.1.2), one can take
ψ (x, t) = f (x)+ (k2x–1– k2l )[g1(t) – g1(0)] + (1– k1x )[g2(t) – g2(0)]
k2– k1– k1k2l (k1<0, k2>0)
4 Mixed boundary value problem: the initial condition is (14.13.2.1) and the boundary
conditions are given in row 4 of Table 14.14 Suppose that the initial and boundary
condi-tions are compatible, i.e., f (0) = g1(0) and f (l) = g2(0) Then, in transformation (14.13.1.2), one can set
ψ (x, t) = f (x) + g1(t) – g1(0) + x
g2(t) – g2(0)
5 Mixed boundary value problem: the initial condition is (14.13.2.1) and the boundary
conditions are given in row 5 of Table 14.14 Suppose that the initial and boundary
condi-tions are compatible, i.e., f (0) = g1(0) and f (l) = g2(0) Then, in transformation (14.13.1.2), one can take
ψ (x, t) = f (x) + (x – l)
g1(t) – g1(0)
+ g2(t) – g2(0)
References for Chapter 14
Akulenko, L D and Nesterov, S V., High Precision Methods in Eigenvalue Problems and Their Applications,
Chapman & Hall/CRC Press, Boca Raton, 2004.
Butkovskiy, A G., Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons,
New York, 1982.
Trang 5Carslaw, H S and Jaeger, J C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.
Constanda, C., Solution Techniques for Elementary Partial Differential Equations, Chapman & Hall/CRC
Press, Boca Raton, 2002.
Courant, R and Hilbert, D., Methods of Mathematical Physics, Vol 2, Wiley-Interscience, New York, 1989 Dezin, A A., Partial Differential Equations An Introduction to a General Theory of Linear Boundary Value
Problems, Springer-Verlag, Berlin, 1987.
Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon Press, New
York, 1965.
Duffy, D G., Transform Methods for Solving Partial Differential Equations, 2nd Edition, Chapman & Hall/CRC
Press, Boca Raton, 2004.
Farlow, S J., Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, New York, 1982 Guenther, R B and Lee, J W., Partial Differential Equations of Mathematical Physics and Integral Equations,
Dover Publications, New York, 1996.
Haberman, R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value
Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.
Hanna, J R and Rowland, J H., Fourier Series, Transforms, and Boundary Value Problems,
Wiley-Interscience, New York, 1990.
Kanwal, R P., Generalized Functions Theory and Technique, Academic Press, Orlando, 1983.
Leis, R., Initial-Boundary Value Problems in Mathematical Physics, John Wiley & Sons, Chichester, 1986 Mikhlin, S G (Editor), Linear Equations of Mathematical Physics, Holt, Rinehart and Winston, New York,
1967.
Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, London, 1977.
Moon, P and Spencer, D E., Field Theory Handbook, Including Coordinate Systems, Differential Equations
and Their Solutions, 3rd Edition, Springer-Verlag, Berlin, 1988.
Morse, P M and Feshbach, H., Methods of Theoretical Physics, Vols 1 and 2, McGraw-Hill, New York,
1953.
Petrovsky, I G., Lectures on Partial Differential Equations, Dover Publications, New York, 1991.
Polyanin, A D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman &
Hall/CRC Press, Boca Raton, 2002.
Sneddon, I N., Fourier Transformations, Dover Publications, New York, 1995.
Stakgold, I., Boundary Value Problems of Mathematical Physics Vols 1 and 2, Society for Industrial & Applied
Mathematics, Philadelphia, 2000.
Strauss, W A., Partial Differential Equations An Introduction, John Wiley & Sons, New York, 1992 Tikhonov, A N and Samarskii, A A., Equations of Mathematical Physics, Dover Publications, New York,
1990.
Vladimirov, V S., Equations of Mathematical Physics, Dekker, New York, 1971.
Zauderer, E., Partial Differential Equations of Applied Mathematics, Wiley-Interscience, New York, 1989 Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, New York, 1997.
Trang 7Nonlinear Partial Differential Equations
15.1 Classification of Second-Order Nonlinear
Equations
15.1.1 Classification of Semilinear Equations in Two Independent
Variables
A second-order semilinear partial differential equation in two independent variables has
the form
a (x, y) ∂
2w
∂x2 +2b (x, y) ∂
2w
∂x∂y + c(x, y) ∂
2w
∂y2 = f
x , y, w, ∂w
∂x, ∂w
∂y
(15.1.1.1) This equation is classified according to the sign of the discriminant
where the arguments of the equation coefficients are omitted for brevity Given a point
(x, y), equation (15.1.1.1) is
parabolic if δ =0,
hyperbolic if δ >0,
elliptic if δ <0
(15.1.1.3)
The reduction of equation (15.1.1.1) to a canonical form on the basis of the solution
of the characteristic equations entirely coincides with that used for linear equations (see Subsection 14.1.1)
The classification of semilinear equations of the form (15.1.1.1) does not depend on their solutions—it is determined solely by the coefficients of the highest derivatives on the left-hand side
15.1.2 Classification of Nonlinear Equations in Two Independent
Variables
15.1.2-1 Nonlinear equations of general form
In general, a second-order nonlinear partial differential equation in two independent
vari-ables has the form
F
x , y, w, ∂w
∂x, ∂w
∂y, ∂
2w
∂x2,
∂2w
∂x∂y, ∂
2w
∂y2
=0 (15.1.2.1)
653