Problem Solving

Problem Solving

Exercise 1

In order to solve the given problems, we uses two main algorithms in order to determine the solution to a differential equation. These algorithms are the Adams method and the Gear method.

The Adams and Gear methods are forms of linear multistep methods. An example of these would be the following:

In the example above, h denotes the step size and the coefficients are determined by the method used. Multistep methods are expansions of more familiar single-step methods used to solve differentials (i.e. Euler, Runge-Kutta, Taylor). Each of these methods requires an initial point in order to calculate the following point. Similarily, multistep methods also require initial points in order to solve the ODE. The number of initial points required depends on which method is used to solve the ODE. Multistep methods typically produce less error than the single-step methods because of multiple initial points.

In order to determine what method to use one must first find the stiffness of the function.

Euler, Taylor and Runge-Kutta methods used points close to the solution value to evaluate derivative functions. The Adams-Bashforth method looks at the derivative at old solution values and uses interpolation ideas along with the current solution and derivative to estimate the new solution [4]. In order to solve an ODE using this method, f(t,y) must be continuous and satisfy Lipschitz condition for the y-variable which states [5]:

for all | h | < e where B and ß are independent of h, ß > 0 and a is an upper bound for all ß for which a finite B exists

Exercise 2This is a basic form of the Adams-Bashforth method. Note that two initial points are required for this method.

There is another Adams method that requires three initial points. The method is solved the same way, however the equation varies a little bit and is referred to as the Adams-Moulton Method.

The coefficients/constraints, ß can be solved for using knowledge of ODE's and other math tools. It was stated earlier that . We can let f(t,y) = ?y, therefore . We can also let if there is a constant step size and s represents a polynomial. Through substitution we find [6]:

We can expand the quadratic using another math identity and ultimately solve for constraints ß1 and ß2. Another method for solving for coefficients ß1,ß2 is mentioned below:

In order to find the coefficient ßj one must first use polynomial interpolation to find the polynomial p of degree s - 1 such that:

for

From this the Lagrange formula for polynomial interpolation yields

Now the polynomial p is a locally good approximation of the right-hand side of the differential equation y' = f(t,y) that is to be solved. Now we must consider the equation y' = p(t) instead. This equation can be solved exactly by simply taking the integral of p.

The Adams-Bashforth method arises when the formula for p is substituted. The coefficients bj turn out to be

The Adams-Bashforth method is typically used ...