Consider solving the differential equation y\' = Ay where A = plusminus 1 and y(

Question

Consider solving the differential equation y' = Ay where A = plusminus 1 and y(0) = 1 using each of the following methods: y_n+1 =1 y_n + h/2(y'_n + 1 + y'_n)(Improved Euler's) y_n + 1 = y_n + h y'_n + 1 (Implicit Euler's) y_n + 1 = y_n - 1 + 2h y'_n (Leapfrog method) y_n + 1 = y_n + h f(x_n + h/2, y_n + h/2 f(x_n, y_n))(RK-2) y_n + 1 = y_n - 1 + h/3(y'_n + 1 + 4y'_n + y'_n - 1)(Milne's) For each of the methods above, discuss the following: Does the method converge as h rightarrow 0? How does the stability of the method depends on values of Ah? Select appropriate step sizes h to demonstrate the stability and/or instabilities of the method, and carry out numerical computations to see if they are consistent with your results from part (b).

Explanation / Answer

a) thus the methods wont converge at h = 0 we have to solve for more than one value so there should be valid values of h other than 0

for convergence lim tends to infinite = 0 it doesnt converge for thier values

b) the method depends for y = Ah by making values in the given rules for increment and decrement values

c) at h = 0.1 is taken as value then for yn yn-1 yn+1 as the values are consistently same for all values in the given theorem at the same value of h

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