9_Partial_diff_equations

9_Partial_diff_equations - Partial Differential...

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Partial Differential Equations (PDE) Dr.B.Santhosh Department of Mechanical Engineering Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Introduction PDE is an equation involving partial derivatives of an unknown function with respect to more than one independent variable PDE’s are important in modeling all types of continuous phenomena in nature Examples Maxwell’s Equations : describing the behavior of an electromagnetic field Navier-Stoke’s Equations : describing the behavior of fluids Linear Elasticity Equations : describe the vibrations in an elastic solids Shrodinger Equation of quantum mechanics Einstein’s Equation of general relativity Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Introduction Notations u t = u t u xy = 2 u x y The independent variable may be space x and time t or two spatial variables x , y Order of a PDE : determined by the highest order partial derivative appearing in PDE One dimensional wave equation (first order PDE) u t = - cu x u t = - c u x Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Introduction Second order PDE’s u t = u xx Heat Equation u tt = u xx Wave Equation u xx + u yy = 0 Laplace Equation Solution of a PDE : Determine a function u whose partial derivatives with respect to the independent variables satisfy the relationship prescribed by a given PDE on a given domain and also satisfy the initial and boundary conditions imposed Solution u can be visualized as a surface over the relevant two dimensional domain in the ( t , x ) or ( x , y ) plane Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Classification of PDE Any second order PDE can be represented in a general form au xx + bu xy + cu yy + du x + eu y + fu + g = 0 The coefficients a , b , .... g may depend on the independent variable but not on the unknown function u The quantity b 2 - 4 ac is known as discriminant The second order linear PDE’s can be classified as b 2 - 4 ac > 0 Hyperbolic . (Eg: Wave equation) b 2 - 4 ac = 0 Parabolic . (Eg: Heat equation) b 2 - 4 ac < 0 Elliptic . (Eg: Laplace equation) Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Hyperbolic PDE Systems governed by hyperbolic PDE are conservative in that the energy in the system is conserved over time Analogous to a linear system of ODE’s whose matrix has purely imaginary eigenvalues The solution is oscillatory neither grows nor decays with time Example : Two dimensional wave equation u tt = u xx Dr.B.Santhosh Department of Mechanical Engineering Partial Differential Equations (PDE)
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Parabolic PDE Are dissipative in that the energy of the system diminishes over time Analogous to a linear system of ODE’s whose matrix has only eigenvalues with negative real part Parabolic PDE’s propagate information instantaneously Parabolic PDE’s have a smoothing effect that over time
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