University of London Solving The First Section of Numerical Analysis Exercise

I’m working on a numerical analysis question and need a sample draft to help me study.

. Recall that, for the Taylor method of order 2, the function T(2)(t, y) is defined asT(2)(t, y) = f(t, y) + h2?f?t (t, y) + h2?f?y (t, y)f(t, y), (1)where we have used the chain rule to evaluate the derivative term. We want to derive a Runge Kutta method by approximating T(2)(t, y) in the forma1f(t, y) + a2f(t + ?1, y + ?1f(t, y)), (2)where a1, a2, ?1, and ?1 are parameters that we must choose.(a) Taylor’s theorem in two variables implies that we may writef(t + ?1, y + ?1f(t, y)) = f(t, y) + ?1?f?t (t, y) + ?1f(t, y)?f?y (t, y) + R1(t + ?1, y + ?1),where R1(t + ?1, y + ?1) is the remainder term. If we want (2) to approximate (1), then wemust choose the parameters a1, a2, ?1, and ?1 such that the matching conditionf(t, y)+h2?f?t (t, y)+h2f(t, y)?f?y (t, y) = a1f(t, y)+a2 f(t, y)+?1?f?t (t, y)+?1f(t, y)?f?y (t, y) holds. By matching the coefficients on like-terms (color-coded for your convenience), derivethe three equations that involve the parameters a1, a2, ?1, and ?1 (but do not solve theseequations). Make sure to distribute a2 into the bracketed expression before matching!(b) You will notice that there are four unknown parameters but only three equations, whichsuggests that we have some flexibility in choosing the parameters. The standard approachis to choose a2 freely and then solve for the remaining three parameters a1, ?1, and ?1. Doso for the choice a2 = 1, and state the name of the Runge-Kutta method from lecturethat corresponds to these parameters.(c) Now choose a2 =23and solve for the remaining parameters a1, ?1, and ?1. This choice ofparameters gives Ralston’s method. (Note: There is another method where you choosea2 =34that is also called Ralston’s method, but we will go with the one above.