Page - 289 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

9.1 Example:TemperatureDevelopment inaRod 289 ∂ ∂x u(L,t)=0, t∈ (0,T], (9.3) u(x,0)= I(x), x∈ [0,L] . (9.4) Mathematically, we assume that at t = 0, the initial condition (9.4) holds and that the PDE (9.1) comes into play for t > 0. Similarly, at the end points, the boundaryconditions(9.2)and (9.3)governuand theequation therefore isvalid for x∈ (0,L). Boundaryandinitial conditionsareneeded! The initial and boundary conditions are extremely important. Without them, the solution is not unique, and no numerical method will work. Unfortunately, many physical applications have one or more initial or boundary conditions as unknowns. Such situations can be dealt with if we have measurements of u, but the mathematical framework is much more complicated. What about the source term g in our example with temperature distribution in a rod? g(x,t) models heat generation inside the rod. One could think of chemical reactions at a microscopic level in some materials as a reason to include g. However, in most applications with temperature evolution, g is zero and heat generation usually takes place at the boundary (as in our example with u(0,t) = s(t)). 9.1.1 AParticularCase Beforecontinuing,wemayconsideranexampleofhowthetemperaturedistribution evolves in the rod. At time t = 0, we assume that the temperature is 10◦C. Then we suddenly apply a device at x = 0 that keeps the temperature at 50◦C at this end. What happens inside the rod? Intuitively, you think that the heat generation at the end will warm up the material in the vicinity of x = 0, and as time goes by, more and more of the rod will be heated, before the entire rod has a temperature of 50◦C (recall that no heat escapes from the surface of the rod). Mathematically, (with the temperature in Kelvin) this example has I(x)= 283 K, except at the end point: I(0) = 323 K, s(t) = 323 K, and g = 0. The figure below shows snapshots from two different times in the evolution of the temperature.