Analog Computation of Green's Function for Integrating Two-Point Boundary Value Problems

A method is described for the analog computation of the solution of linear two-point boundary value problems, i.e., the problem of a linear ordinary differential equation with linear two-point boundary conditions. The nonhomogeneous problem, in particular, is considered; the differential equal for y on the real interval 0≦x≦1 is taken to be nonhomogeneous. The method for its solution is based upon the integral formula for the solution, namely, y(x) = ∫
₀
¹
G(x, t)f(t)dt, where G(x,t) is the Green's function satisfying two-point boundary conditions while f is a known integrable function. The paper describes a method for finding a specific set of initial value problems whose solutions, taken in linear combination, form the Green's function whose second argument (t) is fixed. This technique reduces the two-point boundary value problem to a set of initial value types and makes the continuous computation of the Green's function as a function of x for fixed t amenable to analog means. The application of the described technique to the adjoint boundary value problem yields a means for continuous computation of G as a function of the second argument (t) for fixed values of the first argument (x). This is the required form for the analog evaluation of the integral formula. The theoretical aspects of the method found are stated, after which an example of a simple non-self-adjoint problem from the study of structures is solved on an electronic analog computer.