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Linear Differential Operators 3 Linear Differential Operators Differential equations seem to be well suited as models for systems. Thus an understanding of differential equations is at least as important as an understanding of matrix equations. In Section 1.5 we inverted matrices and solved matrix equations. In this chapter we explore the analogous inversion and solution process for linear differential equations. Because of the presence of boundary conditions, the process of inverting a differential operator is somewhat more complex than the analogous matrix inversion. The notation ordinarily used for the study of differential equations is designed for easy handling of boundary conditions rather than for understanding of differential operators. As a consequence, the concept of the inverse of a differential operator is not widely understood among engineers. The approach we use in this chapter is one that draws a strong analogy between linear differential equations and matrix equations, thereby placing both these types of models in the same conceptual frame- work. The key concept is the Green’s function. It plays the same role for a linear differential equation as does the inverse matrix for a matrix equa- tion. There are both practical and theoretical reasons for examining the process of inverting differential operators. The inverse (or integral form) of a differential equation displays explicitly the input-output relationship of the system. Furthermore, integral operators are computationally and theoretically less troublesome than differential operators; for example, differentiation emphasizes data errors, whereas integration averages them. Consequently, the theoretical justification for applying many of the com- putational procedures of later chapters to differential systems is based on the inverse (or integral) description of the system. Finally, the application of the optimization techniques of Chapters 6-8 to differential systems often depends upon the prior determination of the integral forms of the systems. One of the reasons that matrix equations are widely used is that we have a practical, automatable scheme, Gaussian elimination, for inverting a matrix or solving a matrix equation. It is also possible to invert certain types of differential equations by computer automation. The greatest progress in understanding and automation has been made for linear, 93 94 Linear Differential Operators constant-coefficient differential equations with initial conditions. These equations are good models for many dynamic systems (systems which evolve with time). In Section 3.4 we examine these linear constant- coefficient models in state-space form and also in the form of nth-order differential equations. The inversion concept can be extended to partial differential equations. 3.1 A Differential Operator and Its Inverse Within the process of inverting a differential operator there is an analogue of the elimination technique for matrix inversion. However, the analogy between the matrix equation and the differential equation is clouded by the presence of the boundary conditions. As an example of a linear differential equation and its associated boundary conditions, we use -f” =u with f(O)= art and f(b) = CQ (3.1) Equation (3.1) can be viewed as a description of the relationship between the steady-state temperature distribution and the sources of heat in an insulated bar of length b. The temperature distribution f varies only as a function of position t along the bar. The temperature distribution is controlled partly by u, the heat generated (say, by induction heating) throughout the bar, and partly by constant temperature baths (of temperatures cyr and [x2, respectively) at the two ends t = 0 and t = b. Thus both the distributed input u and the boundary inputs { ai} have practical significance. The concepts of distributed and boundary inputs extend to other ordinary and partial differential equations. A Discrete Approximation of the Differential System In order to obtain a more transparent analogy to matrix equations and thereby clarify the role of the boundary conditions, we temporarily approximate the differential equation by a set of difference equations.