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

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Unformatted text preview: 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 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 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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This note was uploaded on 08/27/2010 for the course ME ME4906 taught by Professor Dr.g.p.zheng during the Spring '10 term at NYU Poly.

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

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