To this end, we set In practice, exponential self-similar solutions are sought using the above existence criterion: if k and m in 15.3.4.2 are known, then the new variables have the form
Trang 1Exponential self-similar solutions exist if equation (15.3.3.3) is invariant under trans-formations of the form
t = ¯ t + ln C, x = Ck¯x, w = Cmw, ¯ (15.3.4.2)
where C > 0 is an arbitrary constant, for some k and m Transformation (15.3.4.2) is a combination of a translation transformation in t and scaling transformations in x and w.
It should be emphasized that these transformations contain an arbitrary parameter C while the equation concerned is independent of C.
Let us find the relation between the parameters α, β in solution (15.3.4.1) and the parameters k, m in the scaling transformation (15.3.4.2) Let w = Φ(x, t) be a solution of
equation (15.3.3.3) Then the function w ¯ = Φ(¯x, ¯t) is a solution of equation (15.3.3.4) In
view of the explicit form of solution (15.3.4.1), we have
¯
w = eα¯tV ( ¯xeβ¯t).
Going back to the original variables, using (15.3.4.2), we obtain
w = Cm–αeαtV C–k–βxeβt
.
Let us require that this solution coincide with (15.3.4.1), which means that the uniqueness
condition for the solution must be satisfied for any C ≠ 0 To this end, we set
In practice, exponential self-similar solutions are sought using the above existence
criterion: if k and m in (15.3.4.2) are known, then the new variables have the form
(15.3.4.1) with parameters (15.3.4.3).
Remark Sometimes solutions of the form (15.3.4.1) are also called limiting self-similar solutions.
Example 1 Let us show that the nonlinear heat equation
∂w
∂t = a ∂
∂x
w n ∂w
∂x
(15.3.4.4) admits an exponential self-similar solution Inserting (15.3.4.2) into (15.3.4.4) yields
C m ∂ w¯
∂ ¯ t = aC mn+m–2k ∂
∂ ¯x
¯
w n ∂ w¯
∂ ¯x
Equating the powers of C gives one linear equation: m = mn + m –2k It follows that k = 12mn , where m
is any number Further using formulas (15.3.4.1) and (15.3.4.3) and setting, without loss of generality, m =2
(this is equivalent to scaling in t), we find the new variables
w = e2 V (ξ), ξ = xe–nt (15.3.4.5)
Substituting these expressions into (15.3.4.4), we arrive at an ordinary differential equation for V (ξ):
a (V n V ξ ) ξ + nξV ξ –2V =0
Example 2 With the above approach, it can be shown that the nonlinear wave equation (15.3.3.11) also
has an exponential self-similar solution of the form (15.3.4.5)
Trang 2676 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
TABLE 15.2 Invariant solutions that may be obtained using combinations of translation and scaling
transformations preserving the form of equations (C is an arbitrary constant, C >0)
No Invariant transformations Form of invariant solutions Example of an equation
1 t= ¯t + Ck, x = ¯x + Cλ w = U (z), z = kx – λt ∂w ∂t = ∂x ∂
f (w) ∂w ∂x
2 t = C ¯ t , x = C k ¯x, w = C m w¯ w = t m U (z), z = xt–k see Table 15.1
3 t= ¯t + ln C, x = C k ¯x, w = C m w¯ w = e mt U (z), z = xe–kt (k =equation (15.3.4.4)1
2mn , m is any)
4 t = C ¯ t , x = ¯x + k ln C, w = C m w¯ w = t m U (z), z = x – k ln t equation (15.3.4.4)
(m = –1/n, k is any)
5 t = C ¯ t , x = C β ¯x, w = ¯ w + α ln C w = U (z) + α ln t, z = xt–β
∂w
∂t = ∂x ∂ e w ∂w ∂x
(α =2β–1, β is any)
6 t = C ¯ t , x = ¯x + β ln C, w = ¯ w + α ln C w = U (z) + α ln t, z = x – β ln t ∂
2w
∂x∂t
2 –∂ ∂x2w2 ∂ ∂t2w2 =0
(α and β are any)
7 t= ¯t + C, x = ¯x + Cλ, w = ¯ w + Ck w = U (z) + kt, z = x – λt ∂w ∂t = f ∂w ∂x ∂2w
∂x2
(k and λ are any)
8 t= ¯t + ln C, x = ¯x + k ln C, w = C m w¯ w = e mt U (z), z = x – kt ∂
2w
∂x∂t
2
–∂ ∂x2w2 ∂ ∂t2w2 =0
(k and m are any)
15.3.4-2 Other solutions obtainable using translation and scaling transformations Table 15.2 lists invariant solutions that may be obtained using combinations of translation and scaling transformations in the independent and dependent variables The transfor-mations are assumed to preserve the form of equations (the given equation is converted into the same equation) Apart from traveling-wave solutions, self-similar solutions, and exponential self-similar solutions, considered above, another five invariant solutions are presented The right column of Table 15.2 gives examples of equations that admit the solutions specified.
