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CH E374 (5)

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Chemical Engineering
CH E374
Joe Mmbaga

CHE 374 Computational Methods in Engineering MODULE 5 Numerical DifferentiationIntegration Differentiation basicsDefinition of the derivative of a function f of x at xx0 dfxfxfx011 limfx0xx0dxxx0xx0 In numerical analysis the differencestays finite does not really go to xxx0zero Derivatives are therefore approximated by differencesfxfx0fx12 0x An important theoretical result known as meanvalue theorem in the interval fxfx0xx there is a pointfor which exactlywithgenerally f0xx0unknownFinite difference approximations Suppose we know the function f at positionsthat are spaced by a distance xxx11iiih We distinguish three approximations to the derivative at xxi evitavForward difference irfxed eurffdfT1iidxxix xxx1ii1i 1Backward difference evitffdfav1iiirfxed eudxxriTxxxx1i i1ievit avirfxed eCentral difference urTffdf11ii2dxxix xxx1ii1i From Figure 1c we get the impression that the central difference approximation is the most accurate This we can prove by means of Taylor expansions23xx4ffxfffOx13 1iiiii26 23xx4ffxfffOx14 1iiiii264OxTheindicates that the leading order of the terms left out of the expansion is 4 Equation 13 can be rewritten such that2ffxx31iifffOx15 iii26x2
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