BMES 372 HW-III: Solving ODEs Numerically

 

Use Matlab for all questions, show code listings, include graphs, run-times and comment what you did (see also Matlab codes for ODE solvers in the weblink file posted in BBLearn)

Important: Do not use Matlab ODE functions (like ODE 45). Instead, implement the numerical methods and calculate all slopes (e.g. k1…. k4) and derivatives explicitly.

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Consider the first order differential equations:

 

dY₁/dt = Y₂    and  dY₂/dt = (1 - Y₁²)Y₂ - Y₁

 

                       with initial conditions:  Y_1 (0) = 2   and   Y_2 (0) = 0

 

1a) Implement and use Euler’s method with a step size of h = 0.1 to find approximate values of the solution for Y_1  for in the interval between t = 0.0 and . Produce a graph showing the numerical solution.

 

1b) Use your Euler implementation with a step size of h = 0.001. Graph the function (can be included in the first figure).

 

2a) Implement and use 4th order Runge-Kutta method with a step size of h = 0.1 to find approximate values between .0 and . Graph the solution (can be included in the first figure).

 

2b) Use 4th order Runge-Kutta method with a step size of .001. Graph the solution (can be included in the first figure).

 

4) Determine the absolute and relative errors (see Chapter 6.4) by comparing the amount of the state variable Y_1 at the last timepoint () of the implementations 1a, 1b and 2a with the implementation 2b.  Comment with a summary statement (3-4 sentences) on the accuracy of the methods investigated, their dependency on different step sizes.

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