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chapter5.page06

chapter5.page06 - x t i the diﬀerence assuming no...

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6 1. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The next screen shot shows a call to the myEuler and tMesh for the equation x 0 = t 2 x + t 2 sin ( t 3 ). It also shows the graph of approxi- mate solution comparing with the exact solution x ( t ) = - 3 10 cos( t 3 ) - 1 10 sin( t 3 ) + 3 10 e 1 10 t 3 Figure 4. Mathcad Euler’s Approximation to x 0 = t 2 x + t 2 sin ( t 3 ) 2.3. Error analysis of Euler’s Method. There are two source of error in every numerical method used to approximate a solution of x 0 ( t ) = f ( t,x ), Local Truncation Error The roundoﬀ error. The local truncation error is due to the method and roundoﬀ error is due to computer that is used. For Euler’s method, we use x i = x i - 1 + f ( t i - 1 ,x i - 1 ) * h to approximate the value of
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Unformatted text preview: x ( t i ), the diﬀerence, assuming no rounding is introduced, is | x i-x ( t i ) | , and is called the local truncation error . The following theorem shows how the local truncation error is depending on f ( t,x ) and h , Theorem 2.1 . Suppose f ( t,x ) , ∂f ( t,x ) ∂t , and ∂f ( t,x ) ∂x are continuous on [a, b], and h = b-1 n is the step size. Furthermore let x ( t ) be the solution of initial value problem x = f ( t,x ) ,x ( a ) = x and x i = x i-1 + f ( t i-1 ,x i-1 ) * h be the approximate of x ( t i ) , and let e i = x ( t i )-x i be...
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