23 In this problem we discuss the global truncation error associated with the Euler method...

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23 In this problem we discuss the global truncation error associated with the Euler method for the initial value problem y f t y y to yo Assuming that the functions f and Jy are continuous in a closed bounded region R of the ty plane that includes the point toyo it can be shown that there exists a constant L such that if t y f t Lly l where t y and 1 are any two points in R with the same t coordinate see Problem 15 of Section 2 8 Further we assume that f is continuous so the solution has a continuous second derivative a Using Eq 20 show that En En hflt tn f n Yn h o a Enl Bh where a 1 hL and max 1 1 1 2 on to tn b Accepting without proof that if E 0 and if E satisfies Eq i then En Bh a 1 a 1 for a 1 show that El ii Equation ii gives a bound for E in terms of h L n and B Notice that for a fixed t this error bound increases with increasing n that is the error bound increases with distance from the starting point fo c Show that 1 hL e hence En sal enkl 1 hL 1 L Bh e 10 1 L i Bh For a fixed point T to nh that is nh is constant and h T to n this error bound is of the form of a constant times h and approaches zero as h 0 Also note that for nhL T to L small the right side of the preceding equation is approximately 12a T

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23 In this problem we discuss the global truncation error associated with the Euler method for the initial value problem y f t y y to yo Assuming that the functions f and Jy are continuous in a closed bounded region R of the ty plane that includes the point toyo it can be shown that there exists a constant L such that if t y f t Lly l where t y and 1 are any two points in R with the same t coordinate see Problem 15 of Section 2 8 Further we assume that f is continuous so the solution has a continuous second derivative a Using Eq 20 show that En En hflt tn f n Yn h o a Enl Bh where a 1 hL and max 1 1 1 2 on to tn b Accepting without proof that if E 0 and if E satisfies Eq i then En Bh a 1 a 1 for a 1 show that El ii Equation ii gives a bound for E in terms of h L n and B Notice that for a fixed t this error bound increases with increasing n that is the error bound increases with distance from the starting point fo c Show that 1 hL e hence En sal enkl 1 hL 1 L Bh e 10 1 L i Bh For a fixed point T to nh that is nh is constant and h T to n this error bound is of the form of a constant times h and approaches zero as h 0 Also note that for nhL T to L small the right side of the preceding equation is approximately 12a T