Investigate the following numerical approximation of the solution of the equatio

Question

Investigate the following numerical approximation of the solution of the equation y'(x) = ay, y(0) = 1, x [0,1]: yn+1 = yn + a delta x(lambda yn + (1 - lambda)yn+1), n = 0,..., N - 1, y0 = 1, where the real number a is a parameter in the equation, real number lambda is a parameter of the numerical approximation, and delta x = 1/N is the (uniform) step size. Two questions to address: (a) is it easier to approximate the solution when a 0? (b) Is there a value of lambda that you would consider the best choice, and how does this best value depend on a? You are welcome to pose and investigate other questions as well. Please support your conclusions with (clearly readable) graphs and/or tables; theoretical analysis is also welcome. It is my intention not to give detailed guidelines for the project: this is your chance to be creative.

Explanation / Answer

it is better to take a negative...

yes it depends on a as the diff equation ha term lambda * yn-yn+1

(time in suffice so not able to write complete) plzz rate..

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