Math 412-501Theory of Partial Differential EquationsLecture 1: Introduction. Heat equationDefinitionsA differential equation is an equation involving anunknown function and certain of its derivatives.An ordinary differential equation (ODE) is anequation involving an unknown function of onevariable and certain of its derivatives.A partial differential equation (PDE) is anequation involving an unknown function of two ormore variables and certain of its partial derivatives.Examplesx2+ 2x + 1 = 0 (algebraic equation)f (2x) = 2(f (x))2− 1 (functional equation)f′(t) + t2f (t) = 4 (ODE)∂u∂x+ 3∂2u∂x∂y− u∂u∂y(not an equation)∂u∂x− 5∂u∂y= u (PDE)u + u2=∂2u∂x∂y(0, 0) (functional-differentialequation)heat equation:∂u∂t= k∂2u∂x2wave equation:∂2u∂t2= c2∂2u∂x2Laplace’s equation:∂2u∂x2+∂2u∂y2= 0In the first two equations, u = u(x, t). In the latterone, u = u(x, y).heat equation:∂u∂t= k∂2u∂x2+∂2u∂y2wave equation:∂2u∂t2= c2∂2u∂x2+∂2u∂y2Laplace’s equation:∂2u∂x2+∂2u∂y2+∂2u∂z2= 0In the first two equations, u = u(x, y, t). In thelatter one, u = u(x, y, z).Heat conduction in a rodu(x, t) = temperaturee(x, t) = thermal energy density (thermal energyper unit volume)Q(x, t) = density of heat sources (heat energy perunit volume generated per unit time)φ(x, t) = heat flux (thermal energy flowing per unitsurface per unit time)φ(x, t) > 0 if heat energy is flowing to the right,φ(x, t) < 0 if heat energy is flowing to the leftConservation of heat energy (in a volume in aperiod of time):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 timeA = area of a sectionheat energy = e(x, t) · A · ∆xrate of change of heat energy =∂∂te(x, t)· A· ∆xheat energy flowing across boundary per unit time= φ(x, t) · A − φ(x + ∆x, t) · Aheat energy generated inside per unit time= Q(x, t) · A · ∆x∂∂te(x, t) · A · ∆x= φ(x, t) · A − φ(x + ∆x, t) · A+ Q(x, t) · A · ∆x∂e(x, t)∂t=φ(x, t) − φ(x + ∆x, t)∆x+ Q(x, t)∂e∂t= −∂φ∂x+ Qc(x) = specific heat or heat capacity (the heatenergy supplied to a unit mass of a substance toraise its temperature one unit)ρ(x) = mass density (mass per unit volume)Thermal energy in a volume is equal to the energy ittakes to raise the temperature of the volume from areference temperature (zero) to its actualtemperature.e(x, t) · A · ∆x = c(x)u(x, t) · ρ(x) · A · ∆xe(x, t) = c(x)ρ(x)u(x, t)cρ∂u∂t= −∂φ∂x+ QFourier’s law of heat conduction:φ = −K0∂u∂x,where K0= K0(x, u) is called the thermalconductivity.Heat equation:cρ∂u∂t=∂∂xK0∂u∂x+ QAssuming K0= const, we havecρ∂u∂t= K0∂2u∂x2+ QAssuming K0= const, c = const, ρ = const(uniform rod), and Q = 0 (no heat sources), weobtain∂u∂t= k∂2u∂x2,where k = K0(cρ)−1is called the thermal
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