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3669201625 - Chapter 5.3 Finite Difference for PDE of 1D in...

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1 Chapter 5.3 Finite Difference for PDE of 1D in Space Parabolic equations are employed to characterize time-variable ( unsteady-state ) problems. Conservation of energy can be used to develop an unsteady-state energy balance for the differential element in a long, thin insulated rod. 2 Heat transfer in a 1D rod Fourier’s law for the definition of heat flux ) / ( dx dT k Q = Models and PDEs
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3 Energy balance together with Fourier’s law of heat conduction yields heat-conduction equation : Parabolic equations can be solved by substituting finite divided differences for the partial derivatives. In contrast to ODEs, we must now consider changes in time as well as in space. Parabolic PDEs are temporally open-ended and involve new issues such as stability. t T x T k = 2 2 4 A grid used for the solution of parabolic PDE using FD Time Positions
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5 Numerical method: Summary 2 2 2 2 . When 0, , diffusivity ρ κ ρ κ v v T T k k q c q x T T x t t c + = = = =
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  • Spring '10
  • Dr.G.P.Zheng
  • Partial differential equation, Parabolic partial differential equation, unsteady heat transfer, Time derivative Tt, Space derivative Txx

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