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AMME2000Assignment 1Yongkang XINGSID:470019281March 27, 2018This report will force on two sections, which are Taylor Series approximations and Fourier Series.Section 1: Taylor Series ApproximationsThe governing equation and two Taylor Series approximations:f(x)=tanh(x)fxx(xi)≈−fi−3+4fi−2−4fi−1+fi+12∆ x2+E(1)fxx(xi)≈−fi−2+16fi−1−30fi+16fi+1−fi+212∆x2+E(2)The leading order error:To evaluate the leading order error, use Taylor Series expansion:fi−3=f(xi)−3∆ x1!fx(xi)+32∆ x22!fxx(xi)−33∆ x33!fxxx(xi)+34∆x44!fx xxx(xi)…fi−2=f(xi)−2∆ x1!fx(xi)+22∆ x22!fxx(xi)−23∆ x33!fxxx(xi)+24∆ x44!fxxxx(xi)…fi−1=f(xi)−∆ x1!fx(xi)+∆x22!fxx(xi)−∆ x33!fxxx(xi)+∆ x44!fxxxx(xi)…Substitute fi−3, fi−2and fi−1into equation 1 and find the leading order error for 4-point finite stencil:E=−512∆x2fx xxx(xi)(3)Same as the above steps, use Taylor Series to expand fi,fi+1and fi+3:fi+1=f(xi)+∆ x1!fx(xi)+∆x22!fxx(xi)+∆ x33!fxxx(xi)+∆ x44!fxxxx(xi)…fi+2=f(xi)+2∆ x1!fx(xi)+22∆ x22!fxx(xi)+23∆ x33!fxxx(xi)++24∆ x44!fxxxx(xi)…fi+3=f(xi)+3∆ x1!fx(xi)+32∆ x22!fxx(xi)+33∆ x33!fxxx(xi)+34∆ x44!fx xxx(xi)…