On Matrix Boundary Value Problems

Introduction. In a recent publication1 a matrix type of boundary value problem was introduced in order to simplify the description of nuclear reactions. It appeared that this type of boundary value problem could find applications in other branches of mathematical physics, and the purpose of the present note is to illustrate them. When we deal with vibrations of continuous media, with problems of heat flow etc., we usually describe the state of the system in terms of a single function which depends on position as well as on the time. As an example, we may mention the lateral displacement of a vibrating string, or the temperature function in case of problems of heat flow. In many problems of vibration and heat conduction, of which examples will be given below, the description of the state by a single function leads to boundary value problems of great difficulty. It is possible though, in some cases, to divide the continuous medium into several regions, and with each region we can associate a function describing its state. These functions can be grouped together in the form of a column matrix or vector, which will then represent the state of the whole system. The mathematical problem we encounter then, is a matrix boundary value problem, which is, in general, much simpler than the one we would have to deal with in the usual formulation. In the present note, we shall discuss two examples of matrix boundary value problems. The first one describing the flow of heat in a cross, illustrates the case where the interactions between the different regions appear through boundary conditions. The second one, dealing with the vibration of systems of plates with intermediate elastic media, illustrates the case where the interactions take place through the equations of motion. We shall obtain the eigenvalues and eigenmatrix functions corresponding to this type of problems, and with the help of them, give a formal solution for any initial conditions. For the discussion of the self-adjoint properties of this type of boundary value problem, and the rigorous derivation of the series expansion theorems, we refer to other publications.2,3,4 1. Flow of heat in a cross. We shall consider the problem of flow of heat in a cross (Fig. la) whose four arms are of the same length I, and of square cross section of area a2, where a <£ I. The material of the cross will have a density p, conductivity k and specific heat c. The lateral sides of the cross will be coated in such a way that the outer conductivity0 can be taken as zero, i.e. there is no radiation. If we tried to deal with this problem as a three-dimensional heat conduction problem in a region bounded by the surface of the cross, we would have a difficult boundary value problem which would not admit a simple solution. Taking into account though, that the smallness of the cross section permits us to assume that the temperature at all points in it is the same, we can describe the state of temperature in the cross in the following fashion: with each bar of the cross we associate its temperature function 6i(x, t), where i = 1, 2, 3, 4 indicates the bar in question, x represents the position of the point on the bar with 0 < x < I as indicated in Fig. la., and t is the time. The temperature state of the whole cross is then described by the column matrix: