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NUMERICAL ANALYSIS of PROCESSES NAP6 Partial differential equations (PDE), classification to hyperbolic, parabolic and eliptic equations Hyperbolic PDE (oscillation of trusses, beams, water hammer) MOC-method of characteristics (compressible flow) Rudolf itn, stav procesn a zpracovatelsk techniky VUT FS 2010

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NAP6 PDE partial differential equations L.Wagner

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NAP6 PDE partial differential equations When a system is described by a greater number of independent variables (spatial coordinates and time) we have to deal with partial differential equations (PDEs). Most PDE are differential equations with maximum second derivatives of the dependent variable (except eg. biharmonic equation deformation of membranes with the fourth derivative): Hyperbolic equation (oscillations and waves, supersonic flow. Characterised by finite velocity of pressure waves) Parabolic equation (evolution problems, e.g. time evolution of a temperature or concentration profile, but also for example evolution of a boundary layer from inlet.) Eliptic equation (steady problems of distribution temperatures, deformations,) Example: Poissons equation Example: Fourier eqauation of heat transfer Example: vibration of an elastic beam, water hammer

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NAP6 PDE partial differential equations Typ of PDE is determined by coefficients of second derivatives (in case of the second order PDE) where a,b,c,f are arbitrary functions of x,y, solution and its first derivatives. Coefficients a,b,c determine characteristics y(x), satisfying equation b 2 -4ac>0 hyperbolic equation (two real roots, therefore two characteristics) b 2 -4ac=0 parabolic equation (one root, one characteristic) b 2 -4ac

NAP6 What is important Type of equation is determined by coefficients at the highest (second) derivatives Characteristics are real if b 2 -4ac>0 (hyperbolic equation) One characteristic if b 2 -4ac=0 (parabolic equation) Real characteristics do not exist if b 2 -4ac