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Unformatted text preview: Improved Euler’s Method Euler’s method was a good first start at approximating solutions to differential equations that we cannot solve. The error (i.e. the difference between ˆ y ( x ) and y ( x ) at a particular x-value) is at most a fixed constant times the step size h . So by halving the step size we can expect to reduce the error by roughly half. The following numerical scheme is an improvement over Euler’s method in that the error is at most a fixed constant times h 2 . This means that by halving h we are actually reducing the error by roughly a factor of four! Improved Euler We will again assume our IVP has the form y = f ( x,y ) , y ( x ) = y and we will again divide the interval [ x ,x + α ] into N equal parts each of length h = α/N so that the points x k = x + kh are equally spaced in the this interval. Now integrate our differential equation form x k to x k +1 to get Z x k +1 x k dy dx dx = Z x k +1 x k f ( x,y ) dx ⇓ y ( x k +1 ) = y ( x k ) + Z x k +1 x k f ( x,y...
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- Equations, Constant of integration, Boundary value problem, yk, xk