Example 3 Let us show that the nonlinear heat equation (15.3.4.4) admits the solution given in the fourth
row of Table 15.2 Perform the transformation
t = C ¯ t, x=¯x + k ln C, w = C m w¯ This gives
C m–1∂ w¯
∂ ¯ t = aC mn+m ∂
∂ ¯x
¯
w n ∂ w¯
∂ ¯x
Equating the powers of C results in one linear equation: m –1= mn + m It follows that m = –1/n, and k is
any number Therefore (see the fourth row in Table 15.2), equation (15.3.4.4) has an invariant solution of the form
w = t–1/n U (z), z = x + k ln t, where k is any number. (15.3.4.6) Substituting (15.3.4.6) into (15.3.4.4) yields the autonomous ordinary differential equation
a (U n U z ) z – kU z + 1
n U =0
To the special case of k =0there corresponds a separable equation that results in a solution in the form of the product of functions with different arguments
The examples considered in Subsections 15.3.2–15.3.4 demonstrate that the existence
of exact solutions is due to the fact the partial differential equations concerned are invariant
Trang 3under some transformations (involving one or several parameters) or, what is the same, possess some symmetries In Section 15.8 below, a general method for the investigation
of symmetries of differential equations (the group-theoretic method) will be described that allows finding similar and more complicated invariant solutions on a routine basis.
15.3.5 Generalized Self-Similar Solutions
A generalized self-similar solution has the form
w(x, t) = ϕ(t)u(z), z = ψ(t)x. (15.3.5.1) Formula (15.3.5.1) comprises the above self-similar and exponential self-similar solutions (15.3.3.1) and (15.3.4.1) as special cases.
The procedure of finding generalized self-similar solutions is briefly as follows: after
substituting (15.3.5.1) into the given equation, one chooses the functions ϕ(t) and ψ(t) so that u(z) satisfies a single ordinary differential equation.
Example A solution of the nonlinear heat equation (15.3.4.4) will be sought in the form (15.3.5.1) Taking
into account that x = z/ψ(t), we find the derivatives
w t = ϕ t u + ϕψ t xu z = ϕ t u+ϕψ t
ψ zu
z, w x = ϕψu z, (w n w x)x = ψ2ϕ n+1(u n u z) z
Substituting them into (15.3.4.4) and dividing by ϕ t, we have
u+ϕψ
t
ϕ t ψ zu
z= ψ
2ϕ n+1
ϕ t (u n u z) z (15.3.5.2)
For this relation to be an ordinary differential equation for u(z), the functional coefficients of zu z and (u n u z) z must be constant:
ϕψ t
ϕ t ψ = a, ψ
2ϕ n+1
The function u(z) will satisfy the equation
u + azu z = b(u n u z) z From the first equation in (15.3.5.3) it follows that
where C1is an arbitrary constant Substituting the resulting expression into the second equation in (15.3.5.3) and integrating, we obtain
C2
b t + C2= – 1
2a+ n ϕ
– 2a–n for a≠–n
2,
C2
b t + C2= ln|ϕ|, for a= –n
2,
(15.3.5.5)
where C2is an arbitrary constant From (15.3.5.4)–(15.3.5.5) we have, in particular,
ϕ (t) = t 2a+n1 , ψ(t) = t 2a+n a at C1=1, C2=0, b= – 1
2a+ n;
ϕ (t) = e2, ψ (t) = e–nt at C1=1, C2=0, b= 1
2.
