Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Six Numerical Differentiation and Integration

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 2 Motivation You encounter differentiation and integration every day! Differentiation: Almost all physical processes/phenomena are best cast in differentiation form Example:Newtons 2 nd law:F = (dv/dt)m Heat conduction:Heat flux = -k’ (dT/dx) Our parachutist problem:dv/dt = (mg – cv)/m Integration: Integration is commonplace in science and engineering Urban area River cross-section Windblow on rocket

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 3 What are Differentiation and Integration? Differentiation :rate of change of a dependent variable with respect to an independent variable. Integration :the integral of the function f(x) with respect to the independent variable x, evaluated between the limits x = a to x = b.

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 4 Why Numerical Methods? Very often, the function f(x) to differentiate or the integrand to integrate is too complex to derive exact analytical solutions. In most cases in engineering, the function f(x) is only available in a tabulated form with values known only at discrete points. Numerical Solution Example: numerical integration

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 5 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!

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 6 Some Often Used Math Derivations Integration Differentiation

Mech300 Numerical Methods, Hong Kong University of Science and Technology. 7 Overall Structure