Handbook of mathematics for engineers and scienteists part 99 ppt

654 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Denote a = ∂F ∂p , b = 1 2 ∂F ∂q , c = ∂F ∂r ,wherep = ∂ 2 w ∂x 2 , q = ∂ 2 w ∂x∂y , r = ∂ 2 w ∂y 2 .(15.1.2.2) Let us select a specific solution w = w(x, y) of equation (15.1.2.1) and calculate a, b, and c by formulas (15.1.2.2) at some point (x, y), and substitute the resulting expressions into (15.1.1.2). Depending on the sign of the discriminant δ, the type of nonlinear equation (15.1.2.1) at the point (x, y) is determined according to (15.1.1.3): if δ = 0, the equation is parabolic, if δ > 0, it is hyperbolic, and if δ < 0, it is elliptic. In general, the coefficients a, b,andc of the nonlinear equation (15.1.2.1) depend not only on the selection of the point (x, y), but also on the selection of the specific solution. Therefore, it is impossible to determine the sign of δ without knowing the solution w(x, y). To put it differently, the type of a nonlinear equation can be different for different solutions at the same point (x, y). A line ϕ(x, y)=C is called a characteristic of the nonlinear equation (15.1.2.1) if it is an integral curve of the characteristic equation a (dy) 2 – 2bdxdy + c (dx) 2 = 0.(15.1.2.3) The form of characteristics depends on the selection of a specific solution. In individual special cases, the type of a nonlinear equation [other than the semilinear equation (15.1.1.1)] may be independent of the selection of solutions. Example. Consider the nonhomogeneous Monge–Amp ` ere equation  ∂ 2 w ∂x∂y  2 – ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 = f(x, y). It is a special case of equation (15.1.2.1) with F (x, y, p, q, r) ≡ q 2 – pr – f(x, y)=0, p = ∂ 2 w ∂x 2 , q = ∂ 2 w ∂x∂y , r = ∂ 2 w ∂y 2 .(15.1.2.4) Using formulas (15.1.2.2) and (15.1.2.4), we find the discriminant (15.1.1.2): δ = q 2 – pr = f(x, y). (15.1.2.5) Here, the relation between the highest derivatives and f(x, y)defined by equation (15.1.2.4) has been taken into account. From (15.1.2.5) and (15.1.1.3) it follows that the type of the nonhomogeneous equation Monge–Amp ` ere at a point (x, y) depends solely on the sign of f(x, y) and is independent of the selection of a particular solution. At the points where f (x, y)=0, the equation is of parabolic type; at the points where f(x, y)>0, the equation is of hyperbolic type; and at the points where f(x, y)<0, it is elliptic. 15.1.2-2. Quasilinear equations. A second-order quasilinear partial differential equation in two independent variables has the form a(x, y, w, ξ, η)p + 2b(x, y, w, ξ, η)q + c(x, y, w, ξ, η)r = f (x, y, w, ξ, η), (15.1.2.6) with the short notation ξ = ∂w ∂x , η = ∂w ∂y , p = ∂ 2 w ∂x 2 , q = ∂ 2 w ∂x∂y , r = ∂ 2 w ∂y 2 . 15.2. TRANSFORMATIONS OF EQUATIONS OF MAT H E M AT I C AL PHYSICS 655 Consider a curve C 0 definedinthex, y plane parametrically as x = x(τ), y = y(τ). (15.1.2.7) Let us fix a set of boundary conditions on this curve, thus defining the initial values of the unknown function and its first derivatives: w = w(τ ), ξ = ξ(τ), η = η(τ)(w  τ = ξx  τ + ηy  τ ). (15.1.2.8) The derivative with respect to τ is obtained by the chain rule, since w = w(x, y). It can be shown that the given set of functions (15.1.2.8) uniquely determines the values of the second derivatives p, q,andr (and also higher derivatives) at each point of the curve (15.1.2.7), satisfying the condition a(y  x ) 2 – 2by  x + c ≠ 0 (y  x = y  τ /x  τ ). (15.1.2.9) Here and henceforth, the arguments of the functions a, b,andc are omitted. Indeed, bearing in mind that ξ = ξ(x, y)andη = η(x, y), let us differentiate the second and the third equation in (15.1.2.8) with respect to the parameter τ: ξ  τ = px  τ + qy  τ , η  τ = qx  τ + ry  τ .