The first pair of functions ϕ(t) and ψ(t) corresponds to a self-similar solution (with any a≠–n/2), and the second pair, to an exponential self-similar solution
Trang 4678 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
15.4 Exact Solutions with Simple Separation of
Variables
15.4.1 Multiplicative and Additive Separable Solutions
Separation of variables is the most common approach to solve linear equations of
mathe-matical physics (see Section 14.4) For equations in two independent variables x, t and a dependent variable w, this approach involves searching for exact solutions in the form of
the product of functions depending on different arguments:
w(x, t) = ϕ(x)ψ(t). (15.4.1.1) The integration of a few classes of first-order nonlinear partial differential equations is based on searching for exact solutions in the form of the sum of functions depending on different arguments (see Paragraph 13.2.1-2):
w(x, t) = ϕ(x) + ψ(t). (15.4.1.2) Some second- and higher-order nonlinear equations of mathematical physics also have
exact solutions of the form (15.4.1.1) or (15.4.1.2) Such solutions are called multiplicative separable and additive separable, respectively.
15.4.2 Simple Separation of Variables in Nonlinear Partial
Differential Equations
In isolated cases, the separation of variables in nonlinear equations is carried out following the same technique as in linear equations Specifically, an exact solution is sought in the form
of the product or sum of functions depending on different arguments On substituting it into the equation and performing elementary algebraic manipulations, one obtains an equation with the two sides dependent on different variables (for equations with two variables) Then one concludes that the expressions on each side must be equal to the same constant quantity,
called a separation constant.
Example 1 The heat equation with a power nonlinearity
∂w
∂t = a ∂
∂x
w k ∂w
∂x
(15.4.2.1) has a multiplicative separable solution Substituting (15.4.1.1) into (15.4.2.1) yields
ϕψ t = aψ k+1(ϕ k ϕ x) x
Separating the variables by dividing both sides by ϕψ k+1, we obtain
ψ t
ψ k+1 = a (ϕ
k ϕ
x) x
The left-hand side depends on t alone and the right-hand side on x alone This is possible only if
ψ t
ψ k+1 = C, a (ϕ
k ϕ
x) x
where C is an arbitrary constant (separation constant) On solving the ordinary differential equations (15.4.2.2),
we obtain a solution of equation (15.4.2.1) with the form (15.4.1.1)
The procedure for constructing a separable solution (15.4.1.1) of the nonlinear equation (15.4.2.1) is
identical to that used in solving linear equations [in particular, equation (15.4.2.1) with k =0] We refer to
similar cases as simple separation of variables.
Trang 5Example 2 The wave equation with an exponential nonlinearity
∂2w
∂t2 = a ∂
∂x
e λw ∂w
∂x
(15.4.2.3)
has an additive separable solution On substituting (15.4.1.2) into (15.4.2.3) and dividing by e λψ, we arrive at the equation
e–λψψ tt = a(e λϕ ϕ x) x,
whose left-hand side depends on t alone and the right-hand side on x alone This is possible only if
e–λψψ tt = C, a (e λϕ ϕ x) x = C, (15.4.2.4)
where C is an arbitrary constant Solving the ordinary differential equations (15.4.2.4) yields a solution of
equation (15.4.2.3) with the form (15.4.1.2)
Example 3 The steady-state heat equation in an anisotropic medium with a logarithmic source
∂
∂x
f (x) ∂w
∂x
+ ∂
∂y
g (y) ∂w
∂y
has a multiplicative separable solution
On substituting (15.4.2.6) into (15.4.2.5), dividing by ϕψ, and rearranging individual terms of the resulting
equation, we obtain
1
ϕ [f (x)ϕ x] x – a ln ϕ = –1
ψ [g(y)ψ y] y + a ln ψ.
The left-hand side of this equation depends only on x and the right-hand only on y By equating both sides to
a constant quantity, one obtains ordinary differential equations for ϕ(x) and ψ(y).
Table 15.3 gives other examples of simple, additive or multiplicative, separable solutions for some nonlinear equations.
15.4.3 Complex Separation of Variables in Nonlinear Partial
Differential Equations
The variables in nonlinear equations often separate more complexly than in linear equations.
We exemplify this below.
