Formulate an initial-boundary value problem modeling heat conduction in a thin homogeneous bar of length L if the left end is kept at temperature zero and the right end is insulated. The initial temperature function is f.

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cheekabooy · Accepted Answer

Step 1  A well-posed problem consist of a differential equation with initial/boundary conditions such that when solved the initial conditions uniquely determine the solution of the equation. Since we are concerned with heat conduction we will (not surprisingly) use the Heat equation it looks like this (in one dimension)  ∂u∂t=k∂2u∂2x  Where k is a constant and is sometimes called the diffusivity of a given medium. Since we have one first derivative with respect to time (+1 condition) and one second derivative with respect to the spatial variable we concluse that we need (at least) three initial conditions in order to full specify a solution. The initial temperature function is (some) f(x) This gives us the first initial condition; namely  u(x, 0)=f(x)  For the boundary condition firstly, the left end of the rod is kept at a constant temperature (of zero degrees). This  means at all times the endpoint will have the value zero, mathematically this is written (assuming the left end is placed at the origin) as  u(0, t)=0  Our last condition states that the right end is insulated, this means no heat can traverse through it, which, in turn, means that the change of temperature on the rightmost end will be zero, we write this as  ∂u∂x(L, t)=0  Thus our well-posed pr (initial-boundary) problem is  ∂u∂x=k∂2u∂2x  u(x, 0)=f(x)  u(0, t)=0  ∂u∂x(L, t)=0

Formulate an initial-boundary value problem modeling heat conduction in a thin h

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