Assignment1.docx - AMME2000 Assignment 1 Yongkang XING SID:470019281 This report will force on two sections which are Taylor Series approximations and

Assignment1.docx - AMME2000 Assignment 1 Yongkang XING...

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AMME2000 Assignment 1 Yongkang XING SID:470019281 March 27, 2018 This report will force on two sections, which are Taylor Series approximations and Fourier Series. Section 1: Taylor Series Approximations The governing equation and two Taylor Series approximations: f ( x ) = tanh ( x ) f xx ( x i ) f i 3 + 4 f i 2 4 f i 1 + f i + 1 2 ∆ x 2 + E (1) f xx ( x i ) f i 2 + 16 f i 1 30 f i + 16 f i + 1 f i + 2 12 ∆x 2 + E (2) The leading order error: To evaluate the leading order error, use Taylor Series expansion: f i 3 = f ( x i ) 3 ∆ x 1 ! f x ( x i ) + 3 2 ∆ x 2 2 ! f xx ( x i ) 3 3 ∆ x 3 3 ! f xxx ( x i ) + 3 4 ∆x 4 4 ! f x xxx ( x i ) f i 2 = f ( x i ) 2 ∆ x 1 ! f x ( x i ) + 2 2 ∆ x 2 2 ! f xx ( x i ) 2 3 ∆ x 3 3 ! f xxx ( x i ) + 2 4 ∆ x 4 4 ! f xxxx ( x i ) f i 1 = f ( x i ) ∆ x 1 ! f x ( x i ) + ∆x 2 2 ! f xx ( x i ) ∆ x 3 3 ! f xxx ( x i ) + ∆ x 4 4 ! f xxxx ( x i ) Substitute f i 3 , f i 2 and f i 1 into equation 1 and find the leading order error for 4-point finite stencil: E = 5 12 ∆x 2 f x xxx ( x i ) ( 3 ) Same as the above steps, use Taylor Series to expand f i ,f i + 1 and f i + 3 : f i + 1 = f ( x i ) + ∆ x 1 ! f x ( x i ) + ∆x 2 2 ! f xx ( x i ) + ∆ x 3 3 ! f xxx ( x i ) + ∆ x 4 4 ! f xxxx ( x i ) f i + 2 = f ( x i ) + 2 ∆ x 1 ! f x ( x i ) + 2 2 ∆ x 2 2 ! f xx ( x i ) + 2 3 ∆ x 3 3 ! f xxx ( x i ) + + 2 4 ∆ x 4 4 ! f xxxx ( x i ) f i + 3 = f ( x i ) + 3 ∆ x 1 ! f x ( x i ) + 3 2 ∆ x 2 2 ! f xx ( x i ) + 3 3 ∆ x 3 3 ! f xxx ( x i ) + 3 4 ∆ x 4 4 ! f x xxx ( x i )