Example 1 Consider the equation with a cubic nonlinearity
∂w
∂t = a ∂
2w
∂x2 + w
∂w
∂x
2
where b >0 We look for exact solutions in the product form We substitute (15.4.1.1) into (15.4.3.1) and
divide the resulting equation by ϕ(x)ψ(t) to obtain
ψ t
ψ = a ϕ
xx
ϕ + ψ2[(ϕ x)2– bϕ2] (15.4.3.2)
In the general case, this expression cannot be represented as the sum of two functions depending on different arguments This, however, does not mean that equation (15.4.3.1) has no solutions of the form (15.4.1.1)
1◦ One can make sure by direct check that the functional differential equation (15.4.3.2) has solutions
ϕ (x) = Cex √ b, ψ (t) = e abt, (15.4.3.3)
where C is an arbitrary constant Solutions (15.4.3.3) for ϕ make the expression in square brackets in (15.4.3.2)
vanish, which allows the separation of variables
Trang 6680 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
TABLE 15.3 Some nonlinear equations of mathematical physics that admit additive or
multiplicative separable solutions (C, C1, and C2are arbitrary constants)
Equation Equation name Form of solutions Determining equations
∂w
∂t = a ∂ ∂x2w2 + bw ln w Heat equation
with source w = ϕ(x)ψ(t)
aϕ xx /ϕ – b ln ϕ = –ψ t /ψ + b ln ψ = C
∂w
∂t = a ∂x ∂ w k ∂w ∂x
+ bw Heat equation
with source w = ϕ(x)ψ(t)
(ψ t – bψ)/ψ k+1=
a (ϕ k ϕ x) x /ϕ = C
∂w
∂t = a ∂x ∂ w k ∂w ∂x
+ bw k+1 Heat equation
with source w = ϕ(x)ψ(t)
ψ t /ψ k+1=
a (ϕ k ϕ x) x /ϕ + bϕ k = C
∂w
∂t = a ∂x ∂ e λw ∂w ∂x
+ b Heat equation
with source w = ϕ(x) + ψ(t) e–λψ(ψ t – b) = a(e λϕ ϕ x) x = C
∂w
∂t = a ∂x ∂ e w ∂w ∂x
+ be w Heat equation
with source w = ϕ(x) + ψ(t) e
–ψ ψ
t = a(e ϕ ϕ x) x + be ϕ = C
∂w
∂t = a ∂ ∂x2w2 + b ∂w ∂x 2 Potential Burgers
equation w = ϕ(x) + ψ(t) ψ t = aϕ xx + b(ϕ x)2= C
∂w
∂t = a ∂w ∂x k ∂2w
∂x2
Filtration equation
w = ϕ(x) + ψ(t),
w = f (x)g(t)
ψ t = a(ϕ x)k ϕ xx = C1,
g t /g k+1= a(f x )k f xx /f = C2
∂w
∂t = F ∂w ∂x ∂2w
∂x2
Filtration equation w = ϕ(x) + ψ(t) ψ t = F (ϕ x )ϕ xx = C
∂2w
∂t2 = a ∂x ∂ w k ∂w ∂x Wave equation
w = ϕ(x)ψ(t) ψ tt /ψ k+1= a(ϕ k ϕ x) x /ϕ = C
∂2w
∂t2 = a ∂x ∂ e λw ∂w ∂x
Wave equation w = ϕ(x) + ψ(t) e–λψψ tt = a(e λϕ ϕ x) x = C
∂2w
∂t2 = a ∂ ∂x2w2 + bw ln w Wave equationwith source w = ϕ(x)ψ(t) ψ
tt /ψ – b ln ψ =
aϕ xx /ϕ + b ln ϕ = C
∂2w
∂x2 + a ∂y ∂ w k ∂w ∂y
=0 Anisotropic steadyheat equation w = ϕ(x)ψ(y) ϕ xx /ϕ k+1 = –a(ψ k ψ y) y /ψ = C
∂2w
∂x2 + a ∂w ∂y ∂ ∂y2w2 =0 Equation of steadytransonic gas flow w = ϕ(x) + ψ(y),
w = f (x)g(y)
ϕ xx = –aψ y ψ yy = C1,
f xx /f = –ag y g yy /g = C2
∂2w
∂x∂y
2
= ∂ ∂x2w2 ∂ ∂y2w2 Monge–Amp`ereequation w = ϕ(x)ψ(y) (ϕ x)2
ϕϕ xx =ψψ(ψ yy
y)2 = C
∂w
∂t = a ∂ ∂x3w3 + b ∂w ∂x 2 Potential
Korteweg-de Vries equation w = ϕ(x) + ψ(t) ψ t = aϕ xxx + b(ϕ x)2= C
∂w
∂y ∂
2w
∂x∂y –∂w ∂x ∂ ∂y2w2 = a ∂ ∂y3w3 Boundary-layerequation w = ϕ(x) + ψ(y),
w = f (x)g(y)
– ϕ x = aψ yyy /ψ yy = C1,
f x = ag yyy [(g y)2– gg yy ]–1= C2
2◦ There is a more general solution of the functional differential equation (15.4.3.2):
ϕ (x) = C1e
√
b + C2e–x
√
b, ψ (t) = e abt C3+4C1C2e2abt – 1/2
