Advanced Calculus fi..

Chapter 10Partial Differential EquationsIn order to obtain a differential equation for u(x, t) we return to the basic equations(10.51) and write them as follows:We assume .yo and XN+, to be fixed and let L = XN+~ - x0, Ax = L/(N + 1). Wethen let N increase indefinitely. The ratio m,/Ax represents an "average density"at position x; it is reasonable to postulate that this approaches as limit a functionp(x) representing density (mass per unit length) at x. The simplest law of frictionwould make h, proportional to m,, so that h, /Ax would approach a function H (x)of dimensions force per unit of length per unit of velocity. For the model of Fig. 10.4the product k2ax represents the tension in one of the springs when it is stretcheda distance Ax. But precisely the same tension must hold in each half of the spring,which is stretched only a distance :AX. Hence if we always use springs of thesame stiffness, k2ax will approach as limit a constant force K'. For the model ofFig. 10.5, one indeed has k2ax = ki(~x- I ), where ki is the actual spring constantfor each spring; therefore k2ax represents precisely the tension in each spring whenall displacements u, are 0; the limiting value K' is precisely the tension in the string.One can writewhere x, is the equilibrium position of P,. In the limit, x, becomes a continuousvariable, and, as in Problem 9 following Section 6.18, the ratio of the seconddifference to (AX)' approaches as "limit" the derivativeThe right-hand members we assume to approach as limit a function F(x, t) representingapplied force per unit length at x. We are thus led to the partial differentialequationa 2 ~ au , a2up(x)- + H(x)- - K- - = F(x, t),at2 at ax2with p(x) 2 0, H(x) 10, K > 0. This is the fundamental partial differential equationto be studied. Certain generalizations will be introduced in later sections, notably byreplacement of a2u/ax2 by the Laplacian V'u :aZuaup- + H- - K ~ V = ~ U F(x, y, Z, t).at2 atThis corresponds to a generalization to motion in 2- or 3-dimensional space. Onecan easily construct an N particle model for this. From the general equation (10.55),one obtains a broader class of equations in which the term - K ~ V is ~ replaced U by amore complicated expression, possibly nonlinear, in u and its derivatives. Two- andthree-dimensional problems can lead to simultaneous partial digerential equations.While the generalizations do introduce complications, the principal problemsand methods reveal. @emselves in Eq. (10.61) and, in fact, in the N particle