Applications of approximate methods of calculation
Teaching Guidance for 14-16
Approximate methods are widely used in pure science. Complex X-ray analysis (of proteins, for example) uses computers to produce maps of the molecules, whose shape is fitted by no simple equation. Approximate methods are used for predicting the structure of all atoms or molecules other than the very simplest, as the equations (Schrodinger's equation) cannot be solved exactly by analytic methods for more than two particles. No analytic equations have been found for astronomical problems involving several comparably sized bodies, and approximate numerical solutions rather than exact equations are used to guide space probes (and very exactly too).
On the other hand, approximate analytic solutions are used to guide numerical methods. If they were not, purely numerical methods would often defeat the largest computers imaginable.
In applied science, the role of approximate numerical methods is even wider. The airflow over an aircraft, the distribution of energy stored thermally in a building, the stresses in a proposed dam or bridge, the magnetic field around a new design of motor armature, or the effect of changing the shape of the hull of a ship are all examples where such methods, nowadays using computers, are the only possible approach in practical problems.
Engineers have pioneered new kinds of numerical calculation, ahead of the mathematicians. They are used for calculations on airflow, vibrations, transfer of energy in solids, deflection of spars of varying thickness, bridges, electrical networks, stability, torsion in beams or shafts, lubrication, structural frameworks, resonances (of aero engines and aircraft wings), stresses in hooks, shock waves, magnetic fields, flow through pipes and nozzles, and many other problems.
The availability of computers, which can quickly do simple repeated calculations, has made it far more practicable to tackle tough problems by approximate methods. Of course, the exact methods of differential and integral calculus are still valued, though more for their elegance, generality, and compactness than for their precision, because approximate methods can be made as exact as one pleases if one takes enough trouble.
Since computers are so much used for this kind of problem, there is merit in handling some problems in physics by teaching in a computational manner rather than an analytic one. Students who meet such methods later on may find them less strange than those who meet only analytic methods. But the main reason for using these methods at school level is that they can offer a better insight into both the meaning of derivatives and the way mathematics models a physical situation than do more formal methods. That is, they are more mathematical than is analysis.