Math 412-501Theory of Partial Differential EquationsLecture 2-1:Higher-dimensional heat equation.PDEs: two variablesheat equation:∂u∂t= k∂2u∂x2wave equation:∂2u∂t2= c2∂2u∂x2Laplace’s equation:∂2u∂x2+∂2u∂y2= 0PDEs: three variablesheat equation:∂u∂t= k∂2u∂x2+∂2u∂y2wave equation:∂2u∂t2= c2∂2u∂x2+∂2u∂y2Laplace’s equation:∂2u∂x2+∂2u∂y2+∂2u∂z2= 0One-dimensional heat equationDescribes heat conduction in a rod:cρ∂u∂t=∂∂xK0∂u∂x+ QK0= K0(x), c = c(x), ρ = ρ(x), Q = Q(x, t).Assuming K0, c, ρ are constant (uniform rod) andQ = 0 (no heat sources), we obtain∂u∂t= k∂2u∂x2where k = K0(cρ)−1.Heat conduction in three dimensionsu(x, y, z, t) =temperature at point (x, y, z) at time te(x, y, z, t) = thermal energy density (thermalenergy per unit volume)Q(x, y, z, t) = density of heat sources (heat energyper unit volume generated per unit time)φ(x, y, z, t) = heat flux~φ(x, y, z, t) is a vector fieldthermal energy flowing per unit surface per unittime =~φ(x, y, z, t) ·~n(x, y, z), where n(x, y, z) isthe unit normal vector of the surfaceHeat fluxc(x, y, z) = specific heat or heat capacity (the heatenergy supplied to a unit mass of a substance toraise its temperature one unit)ρ(x, y, z) = mass density (mass per u nit volume)Thermal energy in a volume is equal to the energy ittakes t o raise the temperature of the volume from areference temperature (zero) to it s actualtemperature.e(x, y, z, t) · ∆V = c(x, y, z)u(x, y, z, t) · ρ(x, y, z) · ∆Ve(x, y, z, t) = c(x, y, z)ρ(x, y, z)u(x, y, z, t)Four quantit ies: u, e, Q, φ.Heat equation should involve only two: u and Q.Heat equation is d erived from two physical laws:• conservation of heat energy,• Fourier’s low of heat conduction.Conservation of heat energy (in a volume in aperiod of t ime):change of heat energy heat energyheat = flowing across + generatedenergy boundary insiderate of heat energy heat energychange of = flowing across + generatedheat boundary inside perenergy per unit time unit timesubregion Rheat energy:ZZZRe(x, y, z, t) dx dy dz =ZZZRe dVrate of change of heat energy:∂∂tZZZRe dV=∂∂tZZZRcρu dVsubregion Rheat energy flowing across boundary per unit time:−II∂R~φ · n dS,where n is the unit outward normal vector of ∂R.heat energy generated in side per unit time:ZZZRQ dV∂∂tZZZRcρu dV= −II∂R~φ· n dS +ZZZRQ dVZZZRcρ∂u∂tdV = −II∂R~φ · n dS +ZZZRQ dVZZZR∇ ·~φ dV =II∂R~φ · n dSwhere~φ = (φx, φy, φz), ∇ ·~φ =∂φx∂x+∂φy∂y+∂φz∂z.(Gauss’ formula) (divergence theorem)∇ ·~φ is called the divergence of vector field φ.ZZZRcρ∂u∂tdV = −ZZZR∇ ·~φ dV +ZZZRQ dVSince R is an arbitrary subregion,cρ∂u∂t= −∇ ·~φ + QFourier’s law of heat conduction:~φ = −K0∇u,where K0= K0(x, u) is the thermal conductivityand ∇u = (∂u∂x,∂u∂y,∂u∂z) is the gradient of u.Heat equation: cρ∂u∂t= ∇ · (K0∇u) + QAssuming K0= const, we havecρ∂u∂t= K0∇2u + Q,where ∇2u = ∇ · (∇u) =∂2u∂x2+∂2u∂y2+∂2u∂z2is theLaplacian of u.Assuming K0, c, ρ = const (uniform medium) andQ = 0 (no heat sources), we obtain∂u∂t= k ∇2u,where k = K0(cρ)−1is called the thermal diffusivity.NotationEach function f : R3→ R is assigned the gradient ( a vectorfield) and the Laplacian (a function). Each vector field~φ : R3→ R3is assigned the divergence (a function).“physical” n otation: ∇ = (∂∂x,∂∂y,∂∂z)gradient: ∇f = (∂f∂x,∂f∂y,∂f∂z)divergence: ∇ ·~φ =∂φx∂x+∂φy∂y+∂φz∂zLaplacian: ∇2f = ∇ · (∇f ) =∂2f∂x2+∂2f∂y2+∂2f∂z2“mathematical” notation:gradient: grad f = (∂f∂x,∂f∂y,∂f∂z)divergence: div~φ =∂φx∂x+∂φy∂y+∂φz∂zLaplacian: ∆f = div(grad f )
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