Ebook Handbook of mathematics for engineers and scientists: Part 2
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Ebook Handbook of mathematics for engineers and scientists: Part 2
Chapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear Equations Ebook Handbook of mathematics for engineers and scientists: Part 2 in Two Independent VariablesA second-order sentilinear partial differential equation in two independent variables has the form,,,. (fiw ,92w,/ ỠU’ ỡw\a(g,y)-gjr + 26(^.y) 4- c(x,y)-X“2 = xty,w,).(15.1.1 1)ax-dxdyay- \ ax ay)This equation is classified according to the sign of the discriminant8-lr-a Ebook Handbook of mathematics for engineers and scientists: Part 2c,(15.1.1.2)where the arguments of the equation coefficients are omitted for brevity. Given a point Cr.y). equation (15.1.1.1) ISparabolicifỔ = 0.hypeEbook Handbook of mathematics for engineers and scientists: Part 2
rbolicifỔ > 0,(15.1.1.3)ellipticifỔ < 0.The reduction of equation (15.1.1.1) to a canonical form on the basis of the solution of the characteristic eqChapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear Equations Ebook Handbook of mathematics for engineers and scientists: Part 21) does not depend on their solutions — it is determined solely by the coefficients of the highest derivatives on (he left-hand side.15.1.2.Classification of Nonlinear Equations in Two Independent Variables15.1.2-1. Nonlinear equations of general form.In general, a second-order nonlinear partial dif Ebook Handbook of mathematics for engineers and scientists: Part 2ferential equation in two independent variables has the formdw dw d2w o^w tffw x'y,Wt dĩ' ~õỹ'~õxĩ' Thxhff w(15.1.2.1)653654Nonlinear Partial DifferenEbook Handbook of mathematics for engineers and scientists: Part 2
tial EquationsDenoteOF ,1 OF OF ,02w02w iFw ,O=7IT’where<7 = 3T^7- r ~ TiTj- (15.1.2.2)dp 2 aq arax- axay ay-Let us select a specific solution U' - w(Chapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear Equations Ebook Handbook of mathematics for engineers and scientists: Part 2.1.1.2). Depending on the sign of the discriminant Ổ. the type of nonlinear equation (15.1.2.1) al the point (:r,7/) is determined according Io (15.1.1.3): if Ó - 0, the equation is parabolic, if Ổ > 0. it is hyperbolic, and if Ỗ < 0. it is elliptic, hl general, the coefficients a, b, and (■ of the Ebook Handbook of mathematics for engineers and scientists: Part 2nonlinear 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, itEbook Handbook of mathematics for engineers and scientists: Part 2
is impossible to determine the sign of Ổ without knowing the solution w(x, y). To pul it differently, the lyjx? of a nonlinear equation can be differeChapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear Equations Ebook Handbook of mathematics for engineers and scientists: Part 2ntegral curve of the characteristic equationa (dy)1 - 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 semihnear equation (15.1.1 1)] may be independent of the selec Ebook Handbook of mathematics for engineers and scientists: Part 2tion of solutions.Example. Consider the nonhomqgcneous Mongc-Ainpcre equation/ iPw Ỹ ipw &w _ .._______.Chapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear Equations Chapter 15Nonlinear Partial Differential Equations15.1.Classification of Second-Order NonlinearEquations15.1.1.Classification of Semilinear EquationsGọi ngay
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