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Twovariable expansion procedures
Applications to partial differential equations
2 other sections not shown
amplitude applied approximate equations assumed asymptotic expansion asymptotic sequence behavior body boundary condition Equation boundary layer boundary-layer equations boundary-value problem characteristic coefficients considered const constants of integration constructed coordinates corresponding curl curl curl curves decay depends dimensionless discussed distinguished limit dominant terms elasticity equation Equation exact solution example expansion Equation expansion procedure expansion valid exponential exponential decay expressed Figure fixed fluid functions hoop stresses ideas initial conditions inner and outer inner expansion integral of Equation intermediate limit inviscid limit process matching condition method motion Navier-Stokes equations nonlinear obtained ordinary differential equations outer expansion outer solution partial differential equations periodic solution perturbation phase plane piston potential flow pressure Reference result right-hand side second-order Section shock waves side of Equation singular small parameter solution of Equation subcharacteristic suitable theory tion transcendentally small transonic uniformly valid upstream infinity variables viscous vorticity zero