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Ebook Handbook of mathematics for engineers and scientists: Part 2

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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 ∂2w ∂2w ∂w ∂w ∂2w + c(x, y) 2 = f x, y, w, , . (15.1.1.1) a(x, y) 2 + 2b(x, y) ∂x ∂x∂y ∂y ∂x ∂y 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 0, it is hyperbolic, and if δ 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 2 (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 = ξx τ + 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 ≠ 0 (yx = yτ /x τ ). (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 .