Perturbation Methods in Applied MathematicsThis book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe matics, Ginn-Blaisdell, 1968. We present the material at a level which assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate level course on the subject. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B. |
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A₁ adiabatic invariant airfoil amplitude approximation assume asymptotic expansion B₁ boundary condition boundary layer C₁ calculate canonical transformation characteristic coefficients consider const constant coordinates corresponding curves d²y defined depend derive differential equations discussed exact solution example expressed F₁ Figure fixed flow function g₁ given h₁ Hamiltonian initial conditions initial value problem inner expansion integral K₁ linear matching method motion multiple variable Navier-Stokes equations obtain oscillator outer expansion outer limit P₁ parameter partial differential equations periodic solutions perturbation phase plane Reference resonance result right-hand side Section shock side of Equation singular slowly varying solution of Equation solve t₁ theory transonic u₁ uniformly valid V₁ variable expansion velocity w₁ w₂ wave zero ε² ди ду дф дх