Handbook of mathematics for engineers and scienteists part 109 potx

724 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Similarly, it can be established that the following special forms of f result in additional operators: 1. f = e w :X 4 = x∂ x + 2∂ w ; 2. f = w k , k ≠ 0,–4/3,–4:X 4 = kx∂ x + 2w∂ w ; 3. f = w –4/3 :X 4 = 2x∂ x – 3w∂ w ,X 5 = x 2 ∂ x – 3xw∂ w ; 4. f = w –4 :X 4 = 2x∂ x – w∂ w ,X 5 = t 2 ∂ t + tw∂ w . The symmetries obtainedwith the procedure presented can beused to find exactsolutions of the differential equations considered (see below). 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions 15.8.3-1. Using symmetries of equations for constructing one-parameter solutions. Suppose a particular solution, w = g(x, y), (15.8.3.1) of a given equation is known. Let us show that any symmetry of the equation defined by a transformation of the form (15.8.1.1) generates a one-parameter family of solutions (except for the cases where the solution is not mapped into itself by the transformations; see Paragraph 15.8.3-2). Indeed, since equation (15.8.2.1) converted to the new variables (15.8.1.1) acquires the same form (15.8.2.2), then the transformed equation (15.8.2.2) has a solution ¯w = g(¯x, ¯y). (15.8.3.2) In (15.8.3.2), going back to the old variables by formulas (15.8.1.1), we obtain a one- parameter solution of the original equation (15.8.2.1). Example 1. The two-dimensional heat equation with an exponential source ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = e w (15.8.3.3) has a one-dimensional solution w =ln 2 x 2 .(15.8.3.4) Equation (15.8.3.3) admits the operator X 3 = y∂ x –x∂ y (see Example 1 in Subsection 15.8.2), which defines rotation in the plane. The corresponding transformation is given in Table 15.7. Replacing x in (15.8.3.4) by ¯x (from Table 15.7), we obtain a one-parameter solution of equation (15.8.3.3): w =ln 2 (x cosε + y sin ε) 2 , where ε is a free parameter. 15.8.3-2. Procedure for constructing invariant solutions. Solution(15.8.3.1)of equation(15.8.2.1) is calledinvariant undertransformations (15.8.1.1) if it coincides with solution (15.8.3.2), which must be rewritten in terms of the old variables using formulas (15.8.1.1). This means that an invariant solution is converted to itself under the given transformation. The basic stages of constructing invariant solutions are outlined below. 15.8. CLASSICAL METHOD OF STUDYING SYMMETRIES OF DIFFERENTIAL EQUATIONS 725 Invariant solutions of equation (15.8.2.1) are sought in the implicit form I(x, y, w)=0. Then I(¯x, ¯y, ¯w)=0.Letusfind a one-parameter transformation with operator (15.8.1.3) whose coordinates are determined from the invariance condition (15.8.2.3) following the procedure described in Subsection 15.8.2. Find two functionally independent integrals (15.8.1.6) of the characteristic system of ordinary differential equations (15.8.1.5). The general solution of the partial differential equation (15.8.1.4) is determined by formula (15.8.1.7). Setting in this formula I = 0 and solving for the invariant I 2 , we obtain I 2 = Φ(I 1 ), (15.8.3.5) where the functions I 1 = I 1 (x, y, w)andI 2 = I 2 (x, y, w) are known,* and the function Φ is to be determined. Relation (15.8.3.5) is the basis for the construction of invariant solutions: solving (15.8.3.5) for w and substituting the resulting expression into (15.8.2.1), we arrive at an ordinary differential equation for Φ. Example 2. A well-known and very important special case of invariant solutions is the self-similar solu- tions (see Subsection 15.3.3); they are based on invariants of scaling groups. The corresponding infinitesimal operator and its invariants are X = ax ∂ ∂x + by ∂ ∂y + cw ∂ ∂w ; I 1 = |y| a |x| –b , I 2 = |w| a |x| –c . Substituting the invariants into (15.8.3.5) gives |w| a |x| –c = Φ  |y| a |x| –b  . On solving this equation for w,we obtain the form of the desired solution, w = |x| c/a Ψ  y|x| –b/a  ,whereΨ(z) is an unknown function. To make it clearer, the general scheme for constructing invariant solutions for evolution second-order equations is depicted in Fig. 15.4. The first-order partial differential equation (15.8.1.4) for finding group invariants is omitted, since the corresponding characteristic system of ordinary differential equations (15.8.1.5) can be immediately used. 15.8.3-3. Examples of constructing invariant solutions to nonlinear equations. Example 3. Consider once again the stationary heat equation with nonlinear source ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = f(w). 