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**Unformatted text preview: **2.7: NUMERICAL APPROXIMATIONS: EULERS 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 ) = y , has a unique solution ( t ) in an interval containing t . We have indi- cated in section 2.4 that the continuity of f and f/y in a rectangular region containing ( t , y ) is sufficient for this to be true. We shall see how a numerical approximation of ( t ) can be made, tabulating (the approximation of) the values of ( t ) at a sequence of mesh points t k s. Eulers method. Recall that the tangent line y = y + prime ( t )( t- t ) = y + f ( t , y )( t- t ) passing through the point ( t , y ) approximates (the graph of) the func- tion ( t ) near t = t . To be more precise, it means if t 1 is close to t , then ( t 1 ) y + f ( t , y )( t 1- t ) . We shall set as y 1 the right-hand side of the last equation , an approxi- mate value of ( 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, ( t 1 ) = ( t ) + integraldisplay t 1 t prime ( s ) ds = ( t ) + integraldisplay t 1 t f ( s, ( s )) ds On the other hand, by the mean value theorem, there exists some t * [ t , t 1 ] such that integraldisplay t 1 t f ( s, ( s )) ds = f ( t * , ( t * ))( t 1- t ) . Date : January 8, 2011. 1 2 KIAM HEONG KWA Hence ( t 1 ) = ( t ) + f ( t * , ( t * ))( t 1- t ) for some t * [ t , t 1 ]. Note that when t 1 approaches t , so does t * , from which it follows that lim t 1 t f ( t * , ( t * )) = f ( t , ( t...

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