* Let b = 4, substitute into (3.1) the finite-difference approximation *The approximation of derivatives by finite differences is a practical numerical approach to the solution of ordinary and partial differential equations. The error owing to the finite- difference approximation can be made as small as desired by using a sufficiently fine approximation to the derivatives. (See Forsythe and Wasow [3.3].) Special techniques are usually used to solve the resulting algebraic equations. See P&C 3.3 and Varga [3.12]. Sec. 3.1 A Differential Operator and Its Inverse 95 and evaluate the equation at t = 1, 2, and 3: -f(O)+2f(l) -f(2) =u(l) -f(l)+2f(2) -f(3) = u(2) - f(2) + 2f(3) - f(4) = u(3) (3.4) f(O) =a f(4) =a: It is obvious that this set of algebraic equations would not be invertible without the boundary conditions. We can view the boundary conditions either as an increase in the number of equations or as a decrease in the number of unknowns. The left side of (3.2), including the boundary conditions, is a matrix multiplication of the general vector (f(0) f( 1) * ’ * f(4))= in the space %5x ‘. The corresponding right-hand side of (3.2) is (u(1) u(2) u(3) (xi (~2)~; the boundary values increase the dimension of the range of definition by two. On the other hand, if we use the boundary conditions to eliminate two variables, we reduce the dimension of the domain of the matrix operator by two, f(0) and f(4) become part of the right-hand side, and the reduced matrix operates on the general vector (f(l) f(2) f(3))’ in 9R,3x ‘. By either the “expanded” or the “reduced” view, the transformation with its boundary conditions is invertible. In the next section we explore the differential equation and its boundary conditions along the same lines as we have used for this discrete approximation. The Role of the Boundary Conditions A differential operator without boundary conditions is like a matrix with fewer rows than columns: it leads to an underdetermined differential equation. In the same manner as in the discrete approximation (3.2), appropriate boundary conditions make a linear differential operator invert- ible. In order that we be able to denote the inverse of (3.1) in a simple manner as we do for matrix equations, we must combine the differential operator - D2 and the two boundary conditions into a single operator on a vector space. We can do so using the “increased equations” view of the boundary conditions. Let f be a function in the space e2(0,b) of twice continuously differentiable functions; then -f” will be in C?(O,b), the space of continuous functions. Define the differential system operator T: e2(0, b)-+ e (0, b) x ‘JR2 by Tf 9 (-f”,f(O),f(b)) (3.3) The system equations become Tf = (u, “1,(x2) (3.4) 96 Linear Differential Operators We are seeking an explicit expression of T- ’ such that f = T- ‘(u, ai, (w,). Because of the abstractness of T, an operation which produces a mixture of a distributed quantity u and discrete quantities {(xi}, it is not clear how to proceed to determine T-l. Standard techniques for solution of differential equations are more consistent with the “decreased unknowns” interpretation of the boundary conditions. Ordinarily, we solve the differential equation, - f” = u, ignoring the boundary conditions. Then we apply the boundary conditions to eliminate the arbitrary constants in the solution. If we think of the operator - D2 as being restricted through the whole solution process to act only on functions which satisfy the boundary conditions, then the “arbitrary” constants in the solution to the differential equation are not arbitrary; rather, they are specific (but unknown) functions of the boundary values, { aj}. We develop this interpretation of the inversion process into an explicit expression for the inverse of the operator T of (3.3). How do we express the “restriction” of -D2 in terms of an operator on a vector space? The set of functions which satisfy the boundary conditions is not a subspace of e2(0,b); it does not include the zero function (unless a 1 =a2=O). The analogue of this set of functions in the three-dimensional arrow space is a plane which does not pass through the origin. We frame the problem in terms of vector space concepts by separating the effects of the distributed and boundary inputs. In point of fact, it is the difference in the nature of these two types of inputs that has prevented the differential equation and the boundary conditions from being expressed as a single equation.* Decompose the differential system (3.1) into two parts, one involving only the distributed input, the other only the boundary inputs: -fpu with f,(O) = fd( b) = 0 (3.5) -fp(j with f,(O) = LYE, fb( b) = a2 (3.6) Equations (3.5) and (3.6) possess unique solutions. By superposition, these solutions combine to yield the unique solution f to (3.1); that is, f = fd+ fb. Each of these differential systems can be expressed as a single operator on a vector space. We invert the two systems separately. We work first with (3.5). The operator -D2 is onto C?(O, b); that is, we can obtain any continuous function by twice differentiating some function in e2(0, b). However, -D2 is singular; the general vector in nullspace (- D2) is of the form f(t) = ci + c,t. We modify the definition of the operator - D2 by reducing its domain. Let ?f be the subspace of functions in e2(0,b) which satisfy the homogeneous boundary conditions of (3.5), *Friedman [3.4] does include the boundary conditions in the differential equation by treating the boundary conditions as delta functions superimposed on the distributed input.
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