(15.1.2.10) On solving relations (15.1.2.6) and (15.1.2.10) for p, q,andr, we obtain formulas for the second derivatives at the points of the curve (15.1.2.7): p = c(x  τ ξ  τ – y  τ η  τ )–2by  τ ξ  τ + f(y  τ ) 2 a(y  τ ) 2 – 2bx  τ y  τ + c(x  τ ) 2 , q = ay  τ ξ  τ + cx  τ η  τ – fx  τ y  τ a(y  τ ) 2 – 2bx  τ y  τ + c(x  τ ) 2 , r = a(y  τ η  τ – x  τ ξ  τ )–2bx  τ η  τ + f(x  τ ) 2 a(y  τ ) 2 – 2bx  τ y  τ + c(x  τ ) 2 . (15.1.2.11) The third derivatives at the points of the curve (15.1.2.7) can be calculated in a similar way. To this end, one differentiates (15.1.2.6) and (15.1.2.11) with respect to τ and then expresses the third derivatives from the resulting relations. This procedure can also be extended to higher derivatives. Consequently, the solution to equation (15.1.2.6) can be represented as a Taylor series about the points of the curve (15.1.2.7) that satisfy condition (15.1.2.9). The singular points at which the denominators in the formulas for the second derivatives (15.1.2.11) vanish satisfy the characteristic equation (15.1.2.3). Conditions of the form (15.1.2.8) cannot be arbitrarily set on the characteristic curves, which are described by equation (15.1.2.3). The additional conditions of vanishing of the numerators in formulas (15.1.2.11) must be used; in this case, the second derivatives will be finite. 15.2. Transformations of Equations of Mathematical Physics 15.2.1. Point Transformations: Overview and Examples 15.2.1-1. General form of point transformations. Let w = w(x, y) be a function of independent variables x and y. In general, a point transformation is defined by the formulas x = X(ξ, η, u), y = Y (ξ, η, u), w = W (ξ, η, u), (15.2.1.1) 656 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS where ξ and η are new independent variables, u = u(ξ, η) is a new dependent variable, and the functions X, Y , W may be either given or unknown (have to be found). A point transformation not only preserves the order or the equation to which it is applied, but also mostly preserves the structure of the equation, since the highest-order derivatives of the new variables are linearly dependent on the highest-order derivatives of the original variables. Transformation (15.2.1.1) is invertible if         ∂X ∂x ∂X ∂y ∂X ∂w ∂Y ∂x ∂Y ∂y ∂Y ∂w ∂W ∂x ∂W ∂y ∂W ∂w         ≠ 0. In the general case, a point transformation (15.2.1.1) reduces a second-order equation with two independent variables F  x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2  = 0 (15.2.1.2) to an equation G  ξ, η, u, ∂u ∂ξ , ∂u ∂η , ∂ 2 u ∂ξ 2 , ∂ 2 u ∂ξ∂η , ∂ 2 u ∂η 2  = 0.(15.2.1.3) If u = u(ξ, η) is a solution of equation (15.2.1.3), then formulas (15.2.1.1) define the corresponding solution of equation (15.2.1.2) in parametric form. Point transformations are employed to simplify equations and their reduction to known equations. 