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Part 2 book “Handbook of mathematics for engineers and scientists” has contents: Nonlinear partial differential equations, integral equations, difference equations and other functional equations, special functions and their properties, calculus of variations and optimization, probability theory, mathematical statistics,… and other contents.
Chapter 15 Nonlinear 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 ∂2w ∂w ∂w ∂2w + c(x, y) , + 2b(x, y) = f x, y, w, 2 ∂x ∂x∂y ∂y ∂x ∂y (15.1.1.1) This equation is classified according to the sign of the discriminant δ = b2 – ac, (15.1.1.2) where the arguments of the equation coefficients are omitted for brevity Given a point (x, y), equation (15.1.1.1) is parabolic hyperbolic elliptic if δ = 0, if δ > 0, if δ < (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 variables has the form F x, y, w, ∂w ∂w ∂ w ∂ w ∂ w , , , , ∂x ∂y ∂x2 ∂x∂y ∂y 653 = (15.1.2.1) 654 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Denote a= ∂F ∂F ∂2w ∂2w ∂2w ∂F , b= , c= , where p = , r= , q= (15.1.2.2) ∂p ∂q ∂r ∂x ∂x∂y ∂y 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, and c 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 – 2b dx dy + c (dx)2 = (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 ∂2w ∂x∂y – ∂2w ∂2w = f (x, y) ∂x2 ∂y It is a special case of equation (15.1.2.1) with F (x, y, p, q, r) ≡ q – pr – f (x, y) = 0, p= ∂2w , ∂x2 q= ∂2w , ∂x∂y r= ∂2w ∂y (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 – 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), with the short notation ξ= ∂w , ∂x η= ∂w , ∂y p= ∂2w , ∂x2 q= ∂2w , ∂x∂y r= ∂2w ∂y (15.1.2.6) 655 15.2 TRANSFORMATIONS OF EQUATIONS OF MATHEMATICAL PHYSICS Consider a curve C0 defined in the x, 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 ), (wτ = ξxτ + ηyτ ) η = η(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, and r (and also higher derivatives) at each point of the curve (15.1.2.7), satisfying the condition a(yx )2 – 2byx + c ≠ (yx = yτ /xτ ) (15.1.2.9) Here and henceforth, the arguments of the functions a, b, and c are omitted Indeed, bearing in mind that ξ = ξ(x, y) and η = η(x, y), let us differentiate the second and the third equation in (15.1.2.8) with respect to the parameter τ : ξτ = pxτ + qyτ , ητ = qxτ + ryτ (15.1.2.10) On solving relations (15.1.2.6) and (15.1.2.10) for p, q, and r, we obtain formulas for the second derivatives at the points of the curve (15.1.2.7): c(xτ ξτ – yτ ητ ) – 2byτ ξτ + f (yτ )2 , a(yτ )2 – 2bxτ yτ + c(xτ )2 ayτ ξτ + cxτ ητ – f xτ yτ , q= a(yτ )2 – 2bxτ yτ + c(xτ )2 a(yτ ητ – xτ ξτ ) – 2bxτ ητ + f (xτ )2 r= a(yτ )2 – 2bxτ yτ + c(xτ )2 p= (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 ∂Y ∂x ∂W ∂x ∂X ∂y ∂Y ∂y ∂W ∂y ∂X ∂w ∂Y ∂w ∂W ∂w ≠ 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 ∂w ∂ w ∂ w ∂ w , , , , ∂x ∂y ∂x2 ∂x∂y ∂y G ξ, η, u, ∂u ∂u ∂ u ∂ u ∂ u , , , , ∂ξ ∂η ∂ξ ∂ξ∂η ∂η =0 (15.2.1.2) to an equation = (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 + x0 , x = k1 ξ, x = ξ cos α – η sin α, y = η + y0 y = k2 η y = ξ sin α + η cos α (translation transformation), (scaling transformation), (rotation transformation) These