I have got a heat flow partial differential equation problem that is giving me a little problem due to the direction the temperature is changing. I have a bar (which lies along the X axis) which is initially at a uniform temperature which (for simplicity sake) we will call zero degrees. At time, t, = 0 and onward one side of the bar is heated so that it is kept at a constant temperature of, say, 100 degrees. I am asked to find the temperature of the bar as a function of the position, x, along the bar, and time t. Setting up the heat flow equation, I have, Del square u = 1/alpha * the first partial derivative of u with respect to time. I can rearrange this and solve for a product solution of u(x, t) as X(x) * T(t). then separate out thr variables into two ordinary differential equations for X(x) and T(t) and get, X’’ + k^2 * X = 0, And, T’ + (k*alpha)^2 * T = 0 The X equation is going to give me some sine and/or cosine solution, and the T equation should give me a exponential solution. Solving for T, T = A * e^-(K * alpha)^2 * t Where A is just some constant. I am ignoring the + exponent solution since this is non-physical, temperature cannot go to infinite as t goes to infinity. But what worries me is that the equation does not satisfy the initial boundary conditions. At t = 0, then entire rod should be at zero. And at t = infinite, the temperature of the rod should be some constant function of x (steady state solution when the two ends of the rod are held at different temperatures). But my equation says that the temperature drops to zero as t goes to infinite and has some maximum value (A) when t = 0. Clearly I need to modify my solution for T(t) so get it to show an increasing temperature which will approach some non-zero value.