= Applied mathematics
<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.<About SIAM by the Society for Industrial and Applied Mathematics>[Our Mission and Advance the Application]{ref}
For a worked example, a uniform-temperature cooling model assumes
$$
\frac{dT}{dt}=-k(T-T_a),\qquad T(0)=T_0,\qquad k>0,
$$
where $T_a$ is a constant ambient <temperature>. This is <Newton's law of cooling>. Setting $u=T-T_a$ gives $u'=-ku$, so <exponential decay> yields
$$
T(t)=T_a+(T_0-T_a)e^{-kt}.
$$
With illustrative values $T_0=100\,{}^{\circ}\mathrm{C}$, $T_a=20\,{}^{\circ}\mathrm{C}$ and $k=0.1\,\mathrm{min}^{-1}$, the predicted <temperature> after ten minutes is $20+80/e\approx49.4\,{}^{\circ}\mathrm{C}$.
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>.
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