transformations correspond to the translation of the origin of coordinates to the point (x0 , y0 ), scaling (extension or contraction) along the x- and y-axes, and the rotation of the coordinate system through the angle α, respectively These transformations not affect the dependent variable (w = u) Linear transformations (15.2.1.4) are employed to simplify linear and nonlinear equations and to reduce equations to the canonical forms (see Subsections 14.1.1 and 15.1.1) 15.2 TRANSFORMATIONS OF EQUATIONS OF MATHEMATICAL PHYSICS 657 Example The nonlinear equation ∂ ∂w ∂w =a wm ∂t ∂x ∂x + xf (t) + g(t) ∂w + h(t)w ∂x can be simplified to obtain ∂u ∂u ∂ = um ∂τ ∂z ∂z with the help of the transformation w(x, t) = u(z, τ )H(t), z = xF (t) + F (t)H m (t) dt, g(t)F (t) dt, τ = H(t) = exp h(t) dt where F (t) = exp f (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 not affect the independent variables (x = ξ and y = η) Combinations of transformations (15.2.1.4) and (15.2.1.5) are also used quite often Example The nonlinear equation ∂w ∂w ∂2w +a = ∂t ∂x2 ∂x + f (x, t) can be reduced to the linear equation ∂u ∂2u = + af (x, t)u ∂t ∂x2 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 Consider the nonlinear second-order equation ∂w ∂w ∂2w =a (15.2.2.2) ∂y ∂x ∂x 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 (differentiation in x), = xw wx (differentiation in y), = xw wy + xy = xww wx2 + xw wxx (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, xy w2 xww xww , wy = – , wxx = – x =– xw xw xw xw Substituting these expressions into (15.2.2.2), we obtain the linear heat equation: wx = ∂x ∂2x =a ∂y ∂w2 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) and v = 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 Rewrite the stationary Khokhlov–Zabolotskaya equation (it arises in acoustics and nonlinear mechanics) ∂w ∂2w ∂ w =0 (15.2.2.4) +a ∂x2 ∂y ∂y as the system of equations ∂v ∂w ∂v ∂w = , –aw = (15.2.2.5) ∂x ∂y ∂y ∂x 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 xw , xv , yw , yv from the resulting relations, we obtain ∂v ∂x ∂w ∂y ∂v ∂y ∂w ∂w ∂v ∂w ∂v ∂x = , =– , =– , = , where J = – (15.2.2.6) ∂w J ∂y ∂v J ∂y ∂w J ∂x ∂v J ∂x ∂x ∂y ∂y ∂x Using (15.2.2.6) to eliminate the derivatives wx , wy , vx , vy from (15.2.2.5), we arrive at the system ∂x ∂x ∂y ∂y = , –aw = (15.2.2.7) ∂v ∂w ∂v ∂w Let us differentiate the first equation in w and the second in v, and then eliminate the mixed derivative ywv As a result, we obtain the following linear equation for the function x = x(w, v): ∂2x ∂2x + aw = ∂w ∂v Similarly, from system (15.2.2.7), we obtain another linear equation for the function y = y(w, v), (15.2.2.8) ∂2y ∂ ∂y + = (15.2.2.9) ∂v ∂w aw ∂w Given a particular solution x = x(w, v) of equation (15.2.2.8), we substitute this solution into system (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 MATHEMATICAL PHYSICS 659 Remark Equation (15.2.2.8) with an arbitrary a admits a simple particular solution, namely, x = C1 wv + C2 w + C3 v + C4 , (15.2.2.10) where C1 , , C4 are arbitrary constants Substituting this solution into system (15.2.2.7), we obtain ∂y ∂y = C1 v + C2 , = –a(C1 w + C3 )w (15.2.2.11) ∂v ∂w Integrating the first equation in (15.2.2.11) yields y = 12 C1 v + C2 v + ϕ(w) Substituting this solution into the second equation in (15.2.2.11), we find the function ϕ(w), and consequently y = 12 C1 v + C2 v – 13 aC1 w3 – 12 aC3 w2 + C5 (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 Consider the Born–Infeld equation ∂w ∂ w ∂w ∂ w ∂w ∂w ∂ w – 1+ +2 = 0, (15.2.2.13) ∂t ∂x2 ∂x ∂t ∂x∂t ∂x ∂t2 which is used in nonlinear electrodynamics (field theory) By introducing the new variables ∂w ∂w ξ = x – t, η = x + t, u = , v= , ∂ξ ∂η equation (15.2.2.13) can be rewritten as the equivalent system ∂u ∂v – = 0, ∂η ∂ξ ∂u ∂v ∂u – (1 + 2uv) + u2 = v2 ∂ξ ∂η ∂η 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 ∂η – = 0, ∂v ∂u (15.2.2.14) ∂ξ ∂η ∂ξ v2 + (1 + 2uv) + u2 = ∂v ∂v ∂u Eliminating η yields the linear second-order equation 1– ∂2ξ ∂ξ ∂2ξ ∂2ξ ∂ξ + (1 + 2uv) + v 2 + 2u + 2v = ∂u2 ∂u∂v ∂v ∂u ∂v Assuming that the solution of interest is in the domain of hyperbolicity, we write out the equation of characteristics (see Subsection 14.1.1): u2 u2 dv – (1 + 2uv) du dv + v du2 = This equation has the integrals r = C1 and s = C2 , where √ √ + 4uv – 1 + 4uv – , s= (15.2.2.15) r= 2v 2u Passing in (15.2.2.14) to the new variables (15.2.2.15), we obtain ∂ξ ∂η + = 0, r2 ∂r ∂r (15.2.2.16) ∂ξ ∂η + s2 = ∂s ∂s Eliminating η yields the simple equation ∂2ξ = 0, ∂r∂s 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 transformations is the dependence of the original variables on the new variables and their first derivatives: ∂u ∂u , , ∂ξ ∂η (15.2.3.1) where u = u(ξ, η) The functions X, Y , and W 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, x = X ξ, η, u, ∂u ∂u , , ∂ξ ∂η y = Y ξ, η, u, ∂u ∂u ∂w = U ξ, η, u, , , ∂x ∂ξ ∂η ∂u ∂u , , ∂ξ ∂η w = W ξ, η, u, ∂w ∂u ∂u = V ξ, η, u, , ∂y ∂ξ ∂η (15.2.3.2) Contact transformations (15.2.3.1)–(15.2.3.2) 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 equation in two independent variables F x, y, w, ∂w ∂w ∂ w ∂ w ∂ w , , , , ∂x ∂y ∂x2 ∂x∂y ∂y =0 (15.2.3.3) to an equation of the form G ξ, η, u, ∂u ∂u ∂ u ∂ u ∂ u , , , , ∂ξ ∂η ∂ξ ∂ξ∂η ∂η = (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 ∂u ∂u , y= , w = xξ + yη – u, (15.2.3.5) x= ∂ξ ∂η 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 = ξ, ∂x ∂w = η ∂y * Prior to reading this section, it is recommended that Subsection 12.1.8 be read first (15.2.3.6) 661 15.2 TRANSFORMATIONS OF EQUATIONS OF MATHEMATICAL PHYSICS With (15.2.3.5)–(15.2.3.6), we find the second derivatives ∂2u ∂2w = J , ∂x2 ∂η where J= ∂2w ∂2w – ∂x2 ∂y ∂2w ∂2u = –J , ∂x∂y ∂ξ∂η ∂2w ∂x∂y , ∂2u ∂2w = J , ∂y ∂ξ ∂2u ∂2u = – J ∂ξ ∂η ∂2u ∂ξ∂η The Legendre transformation (15.2.3.5), with J ≠ 0, allows us to rewrite a general second-order equation with two independent variables F x, y, w, ∂w ∂w ∂ w ∂ w ∂ w , , , , ∂x ∂y ∂x2 ∂x∂y ∂y =0 (15.2.3.7) in the form F ∂u ∂2u ∂2u ∂2u ∂u ∂u ∂u , ,ξ +η – u, ξ, η, J , –J ,J ∂ξ ∂η ∂ξ ∂η ∂η ∂ξ∂η ∂ξ = (15.2.3.8) Sometimes equation (15.2.3.8) may be simpler than (15.2.3.7) Let u = u(ξ, η) be a solution of equation (15.2.3.8) Then the formulas (15.2.3.5) define the corresponding solution of equation (15.2.3.7) in parametric form Remark The Legendre transformation may result in the loss of solutions for which J = Example The