,
where C1, C2, and C3are arbitrary constants The function ϕ = ϕ(x) is such that both x-dependent expressions
in (15.4.3.2) are constant simultaneously:
ϕ xx /ϕ= const, (ϕ x)2– bϕ2= const
It is this circumstance that makes it possible to separate the variables
Example 2 Consider the second-order equation with a quadratic nonlinearity
∂w
∂y
∂2w
∂x2 + a ∂w
∂x
∂2w
∂y2 = b ∂
3w
∂x3 + c ∂
3w
∂y3 (15.4.3.4)
Trang 7We look for additive separable solutions
Substituting (15.4.3.5) into (15.4.3.4) yields
g y f xx + af x g yy = bf xxx + cg yyy (15.4.3.6) This expression cannot be rewritten as the equality of two functions depending on different arguments
It can be shown that equation (15.4.3.4) has a solution of the form (15.4.3.5):
w = C1e–aλx+cλ
a x + C2e λy – abλy + C3,
where C1, C2, C3, and λ are arbitrary constants The mechanism of separation of variables is different here:
both nonlinear terms on the left-hand side in (15.4.3.6) contain terms that cannot be rewritten in additive form but are equal in magnitude and have unlike signs In adding, the two terms cancel out, thus resulting in separation of variables:
g y f xx = C1C2a2λ3e λy–aλx – C1b (aλ)3e–aλx
+
af x g yy = –C1C2a2λ3e λy–aλx + C2cλ3e λy
g y f xx + af x g yy = –C1b (aλ)3e–aλx+ C2cλ3e λy = bf xxx + cg yyy .
Example 3 Consider the second-order equation with a cubic nonlinearity
(1+ w2)
∂2w
∂x2 +∂
2w
∂y2
–2w
∂w
∂x
2
–2w
∂w
∂y
2
= aw(1– w2) (15.4.3.7)
We seek an exact solution of this equation in the product form
Substituting (15.4.3.8) into (15.4.3.7) yields
(1+ f2g2)(gf xx + f g yy) –2f g[g2(f x )2+ f2(g y)2] = af g(1– f2g2) (15.4.3.9) This expression cannot be rewritten as the equality of two functions with different arguments Nevertheless, equation (15.4.3.7) has solutions of the form (15.4.3.8) One can make sure by direct check that the functions
f = f (x) and g = g(y) satisfying the nonlinear ordinary differential equations
(f x )2= Af4+ Bf2+ C, (g y )2= Cg4+ (a – B)g2+ A, (15.4.3.10)
where A, B, and C are arbitrary constants, reduce equation (15.4.3.9) to an identity; to verify this, one should use the relations f xx =2Af3+ Bf and g yy =2Cg3+ (a – B)g that follow from (15.4.3.10).
Remark By the change of variable u =4arctan w equation (15.4.3.7) can be reduced to a nonlinear heat
equation with a sinusoidal source,Δu = a sin u.
The examples considered above illustrate some specific features of separable solutions
to nonlinear equations Section 15.5 outlines fairly general methods for constructing similar and more complicated solutions to nonlinear partial differential equations.
15.5 Method of Generalized Separation of Variables
15.5.1 Structure of Generalized Separable Solutions
15.5.1-1 General form of solutions The classes of nonlinear equations considered.
To simplify the presentation, we confine ourselves to the case of mathematical physics
equa-tions in two independent variables x, y and a dependent variable w (one of the independent
variables can play the role of time).