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ISBN-10: 0126741506

ISBN-13: 9780126741506

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Additional resources for Introduction to the Theory of Infinitesimals

Example text

This is a more or less standard problem in which the model is completely specified and the future effect of initial conditions is calculateda classic direct problem. But how are the concentration of the polluted water seeping into the cistern and the rate of seepage determined? In the real world, the seepage rate and the concentration of pollution in the groundwater would probably be obtained by measurements of the state of the cistern itself. These parameters specify the model and in real applications it is often the model itself which is uncertain.

In particular, x(t) = 0 if y(s) = 0. For a given c > 0, let y,(s) = csin(s/< 2 ). Then I y,(s) I:S c. Show that the solution x,(s) satisfies maxt I x,(t) 12: 1/c. 0 We conclude by noticing that there is a type of Volterra equation, the Volterra integral equation of the second kind, which in an appropriate setting always has a unique, stable solution. This equation has the form x(s) = y(s) + 1' k(s, t)x(t)dt. g. 27) has a unique solution x( s) which is continuous for 0 :S s :S 1. 27), then { Xn} converges uniformly to x.

Namely, suppose that the water level y can be measured, but the cross-sectional area A(y) is inaccessible. 3 (a sealed institutional coffee urn with unknown internal geometry is an appropriate visualization). 6). 1!.. 7) dt (note that in practice there is no region where 'fjf = 0 and hence A(y) 1s m principle identifiable). Although (3. 7) uniquely determines A(y), we see that practical determination of A(y) will be at the mercy of the instabilities inherent in the differentiation of the measured water level y(t).