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2.7 - 2.7 NUMERICAL APPROXIMATIONS EULERS METHOD KIAM HEONG...

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2.7: NUMERICAL APPROXIMATIONS: EULER’S METHOD KIAM HEONG KWA Unless otherwise stated, we shall assume that the first-order initial value problem (1) dy dt = f ( t, y ) , y ( t 0 ) = y 0 , has a unique solution φ ( t ) in an interval containing t 0 . We have indi- cated in section 2.4 that the continuity of f and ∂f/∂y in a rectangular region containing ( t 0 , y 0 ) is sufficient for this to be true. We shall see how a numerical approximation of φ ( t ) can be made, tabulating (the approximation of) the values of φ ( t ) at a sequence of mesh points t k ’s. Euler’s method. Recall that the tangent line y = y 0 + φ prime ( t 0 )( t - t 0 ) = y 0 + f ( t 0 , y 0 )( t - t 0 ) passing through the point ( t 0 , y 0 ) approximates (the graph of) the func- tion φ ( t ) near t = t 0 . To be more precise, it means if t 1 is close to t 0 , then φ ( t 1 ) y 0 + f ( t 0 , y 0 )( t 1 - t 0 ) . We shall set as y 1 the right-hand side of the last equation , an approxi- mate value of φ ( t 1 ). More rigorously, the approximation can be obtained as follows pro- vided the rate function f is continuous. By the fundamental theorem of calculus, φ ( t 1 ) = φ ( t 0 ) + integraldisplay t 1 t 0 φ prime ( s ) ds = φ ( t 0 ) + integraldisplay t 1 t 0 f ( s, φ ( s )) ds On the other hand, by the mean value theorem, there exists some t * [ t 0 , t 1 ] such that integraldisplay t 1 t 0 f ( s, φ ( s )) ds = f ( t * , φ ( t * ))( t 1 - t 0 ) . Date : January 8, 2011. 1

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2 KIAM HEONG KWA Hence φ ( t 1 ) = φ ( t 0 ) + f ( t * , φ ( t * ))( t 1 - t 0 ) for some t * [ t 0 , t 1 ]. Note that when t 1 approaches t 0 , so does t * , from which it follows that lim t 1
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