Differential Equation Assignment

Question 1: Runge{Kutta Method

Consider an autonomous system of n coupled rst order differential equations in a vector form:

2. (5 pts) Write a MatLab.m file to implement this algorithm to solve an autonomous first order system of equations. The vectors f and u0, the step size t and the number of steps should be inputs to the function.

3. (4 pt) Given that the exact solution of (1) is

yexact = (ln x)2 + ln x:

Use your MatLab function numerically determine how the error at x = 1:8 depends on h. Choosing suitable values for step-size h, plot log jerrorj versus log h. Use your plot to determine a relationship between the error and h. How accurate is the method? Is this what you would expect? Please discuss.

Note: Submission for this question includes a brief description of the computer program, program listing and output printouts (in the report). Comment on the result. The program (all m-files) must be submitted so that assessor can verify results.

Question 2. The KdV Equation (Korteweg and de Vries, 1895)

ut - 6uux + uxxx = 0;

which models Russell's observation. The term uux describes the sharpening of the wave and uxxx the dispersion (i.e., waves with dierent wave lengths propagate with dierent velocities). When these two terms balance a propagating wave with unchanged form is the result. These waves with unchanged form are called solitons. To solve this equation numerically, we first use the finite dierence approximations for the space derivatives. Divide the numerical domain into N points with subinterval x. Then using the second order finite dierence approximation to approximate the first and third order derivatives terms in (3),

1. (5 pts) Write a program that solves the equation in the discretization form using RK4 above. Solve it in the region ????10 x 10 with a grid size x = 0:1, and use periodic boundary conditions:

x(????10) = x(10) (6)

Integrate from t = 0 to t = 2, using an appropriate time-step that satises the stability condition. For each of the initial conditions below, plot the solution at t = 2 and comment on the results.

Note: Submission for this question includes a brief description of the computer pro- gram, program listing and output printouts (in the report). Comment on the result. The program (all m-les) must be submitted so that assessor can verify results.

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