Advanced Calculus fi..

Chapter 10 Partial Differential Equations 703For problems with variable coefficients or problems in two or three dimensionsconcerning regions of inappropriate shape, the methods described before will ingeneral fail to produce solutions in a form suitable for numerical applications. Similarremarks apply to classes of differential equations more general than those consideredhere, in particular nonlinear equations. Although the theoretical aspects of the subjectare highly developed and one can often establish existence of solutions, this is notalways sufficient for the needs of physics. (Such an existence theorem is the famousCauchy-Kowaleski theorem for initial value problems; see Chapter 3 of the secondbook by John listed at the end of the chapter.)Accordingly, a variety of numerical methods have been devised for explicit determinationof solutions satisfying given boundary conditions and initial conditions.We consider briefly some of these methods.The first method consists simply in a reversal of the limitprocess of Section 10.5.We replace the derivative a2u/ax2 by the difference expressionwhere u, = u(x,). From the differential equationwe are thus led to the system of equations - .d2u,dt2where a = 1,. . . , N anddu,dtm,- + h, - - k (&+I - 2~4, + u,-~) = F,(t), (10.139)Equations (10.139) can be handled completely by the methods of Section 9.6. Thetools required are basically algebraic, in particular the solution of simultaneousequations. In order that (10.139) be an accurate approximation to (10.138) it is necessarythat N be large; this makes the algebraic problems far from trivial, at least asfar as time requirements are concerned.Initial value problems. If one seeks a particular solution of (10.138) satisfyinggiven initial conditions and boundary conditions (values of uo and u ~+~), one canset up the approximating equations (10.139) and solve these numerically (see thebook by Isaacson and Keller and that by Iserles listed at the end of the chapter).These measures are even more appropriate if nonlinearity is present, for example, ifa2u/ax2 is replaced by its square.Characteristic value problems. Determination of normal modes for a wave equationor heat equation obtained from (10.138) leads in general to a Sturm-Liouvilleproblem (10.127). This can be attacked by considering the approximating problem(10.139), for which determination of normal modes is an algebraic problem; one