The One-Dimensional Heat Equation - Trinity University

The heat equationHomogeneous Dirichlet conditionsInhomogeneous Dirichlet conditionsThe One-Dimensional heat EquationR. C. DailedaTrinity UniversityPartial Differential EquationsFebruary 25, 2014 Daileda1-D heat EquationThe heat equationHomogeneous Dirichlet conditionsInhomogeneous Dirichlet conditionsIntroductionGoal:Model heat flow in a One-Dimensional object (thin rod).Set up:Place rod of lengthLalongx-axis, one end at origin:xL0heated rodLetu(x,t) = temperature in rod at positionx, timet.(Ideal) Assumptions:Rod is given some initial temperature distributionf(x) alongits is perfectly insulated, heat only moves internal heat sources or heat EquationThe heat equationHomogeneous Dirichlet conditionsInhomogeneous Dirichlet conditionsThe heat EquationOne can show thatusatisfies theone- dimensional heat equationut= :This can be derived via conservation of energy and Fourier slaw of heat conduction (see textbook pp. 143-144).The constantc2is thethermal diffusivity:K0= thermal conductivity,c2=K0s ,s= specific heat , = heat EquationThe heat equationHomogeneous Dirichlet conditionsInhomogeneous Dirichlet conditionsInitial and Boundary ConditionsTo completely determineuwe must also specify:Initial conditions:The initial temperature profileu(x,0) =f(x) for 0<x< conditions:Specific behavior atx0= 0, temperature:u(x0,t) =Tfort> end:ux(x0,t) = 0 fort> end:ux(x0,t) =Au(x0,t) fort> heat EquationThe heat

The heat equation Homogeneous Dirichlet conditions Inhomogeneous Dirichlet conditions TheHeatEquation One can show that u satisfies the one-dimensional heat equation u t = c2u xx. Remarks: This can be derived via conservation of energy and Fourier’s law of heat conduction (see textbook pp. 143-144). The constant c2 is the thermal diffusivity: K

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  Heat, Equations, Dimensional, Heat equation, Dimensional heat equation

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