1 ◦ . Let us dwell on the case f(w)=w k , where the equation admits an additional operator (see Example 1 from Subsection 15.8.2): X 4 = x∂ x + y∂ y + 2 1 – k w∂ w . In order to find invariants of this operator, we have to consider the linear first-order partial differential equation X 4 I = 0, or, in detailed form, x ∂I ∂x + y ∂I ∂y + 2 1 – k w ∂I ∂w = 0. The corresponding characteristic system of ordinary differential equations, dx x = dy y = 1 – k 2 dw w , has the first integrals y/x = C 1 , x 2/(k–1) w = C 2 , * Usually, the invariant that is independent of w is taken to be I 1 . 726 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Calculate the coordinates of the prolonged operator Derive the determining system of PDEs Solve the characteristic system Figure 15.4. An algorithm for constructing invariant solutions for evolution second-order equations. Notation: ODE stands for ordinary differential equation and PDE stands for partial differential equation; ξ = ξ(x, t, w), η = η(x, t, w), ζ = ζ(x, t, w); ζ 1 , ζ 2 ,andζ 11 are the coordinates of the prolonged operator, which are defined by formulas (15.8.1.9) and (15.8.1.14) with y = t. where C 1 , C 2 are arbitrary constants. Therefore, the functions I 1 = y/x and I 2 = x 2/(k–1) w are invariants of the operator X 4 . Assuming that I 2 = Φ(I 1 ) and expressing w,wefind the form of the invariant (self-similar) solution: w = x –2/(k–1) Φ(y/x). (15.8.3.6) Substituting (15.8.3.6) into the original equation (15.8.2.6) yields a second-order ordinary differential equation for Φ(z): (k – 1) 2 (z 2 + 1)Φ  zz + 2(k 2 – 1)zΦ  z + 2(k + 1)Φ –(k – 1) 2 Φ k = 0, where z = y/x. 2 ◦ . The functions u = x 2 +y 2 and w are invariants of the operator X 3 for the nonlinear heat equation concerned. The substitutions w = w(u)andu = x 2 + y 2 lead to an ordinary differential equation describing solutions of the original equation which are invariant under rotation: uw  uu + w  u = 1 4 f(w). Remark. In applications, the polar radius r =  x 2 + y 2 is normally used as an invariant instead of u = x 2 + y 2 . 15.8. CLASSICAL METHOD OF STUDYING SYMMETRIES OF DIFFERENTIAL EQUATIONS 727 Example 4. Consider the nonlinear nonstationary heat equation (15.8.2.14). 1 ◦ . For arbitrary f(w), the equation admits the operator (see Example 2 from Subsection 15.8.2) X 3 = 2t∂ t + x∂ x . Invariants of X 3 are found for the linear first-order partial differential equation X 3 I = 0,or 2t ∂I ∂t + x ∂I ∂x + 0 ∂I ∂w = 0. The associated characteristic system of ordinary differential equations, dx x = dt 2t = dw 0 , has the first integrals xt –1/2 = C 1 , w = C 2 , where C 1 and C 2 are arbitrary constants. Therefore, the functions I 1 = xt –1/2 and I 2 = w are invariants of the operator X 3 . Assuming I 2 = Φ(I 1 ), we get w = Φ(z), z = xt –1/2 ,(15.8.3.7) where Φ(z) is to be determined in the subsequent analysis. Substituting (15.8.3.7) in the original equation (15.8.2.14) yields the second-order ordinary differential equation 2[f(Φ)Φ  z ]  z + zΦ  z = 0, which describes an invariant (self-similar) solution. 2 ◦ . Let us dwell on the case f(w)=w k , where the equation admits the operator X 4 = kx∂ x + 2w∂ w . The invariants are described by the first-order partial differential equation X 4 I = 0,or 0 ∂I ∂t + kx ∂I ∂x + 2w ∂I ∂w = 0. The associated characteristic system of ordinary differential equations, dt 0 = dx kx = dw 2w , has the first integrals t = C 1 , x –2/k w = C 2 , where C 1 , C 2 are arbitrary constants. Therefore, I 1 = t and I 2 = x –2/k w are invariants of the operator X 4 . Assuming I 2 = θ(I 1 ) and expressing w,weget w = x 2/k θ(t), (15.8.3.8) where θ(t) is to be determined in the subsequent analysis. Substituting (15.8.3.8) in the original equation (15.8.2.14) with f(w)=w k gives the first-order ordinary differential equation 2kθ  t = 2(k + 2)θ k+1 . Integrating yields θ(t)=  A – 2(k + 2) k t  –1/k , where A is an arbitrary constant. Hence, the solution of equation (15.8.2.14) with f(w)=w k , which is invariant under scaling, has the from w(x, t)=x 2/k  A – 2(k + 2) k t  –1/k . 