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Primary institutional description by the Society for Industrial and Applied Mathematics.

Our Mission, Advance the Application, and What Is Applied Mathematics: mathematical and computational methods connecting scientific and practical problems. No changing membership counts or current officeholders reused.

Applied mathematics develops and uses mathematical models, analysis and computation to investigate problems in science, engineering and other fields. Choosing assumptions, deriving consequences and comparing predictions with observations are connected parts of the work. The Society for Industrial and Applied Mathematics describes the field through this cooperation between mathematical methods and practical scientific problems.^[ref]

For a worked example, a uniform-temperature cooling model assumeswhere $T_a$ is a constant ambient temperature. This is Newton's law of cooling. Setting $u=T-T_a$ gives $u^{\prime }=-ku$, so exponential decay yieldsWith illustrative values $T_0=100{}^{\circ }\mathrm{C}$, $T_a=20{}^{\circ }\mathrm{C}$ and $k=0.1{\min }^{-1}$, the predicted temperature after ten minutes is $20+80/e\approx 49.4{}^{\circ }\mathrm{C}$.

Solving the ordinary differential equation establishes a consequence of the assumptions. Testing the cooling prediction asks whether those assumptions fit the object and conditions. Parameter estimation uses measurements to infer $k$; disagreement can also reveal missing spatial variation or changing surroundings. This separates a correct mathematical derivation from the empirical adequacy of its mathematical model.