AMME2000 - Analytic Solution to The Heat Equation - Science Assignment Help

Assignment Task

Part 1: Analytic Solution to the Heat Equation

In order to solve this problem analytically you will need to separate your solution into a homogeneous and steady-state solution. It is recommended that you keep your boundary and initial conditions expressed as variables.

1. Using the information given in the Introduction, briefly summarise the problem information. (5%).

Be sure to include the following:

• Governing equation

• Order and type of the PDE

• Initial condition

• Boundary conditions

• Type of solution you expect to obtain

2. Derive the steady state analytic solution for the temperature as a function of ocean depth, showing all working. Produce a plot comparing the initial condition, steady-state solution and the initial condition that will be used for the homogeneous problem. All three plots should be overlaid. (5%)

3. Using the method of Separation of Variables you’ve seen in the lectures and tutorials, derive a solution to the heat equation for this problem in terms of unknown Fourier coefficients (i.e. do not apply the boundary conditions yet). In your working, clearly indicate: the spatial, F(x), and temporal, G(t), functions and the eigenvalues.

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