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MTH6151: Partial differential equations

Coursework 6

        The aim of this coursework is to help you understand and master the concepts developed in the lectures. You are strongly encouraged to attempt all the questions. The problems will be discussed in the tutorial session (fourth lecture) of Week 8. If you have any particular question you would like to have discussed during the tutorial please post your request ahead of the tutorial in the module’s forum in QM+. Solutions will be posted in the beginning of Week 9.

Exercise 1. Consider for the wave equation

with boundary conditions

and the initial conditions

Find, using the method of separation of variables, the solution explicitly in series form. HINT: look at question 3 in coursework 5.

Exercise 2. Consider the wave equation

on the half-line  with boundary condition

and initial conditions

In this problem we construct the solution to the above problem using even extensions. For this, let

and

(i) Show that and are even functions. Show that is an odd function.

(ii) Use D’Alembert’s formula to write down the solution to the problem

(iii) Compute Hint: recall question 1 in coursework 5.

(iv) Show that HINT: use that and are even and that is odd. What can you conclude about  in relation to the original problem for on the half line?

(v) Provide an interpretation of what is happening with the string and how reflection works in this case.

Exercise 3. Solve

in the rectangle

with boundary conditions

Exercise 4. Solve

in the rectangle

with boundary conditions

Exercise 5. Find the harmonic function in the square

with the boundary conditions