The previous chapter has discussed the solution of differential equations of the “initial value” type, where all the values needed to specify a specific solution are given at one specific initial value of the independent variable. Many time dependent differential equations in engineering are of this type where some dependent variable is governed by a differential equation in time and the initial conditions are specified at some initial time that can usually be taken as t = 0. For such problems, the differential equation can then be integrated into the future and in principle to any desired value of time. The previous chapter has developed several general computer algorithms and software packages for addressing such problems. The developed code can be applied to nonlinear differential equations just as easily as linear differential equations although as with all nonlinear problems iterative approaches must be used in obtaining a solution. One of the features of the numerical solution of such problems, either linear or nonlinear, is that the relative error in the solution tends to increase as such equations are integrated further into the future or further from the initial starting point.
This chapter addresses a different type of differential equation problem, the so called “boundary value” problem. For this class of problems, specific values of the dependent variable (can be either values or derivatives) are not specified at one particular point but are specified at two different values of the independent variable. Engineering problems of this type usually involve some spatial independent variable as opposed to time as the independent variable. For example for a second- order differential equation, the values of the independent variable may be specified at two values of x such as x = 0 and x = L. Again this chapter addresses differential equations of only one independent variable. Partial differential equations involving two or more independent variables are discussed in the next two chapters.