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The only reason for scientists and engineers (i.e. non-pure-mathematicians) to learn ordinary differential equations is so they can move on to partial different

The only reason for scientists and engineers (i.e. non-pure-mathematicians) to learn ordinary differential equations is so they can move on to partial differential equations. This is because: > "Most of the natural laws of physics, such as Maxwell's equations, Newton's law of cooling, the Navier-Stokes equations, Newton's equations of motion, and Schrodinger's equation of quantum mechanics, are stated (or can be) in terms of PDEs, that is these laws describe physical phenomena by relating space and time derivatives. Derivatives occur in these equations because the derivatives represent natural things (like velocity, acceleration, force, friction, flux, current). Hence, we have equations relating partial derivatives of some unknown quantity that we would like to find." For the vast majority of people working with DEs, they'll have two major goals: (1) Take a physical problem they wish to model and construct (formulate) the PDE (the actual mathematical model), and (2) Solve the PDE, taking into account the initial value and boundary conditions. Going through a introductory course in ODEs is a valuable process because a common method of solving a PDE is reducing it to an ODE. However, with real-world problems, the actual approach will almost always be computational: > Numerical methods: These methods change a PDE to a system of difference equations that can be solved by means of iterative techniques on a computer; in many cases this is the only technique that will work. In addition to methods that replace PDEs by difference equations, there are other methods that attempt to approximate solutions by polynomial surfaces - spline approximations. A related common approach are perturbation methods, which transform a nonlinear problem into a series of linear ones that approximate the original nonlinear equation, and to which in turn numerical methods can be applied. Wikipedia has a good writeup: https://en.wikipedia.org/wiki/Perturbation_theory https://en.wikipedia.org/wiki/Perturbation_theory Quotes from a very good source: "Partial Differential Equations for Scientists and Engineers, Stanley J. Farnow, 1993, Dover" Pure mathematicians tend to oppose the introduction of approximate and computational methods early on, but there's no reason they can't be taught side-by-side with the traditional 'exact solutions' material, and it would probably greatly interest and benefit students.

meltyness 4y ago
As I read, I begin to share your blame. That what looked so promising in the 19th century really should have been immediately quashed in the 20th when computing theory reared its head. The types of assertions Rota chides -- must have a closed form solution, composed strictly of non-infinite, occasionally wonderfully simple solutions, and asserting that they must exist through deductive route -- all hallmarks. I thought pure mathematics was over?
jbay808 4y ago
As an engineer, approximate analytical solutions are often so much more valuable than numerical solutions. I wish approximation methods were taught more. They allow playing "what if?" exercises much more quickly, or finding important scaling laws. For example, the Tsiolkovsky rockey equation says that delta v goes with the log of propellant mass. That's so useful to an engineer, to quickly estimate feasibility before diving deep into simulations! Or, the stiffness of a beam goes as thickness cubed, divided by length to the fourth power. Or the resonant frequency of an oscillator goes as the square root of 1/LC. Relationships like that are immediately useful for predicting problems and understanding what a solution looks like. Of course sometimes numerical solutions are necessary. But usually, you run numerical simulations after most of the major design parameters are already dialed-in, and you just want to refine the solution or predict its behaviour. Searching only numerically for an optimum design is kind of a blind search, and no matter how fast your simulation, it gets increasingly difficult in higher dimensions. L And it often needs to start all over again when the boss says "what if we double the payload? How much will that cost us?"

meltyness 4y ago

As I read, I begin to share your blame. That what looked so promising in the 19th century really should have been immediately quashed in the 20th when computing theory reared its head. The types of assertions Rota chides -- must have a closed form solution, composed strictly of non-infinite, occasionally wonderfully simple solutions, and asserting that they must exist through deductive route -- all hallmarks. I thought pure mathematics was over?

jbay808 4y ago

As an engineer, approximate analytical solutions are often so much more valuable than numerical solutions. I wish approximation methods were taught more. They allow playing "what if?" exercises much more quickly, or finding important scaling laws. For example, the Tsiolkovsky rockey equation says that delta v goes with the log of propellant mass. That's so useful to an engineer, to quickly estimate feasibility before diving deep into simulations! Or, the stiffness of a beam goes as thickness cubed, divided by length to the fourth power. Or the resonant frequency of an oscillator goes as the square root of 1/LC. Relationships like that are immediately useful for predicting problems and understanding what a solution looks like. Of course sometimes numerical solutions are necessary. But usually, you run numerical simulations after most of the major design parameters are already dialed-in, and you just want to refine the solution or predict its behaviour. Searching only numerically for an optimum design is kind of a blind search, and no matter how fast your simulation, it gets increasingly difficult in higher dimensions. L And it often needs to start all over again when the boss says "what if we double the payload? How much will that cost us?"