728 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 15.8 Operators, invariants, and solution structures admitted by the nonlinear nonstationary heat equation (15.8.2.14) Function f (w) Operators Invariants Solution structure Arbitrary X 1 = ∂ x , X 2 = ∂ t , X 3 = 2t∂ t + x∂ x I 1 = t, I 2 = w, I 1 = x, I 2 = w, I 1 = x 2 /t, I 2 = w w = w(t) = const, w = w(x), w = w(z), z = x 2 /t e w X 4 = x∂ x + 2∂ w I 1 = t, I 2 = w – 2 ln|x| w = 2 ln |x| + θ(t) w k (k ≠ 0,– 4 3 ) X 4 = kx∂ x + 2w∂ w I 1 = t, I 2 = w|x| –k/2 w = |x| k/2 θ(t) w –4/3 X 4 = 2x∂ x – 3w∂ w , X 5 = x 2 ∂ x – 3xw∂ w I 1 = t, I 2 = wx 2/3 , I 1 = t, I 2 = wx 3 w = x –2/3 θ(t), w = x –3 θ(t) Table 15.8 summarizes the symmetries of equation (15.8.2.14) (see Example 2 from Subsection 15.8.2 and Example 4 from Subsection 15.8.3). Example 5. Consider the nonlinear wave equation (15.8.2.15). For arbitrary f(w), this equation admits the following operator (see Example 3 from Subsection 15.8.2): X 3 = t∂ t + x∂ x . The invariants are found from the linear first-order partial differential equation X 3 I 1 = 0,or t ∂I ∂t + x ∂I ∂x + 0 ∂I ∂w = 0. The associated characteristic system of ordinary differential equations dx x = dt t = dw 0 admits the first integrals xt –1 = C 1 , w = C 2 , where C 1 , C 2 are arbitrary constants. Therefore, I 1 = xt –1 and I 2 = w are invariants of the operator X 3 . Taking I 2 = Φ(I 1 ), we get w = Φ(y), y = xt –1 .(15.8.3.9) The function Φ(y) is found by substituting (15.8.3.9) in the original equation (15.8.2.15). This results in the ordinary differential equation [f(Φ)Φ  y ]  y =(y 2 Φ  y )  y , which defines aninvariant(self-similar)solution. Thisequationhas the obviousfirst integral f(Φ)Φ  y =y 2 Φ  y +C. Table 15.9 summarizes the symmetries of equation (15.8.2.15) (see Example 3 from Subsection 15.8.2 and Example 5 from Subsection 15.8.3). 15.8.3-4. Solutions induced by linear combinations of admissible operators. If a given equation admits N operators, then we have N associated different invariant solutions. However, when dealing with operators individually, one may overlook solutions that are invariant under a linear superposition of the operators; such solutions may have a significantly different form. In order to find all types of invariant solutions, one should study all possible linear combinations of the admissible operators. Example 6. Consider once again the nonlinear nonstationary heat equation (15.8.2.14). 15.8. CLASSICAL METHOD OF STUDYING SYMMETRIES OF DIFFERENTIAL EQUATIONS 729 TABLE 15.9 Operators, invariants, and solution structures admitted by the nonlinear wave equation (15.8.2.15) Functions f(w) Operators Invariants Solution structure Arbitrary X 1 = ∂ x , X 2 = ∂ t , X 3 = t∂ t + x∂ x I 1 = t, I 2 = w, I 1 = x, I 2 = w, I 1 = x/t, I 2 = w w = w(t), w = w(x), w = w(z), z = x/t e w X 4 = x∂ x + 2∂ w I 1 = t, I 2 = w – 2 ln|x| w = 2 ln |x| + θ(t) w k (k ≠ 0,– 4 3 ,–4) X 4 = kx∂ x + 2w∂ w I 1 = t, I 2 = w|x| –k/2 w = |x| k/2 θ(t) w –4/3 X 4 = 2x∂ x – 3w∂ w , X 5 = x 2 ∂ x – 3xw∂ w I 1 = t, I 2 = wx 2/3 , I 1 = t, I 2 = wx 3 w = x –2/3 θ(t), w = x –3 θ(t) w –4 X 4 = 2x∂ x – w∂ w , X 5 = t 2 ∂ t + tw∂ w I 1 = t, I 2 = w|x| 1/2 , I 1 = x, I 2 = w/t w = |x| –1/2 θ(t), w = tθ(x) 1 ◦ . For arbitrary f(w), this equation admits three operators (see Table 15.8): X 1 = ∂ t , X 2 = ∂ x , X 3 = 2t∂ t + x∂ x . The respective invariant solutions are w = Φ(x), w = Φ(t), w = Φ(x 2 /t). However, various linear combinations give another operator, X 1,2 = X 1 + aX 2 = ∂ t + a∂ x , where a ≠ 0 is an arbitrary constant. The solution invariant under this operator is written as w = Φ(x – at). It is apparent that solutions of this type (traveling waves) are not contained in the invariant solutions associated with the individual operators X 1 , X 2 ,andX 3 . 