15.2.1-2. Linear transformations. Linear point transformations (or simply linear transformations), x = X(ξ, η), y = Y (ξ, η), w = f (ξ, η)u + g(ξ, η), (15.2.1.4) are most commonly used. The simplest linear transformations of the independent variables are x = ξ + x 0 , y = η + y 0 (translation transformation), x = k 1 ξ, y = k 2 η (scaling transformation), x = ξ cos α – η sin α, y = ξ sin α + η cos α (rotation transformation). These transformations correspond to the translation of the origin of coordinates to the point (x 0 , y 0 ), scaling (extension or contraction) along the x-andy-axes, and the rotation of the coordinate system through the angle α, respectively. These transformations do not affect the dependent variable (w = u). Linear transformations (15.2.1.4) are employed to simplify linear and nonlinear equa- tions and to reduce equations to the canonical forms (see Subsections 14.1.1 and 15.1.1). 15.2. TRANSFORMATIONS OF EQUATIONS OF MAT H E M AT I C AL PHYSICS 657 Example 1. The nonlinear equation ∂w ∂t = a ∂ ∂x  w m ∂w ∂x  +  xf(t)+g(t)  ∂w ∂x + h(t)w can be simplified to obtain ∂u ∂τ = ∂ ∂z  u m ∂u ∂z  with the help of the transformation w(x, t)=u(z, τ )H(t), z = xF (t)+  g(t)F (t) dt, τ =  F 2 (t)H m (t) dt, where F (t)=exp   f(t) dt  , H(t)=exp   h(t) dt  . 15.2.1-3. Simple nonlinear point transformations. Point transformations can be used for the reduction of nonlinear equations to linear ones. The simplest nonlinear transformations have the form w = W (u)(15.2.1.5) and do not affect the independent variables (x = ξ and y = η). Combinations of transforma- tions (15.2.1.4) and (15.2.1.5) are also used quite often. Example 2. The nonlinear equation ∂w ∂t = ∂ 2 w ∂x 2 + a  ∂w ∂x  2 + f(x, t) can be reduced to the linear equation ∂u ∂t = ∂ 2 u ∂x 2 + af(x, t)u for the function u = u(x, t) by means of the transformation u =exp(aw). 15.2.2. Hodograph Transformations (Special Point Transformations) In some cases, nonlinear equations and systems of partial differential equations can be simplified by means of the hodograph transformations, which are special cases of point transformations. 15.2.2-1. One of the independent variables is taken to be the dependent one. For an equation with two independent variables x, y and an unknown function w = w(x, y), the hodograph transformation consists of representing the solution in implicit form x = x(y, w)(15.2.2.1) [or y = y(x, w)]. Thus, y and w are treated as independent variables, while x is taken to be the dependent variable. The hodograph transformation (15.2.2.1) does not change the order of the equation and belongs to the class of point transformations (equivalently, it can be represented as x = w, y = y, w = x). 658 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 1. Consider the nonlinear second-order equation ∂w ∂y  ∂w ∂x  2 = a ∂ 2 w ∂x 2 .(15.2.2.2) Let us seek its solution in implicit form. Differentiating relation (15.2.2.1) with respect to both variables as an implicit function and taking into account that w = w(x, y), we get 1 = x w w x (differentiation in x), 0 = x w w y + x y (differentiation in y), 0 = x ww w 2 x + x w w xx (double differentiation in x), where the subscripts indicate the corresponding partial derivatives. We solve these relations to express the “old” derivatives through the “new” ones, w x = 1 x w , w y =– x y x w , w xx =– w 2 x x ww x w =– x ww x 3 w . Substituting these expressions into (15.2.2.2), we obtain the linear heat equation: ∂x ∂y = a ∂ 2 x ∂w 2 . 