equation of steady-state transonic gas flow a ∂w ∂ w ∂ w + =0 ∂x ∂x2 ∂y is reduced by the Legendre transformation (15.2.3.5) to the linear equation with variable coefficients aξ ∂2u ∂2u + = ∂η ∂ξ Example The Legendre transformation (15.2.3.5) reduces the nonlinear equation f ∂w ∂w , ∂x ∂y ∂2w ∂w ∂w +g , ∂x2 ∂x ∂y ∂2w ∂w ∂w +h , ∂x∂y ∂x ∂y ∂2w =0 ∂y to the following linear equation with variable coefficients: f (ξ, η) ∂2u ∂2u ∂2u – g(ξ, η) = + h(ξ, η) ∂η ∂ξ∂η ∂ξ 15.2.3-3 Euler transformation The Euler transformation belongs to the class of contact transformations and is defined by the relations ∂u , y = η, w = xξ – u (15.2.3.9) x= ∂ξ Differentiating the last relation in (15.2.3.9) with respect to x and y and taking into account the other two relations, we find that ∂w = ξ, ∂x ∂w ∂u =– ∂y ∂η (15.2.3.10) 662 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Differentiating these expressions in x and y, we find the second derivatives: wxx = , uξξ wxy = – uξη , uξξ wyy = u2ξη – uξξ uηη uξξ (15.2.3.11) The subscripts indicate the corresponding partial derivatives The Euler transformation (15.2.3.9) is employed in finding solutions and linearization of certain nonlinear partial differential equations The Euler transformation (15.2.3.9) allows us to reduce a general second-order equation with two independent variables F x, y, w, ∂w ∂w ∂ w ∂ w ∂ w , , , , ∂x ∂y ∂x2 ∂x∂y ∂y =0 (15.2.3.12) to the equation uξη u2ξη – uξξ uηη ,– , F uξ , η, ξuξ – u, ξ, –uη , uξξ uξξ uξξ = (15.2.3.13) In some cases, equation (15.2.3.13) may become simpler than equation (15.2.3.12) Let u = u(ξ, η) be a solution of equation (15.2.3.13) Then formulas (15.2.3.9) define the corresponding solution of equation (15.2.3.12) in parametric form Remark The Euler transformation may result in the loss of solutions for which wxx = Example The nonlinear equation ∂w ∂ w +a=0 ∂y ∂x2 is reduced by the Euler transformation (15.2.3.9)–(15.2.3.11) to the linear heat equation ∂u ∂2u =a ∂η ∂ξ Example The nonlinear equation ∂w ∂ w ∂ 2w =a ∂x∂y ∂y ∂x2 (15.2.3.14) can be linearized with the help of the Euler transformation (15.2.3.9)–(15.2.3.11) to obtain ∂u ∂2u =a ∂ξ∂η ∂η Integrating this equation yields the general solution u = f (ξ) + g(η)eaξ , (15.2.3.15) where f (ξ) and g(η) are arbitrary functions Using (15.2.3.9) and (15.2.3.15), we obtain the general solution of the original equation (15.2.3.14) in parametric form: w = xξ – f (ξ) – g(y)eaξ , x = fξ (ξ) + ag(y)eaξ Remark In the degenerate case a = 0, the solution w = ϕ(y)x + ψ(y) is lost, where ϕ(y) and ψ(y) are arbitrary functions; see also the previous remark ... in (15 .2. 2.11), we find the function ϕ(w), and consequently y = 12 C1 v + C2 v – 13 aC1 w3 – 12 aC3 w2 + C5 (15 .2. 2. 12) Formulas (15 .2. 2.10) and (15 .2. 2. 12) define an exact solution of equation... ξτ + f (yτ )2 , a(yτ )2 – 2bxτ yτ + c(xτ )2 ayτ ξτ + cxτ ητ – f xτ yτ , q= a(yτ )2 – 2bxτ yτ + c(xτ )2 a(yτ ητ – xτ ξτ ) – 2bxτ ητ + f (xτ )2 r= a(yτ )2 – 2bxτ yτ + c(xτ )2 p= (15.1 .2. 11) The... ∂2w ∂2w = f (x, y) ∂x2 ∂y It is a special case of equation (15.1 .2. 1) with F (x, y, p, q, r) ≡ q – pr – f (x, y) = 0, p= ∂2w , ∂x2 q= ∂2w , ∂x∂y r= ∂2w ∂y (15.1 .2. 4) Using formulas (15.1 .2. 2)