2 ◦ .Iff(w)=e w , apart from the above three operators, there is another one, X 4 = x∂ x + 2∂ w (see Table 15.8). In this case, the linear combination X 3,4 = X 3 + aX 4 = 2t∂ t +(a + 1)x∂ x + 2a∂ w gives another invariant solution, w = Φ(ξ)+a lnt, ξ = xt a+1 2 , where the function Φ = Φ(ξ) satisfies the ordinary differential equation (e Φ Φ  ξ )  ξ + 1 2 (a + 1)ξΦ  ξ = a. 3 ◦ .Iff(w)=w k (k ≠ 0,–4/3), apart from the three operators from 1 ◦ , there is another one X 4 = kx∂ x +2w∂ w . The linear combination X 3,4 = X 3 + aX 4 = 2t∂ t +(ak + 1)x∂ x + 2aw∂ w generates the invariant (self-similar) solution w = t a Φ(ζ), ζ = xt ak+1 2 , where the function Φ = Φ(ζ) satisfies the ordinary differential equation (Φ k Φ  ζ )  ζ + 1 2 (ak + 1)ζΦ  ζ = aΦ. The invariant solutions presented in Items 1 ◦ –3 ◦ are not listed in Table 15.8. It is clearly important to consider solutions induced by linear combinations of admissible operators. 730 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.8.4. Some Generalizations. Higher-Order Equations 15.8.4-1. One-parameter Lie groups of point transformations. Group generator. Here we will be considering functions dependent on n + 1 variables, x 1 , , x n , w.The brief notation x =(x 1 , , x n ) will be used. The set of invertible transformations of the form T ε =  ¯x i = ϕ i (x, w, ε), ¯x i | ε=0 = x i , ¯w = ψ(x, w, ε), ¯w| ε=0 = w, (15.8.4.1) where ϕ i and ψ are sufficiently smooth functions of their arguments (i = 1, , n)andε is a real parameter, is called a one-parameter continuous point group of transformations G if for any ε 1 and ε 2 the relation T ε 1 ◦ T ε 2 =T ε 1 +ε 2 holds, that is, the successive application (composition) of two transformations of the form (15.8.4.1) with parameters ε 1 and ε 2 is equivalent to a single transformation of the same form with parameter ε 1 + ε 2 . Further on, we consider local one-parameter continuous Lie groups of point transfor- mations (or, for short, point groups), corresponding to the infinitesimal transformation (15.8.4.1) as ε → 0. The expansion of (15.8.4.1) into Taylor series in the parameter ε about ε = 0 to the first order gives ¯x i  x i + εξ i (x, w), ¯w  w + εζ(x, w), (15.8.4.2) where ξ i (x, w)= ∂ϕ i (x, w, ε) ∂ε    ε=0 , ζ(x, w)= ∂ψ(x, w, ε) ∂ε    ε=0 . The linear first-order differential operator X=ξ i (x, w) ∂ ∂x i + ζ(x, w) ∂ ∂w (15.8.4.3) corresponding to the infinitesimal transformation (15.8.4.2) is called a group generator (or an infinitesimal operator). In formula (15.8.4.3), summation is assumed over the repeated index i. T HEOREM (LIE). Suppose the coordinates ξ i (x, w) and ζ(x, w) of the group generator (15.8.4.3) are known. Then the one-parameter group of transformations (15.8.4.1) can be completely recovered by solving the Lie equations dϕ i dε = ξ i (ϕ, ψ), dψ dε = ζ(ϕ, ψ)(i = 1, , n) with the initial conditions ϕ i | ε=0 = x i , ψ| ε=0 = w. Here, the short notation ϕ =(ϕ 1 , , ϕ n ) has been used. Remark. The widely known terms “Lie group analysis of differential equations,” “group-theoretic meth- ods,” and others are due to the prevailing concept of a local one-parameter Lie group of point transformations. However, in this book, we prefer to use the term “method of symmetry analysis of differential equations.” . successive application (composition) of two transformations of the form (15.8.4.1) with parameters ε 1 and ε 2 is equivalent to a single transformation of the same form with parameter ε 1 + ε 2 . Further. algorithm for constructing invariant solutions for evolution second-order equations. Notation: ODE stands for ordinary differential equation and PDE stands for partial differential equation; ξ = ξ(x,. used. The set of invertible transformations of the form T ε =  ¯x i = ϕ i (x, w, ε), ¯x i | ε=0 = x i , ¯w = ψ(x, w, ε), ¯w| ε=0 = w, (15.8.4.1) where ϕ i and ψ are sufficiently smooth functions of their