15.2.2-2. Method of conversion to an equivalent system of equations. In order to investigate equations with the unknown function w = w(x, y), it may be useful to convert the original equation to an equivalent system of equations for w(x, y)andv = v(x, y) (the elimination of v from the system results in the original equation) and then apply the hodograph transformation x = x(w, v), y = y(w, v), (15.2.2.3) where w, v are treated as the independent variables and x, y as the dependent variables. Let us illustrate this by examples of specific equations of mathematical physics. Example 2. Rewrite the stationary Khokhlov–Zabolotskaya equation (it arises in acoustics and nonlinear mechanics) ∂ 2 w ∂x 2 + a ∂ ∂y  w ∂w ∂y  = 0 (15.2.2.4) as the system of equations ∂w ∂x = ∂v ∂y ,–aw ∂w ∂y = ∂v ∂x .(15.2.2.5) We now take advantage of the hodograph transformation (15.2.2.3), which amounts to taking w, v as the independent variables and x, y as the dependent variables. Differentiating each relation in (15.2.2.3) with respect to x and y (as composite functions) and eliminating the partial derivatives x w , x v , y w , y v from the resulting relations, we obtain ∂x ∂w = 1 J ∂v ∂y , ∂x ∂v =– 1 J ∂w ∂y , ∂y ∂w =– 1 J ∂v ∂x , ∂y ∂v = 1 J ∂w ∂x ,whereJ = ∂w ∂x ∂v ∂y – ∂w ∂y ∂v ∂x .(15.2.2.6) Using (15.2.2.6) to eliminate the derivatives w x , w y , v x , v y from (15.2.2.5), we arrive at the system ∂y ∂v = ∂x ∂w ,–aw ∂x ∂v = ∂y ∂w .(15.2.2.7) Let us differentiate the first equation in w and the second in v, and then eliminate the mixed derivative y wv .As a result, we obtain the following linear equation for the function x = x(w, v): ∂ 2 x ∂w 2 + aw ∂ 2 x ∂v 2 = 0.(15.2.2.8) Similarly, from system (15.2.2.7), we obtain another linear equation for the function y = y(w, v), ∂ 2 y ∂v 2 + ∂ ∂w  1 aw ∂y ∂w  = 0.(15.2.2.9) Given a particular solution x = x(w, v) of equation (15.2.2.8), we substitute this solution into sys- tem (15.2.2.7) and find y = y(w, v) by straightforward integration. Eliminating v from (15.2.2.3), we obtain an exact solution w = w(x, y) of the nonlinear equation (15.2.2.4). 15.2. TRANSFORMATIONS OF EQUATIONS OF MAT H E M AT I C AL PHYSICS 659 Remark. Equation (15.2.2.8) with an arbitrary a admits a simple particular solution, namely, x = C 1 wv + C 2 w + C 3 v + C 4 , (15.2.2.10) where C 1 , , C 4 are arbitrary constants. Substituting this solution into system (15.2.2.7), we obtain ∂y ∂v = C 1 v + C 2 , ∂y ∂w =–a(C 1 w + C 3 )w. (15.2.2.11) Integrating the first equation in (15.2.2.11) yields y = 1 2 C 1 v 2 + C 2 v + ϕ(w). Substituting this solution into the second equation in (15.2.2.11), we find the function ϕ(w), and consequently y = 1 2 C 1 v 2 + C 2 v – 1 3 aC 1 w 3 – 1 2 aC 3 w 2 + C 5 . (15.2.2.12) Formulas (15.2.2.10) and (15.2.2.12) define an exact solution of equation (15.2.2.4) in parametric form (v is the parameter). In a similar way, one can construct more complex solutions of equation (15.2.2.4) in parametric form. Example 3. Consider the Born–Infeld equation  1 –  ∂w ∂t  2  ∂ 2 w ∂x 2 + 2 ∂w ∂x ∂w ∂t ∂ 2 w ∂x∂t –  1 +  ∂w ∂x  2  ∂ 2 w ∂t 2 = 0, (15.2.2.13) which is used in nonlinear electrodynamics (fi eld theory). By introducing the new variables ξ = x – t, η = x + t, u = ∂w ∂ξ , v = ∂w ∂η , equation (15.2.2.13) can be rewritten as the equivalent system ∂u ∂η – ∂v ∂ξ = 0, v 2 ∂u ∂ξ –(1 + 2uv) ∂u ∂η + u 2 ∂v ∂η = 0. The hodograph transformation, where u, v are taken to be the independent variables and ξ, η the dependent ones, leads to the linear system ∂ξ ∂v – ∂η ∂u = 0, v 2 ∂η ∂v +(1 + 2uv) ∂ξ ∂v + u 2 ∂ξ ∂u = 0. (15.2.2.14) Eliminating η yields the linear second-order equation u 2 ∂ 2 ξ ∂u 2 +(1 + 2uv) ∂ 2 ξ ∂u∂v + v 2 ∂ 2 ξ ∂v 2 + 2u ∂ξ ∂u + 2v ∂ξ ∂v = 0. Assuming that the solution of