Part 06

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Part Six Numerical Differentiation and Integration

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Motivation You encounter differentiation and integration every day! Differentiation: Almost all physical processes/phenomena are best cast in differentiation form Example:

Newtons 2nd law: Heat conduction: Our parachutist problem:

F = (dv/dt)m Heat flux = -k’ (dT/dx) dv/dt = (mg – cv)/m

Integration: Integration is commonplace in science and engineering

Urban area

River cross-section

Windblow on rocket

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

What are Differentiation and Integration? Differentiation:

rate of change of a dependent variable with respect to an independent variable.

f ( x i + ∆x ) − f ( x i ) dy = lim dx ∆x →0 ∆x

Integration:

the integral of the function f(x) with respect to the independent variable x, evaluated between the limits x = a to x = b.

I = ∫ f ( x )dx b

a

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Why Numerical Methods? •

Very often, the function f(x) to differentiate or the integrand to integrate is too complex to derive exact analytical solutions.

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In most cases in engineering, the function f(x) is only available in a tabulated form with values known only at discrete points.

Example: numerical integration

Numerical Solution

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Examples of Numerical Differentiation and Integration

Differentiation

Integration

There exist much more efficient and accurate numerical methods than these two! They are the ones we are to learn! 5

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Some Often Used Math Derivations

Differentiation

Integration

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Mech300 Numerical Methods, Hong Kong University of Science and Technology.

Overall Structure

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