Analysis of previous exams

Examination aids

C: Specified printed and hand-written support material is allowed. A specific basic calculator is allowed.

Format

8-10 problems, each with 1-3 subproblems.

Content

Laplace transform

• Compute the Laplace transform of ...
• Use to find the solution of integral equation
• Use to find the solution of ordinary differential equation
• Compute inverse ...

Fourier transform

• Compute the Fourier transform of ...
• Verify (complex) Fourier series expansion
• Show that ...
• Find the Fourier cosinus series of ...
• Compute the sum ...

Partial Derivative

• Show that the heat kernel ... satisfies ...
• Solve the following heat equation ... with the boundary conditions ... and the initial condition ...

Polynomial Interpolation (Lagrange / Newton)

• Find ... s.t. the polynomial interpolates the points

Numerical Integration

• Apply the quadrature rule to the integral ...
• Find the degree of precision of the quadrature rule

Wave equation

• Initial value problem
• d'Alembert's solution
• Show that function satisfies
• Crank–Nicolsons method

Heat equation

• Find the Fourier sine series solution
• Write an explicit difference scheme
• Compute approximate solutions

Simpson's method

• Apply to integral
• Determine the degree of precision
• Use to approximate the integral

Newton iteration

• Show that the solution is unique
• Compute $x_1$

Ordinary Differential Equations (Runge-Kutta method / Heun's method / Euler explicit method ?)

• Do one step
• Find the exact solution
• Compute the error
• Find the stability function
• Find the (corresponding) stability interval

Numerical Solution of Nonlinear Equations

• Write the fixed point iteration scheme which is implemented here
• Suggest an appropriate stopping criterium
• Estimate the rate of convergence for this iteration scheme
• Compute (N) fixed point interations
• Compute (N) iterations of Jacobi's method

Finite difference scheme

• Set up a finite difference scheme for the two point boundary value problem
• Set up the finite difference scheme for a general N in the form AU = b