interest is in the domain of hyperbolicity, we write out the equation of characteristics (see Subsection 14.1.1): u 2 dv 2 –(1 + 2uv) du dv + v 2 du 2 = 0. This equation has the integrals r = C 1 and s = C 2 ,where r = √ 1 + 4uv – 1 2v , s = √ 1 + 4uv – 1 2u . (15.2.2.15) Passing in (15.2.2.14) to the new variables (15.2.2.15), we obtain r 2 ∂ξ ∂r + ∂η ∂r = 0, ∂ξ ∂s + s 2 ∂η ∂s = 0. (15.2.2.16) Eliminating η yields the simple equation ∂ 2 ξ ∂r∂s = 0, whose general solution is the sum of two arbitrary functions with different arguments: ξ = f(r)+g(s). The function η is determined from system (15.2.2.16).  In Paragraph 15.14.4-4, the hodograph transformation is used for the linearization of gas-dynamic systems of equations. 660 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.2.3. Contact Transformations. ∗ Legendre and Euler Transformations 15.2.3-1. General form of contact transformations. Consider functions of two variables w = w(x, y). A common property of contact trans- formations is the dependence of the original variables on the new variables and their first derivatives: x = X  ξ, η, u, ∂u ∂ξ , ∂u ∂η  , y = Y  ξ, η, u, ∂u ∂ξ , ∂u ∂η  , w = W  ξ, η, u, ∂u ∂ξ , ∂u ∂η  , (15.2.3.1) where u = u(ξ, η). The functions X, Y ,andW in (15.2.3.1) cannot be arbitrary and are selected so as to ensure that the first derivatives of the original variables depend only on the transformed variables and, possibly, their first derivatives, ∂w ∂x = U  ξ, η, u, ∂u ∂ξ , ∂u ∂η  , ∂w ∂y = V  ξ, η, u, ∂u ∂ξ , ∂u ∂η  .(15.2.3.2) Contact transformations (15.2.3.1)–(15.2.3.2) do not increase the order of the equations to which they are applied. In general, a contact transformation (15.2.3.1)–(15.2.3.2) reduces a second-order equa- tion in two independent variables F  x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2  = 0 (15.2.3.3) to an equation of the form G  ξ, η, u, ∂u ∂ξ , ∂u ∂η , ∂ 2 u ∂ξ 2 , ∂ 2 u ∂ξ∂η , ∂ 2 u ∂η 2  = 0.(15.2.3.4) In some cases, equation (15.2.3.4) turns out to be more simple than (15.2.3.3). If u = u(ξ, η) is a solution of equation (15.2.3.4), then formulas (15.2.3.1) define the corresponding solution of equation (15.2.3.3) in parametric form. Remark. It is significant that the contact transformations are defined regardless of the specific equations. 15.2.3-2. Legendre transformation. An important special case of contact transformations is the Legendre transformation defined by the relations x = ∂u ∂ξ , y = ∂u ∂η , w = xξ + yη – u,(15.2.3.5) where ξ and η are the new independent variables, and u = u(ξ, η) is the new dependent variable. Differentiating the last relation in (15.2.3.5) with respect to x and y and taking into account the other two relations, we obtain the first derivatives: ∂w ∂x = ξ, ∂w ∂y = η.(15.2.3.6) * Prior to reading this section, it is recommended that Subsection 12.1.8 be read first. . Transformations of Equations of Mathematical Physics 15.2.1. Point Transformations: Overview and Examples 15.2.1-1. General form of point transformations. Let w = w(x, y) be a function of independent. t)u for the function u = u(x, t) by means of the transformation u =exp(aw). 15.2.2. Hodograph Transformations (Special Point Transformations) In some cases, nonlinear equations and systems of partial. linearization of gas-dynamic systems of equations. 660 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.2.3. Contact Transformations. ∗ Legendre and Euler Transformations 15.2.3-1. General form of contact