M257-316Notes_Lecture17

M257-316Notes_Lecture17 - Chapter 13 Lecture 17 - Solving...

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Chapter 13 Lecture 17 - Solving the heat equation using fnite diFerence methods 13.1 Approximating the Derivatives o± a ²unction by ²inite DiFerences Recall that the derivative of a function was deFned by taking the limit of a di±erence quotient: f 0 ( x ) = lim Δ x 0 f ( x x ) f ( x ) Δ x . (13.1) Now to use the computer to solve di±erential equations we go in the opposite direction - we replace derivatives by appropriate di±erence quotients. If we assume that the function can be di±erentiated many times then Taylor’s Theorem is a very useful device in determining the appropriate di±erence quotient to use. ²or example consider f ( x x )= f ( x )+Δ xf 0 ( x )+ Δ x 2 2! f 0 ( x Δ x 3 3! f (3) ( x Δ x 4 4! f (4) ( x ... (13.2) Re-arranging terms in (2) and dividing by Δ x we obtain f ( x x ) f ( x ) Δ x = f 0 ( x Δ x 2 f 0 ( x Δ x 2 3! f (3) ( x .... 85
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Lecture 17 - Solving the heat equation using fnite diFerence methods If we take the limit Δ x 0 then we recover (1). But for our purposes it is more useful to retain the approximation f ( x x ) f ( x ) Δ x = f 0 ( x )+ Δ x 2 f 0 ( ξ ) (13.3) = f 0 ( x O x ) . We retain the term Δ x 2 f 0 ( ξ ) in (3) as a measure of the error involved when we approximate f 0 ( x ) by the diFerence quotient ( f ( x x ) f ( x ) ) / Δ x . Notice that this error depends on how large f 0 is in the interval [ x, x x ] (i.e. on the smoothness of f ) and on the size of Δ x . Since we like to focus on that part of the error we can control we say that the error term is of the order Δ x – denoted by O x ). Technically a term or function E x )i s O x f E x ) Δ x Δ x 0 const . Now the diFerence quotient (3) is not the only one that can be used to approximate f 0 ( x ). Indeed if we consider the expansion of f ( x Δ x ): f ( x Δ x )= f ( x ) Δ xf 0 ( x Δ x 2 2!
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M257-316Notes_Lecture17 - Chapter 13 Lecture 17 - Solving...

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