General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions

Abstract

In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor α, 0≤α≤1, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation Δx(t)=αA(t) leading from the nominal functionx(t) to the varied function $$\tilde x$$ (t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system. The scaling factor (or stepsize) α is determined by a one-dimensional search starting from α=1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if α is sufficiently small, it is guaranteed that $$\tilde P$$ <P. Convergence to the desired solution is achieved when the inequalityP≤ɛ is met, where ɛ is a small, preselected number. In the present technique, the entire functionx(t) is updated according to $$\tilde x$$ (t)=x(t)+αA(t). This updating procedure is called Scheme (a). For comparison purposes, an alternate procedure, called Scheme (b), is considered: the initial pointx(0) is updated according to $$\tilde x$$ (0)=x(0)+αA(0), and the new nominal function $$\tilde x$$ (t) is obtained by forward integration of the nonlinear differential system. In this connection, five numerical examples are presented; they illustrate (i) the simplicity as well as the rapidity of convergence of the algorithm, (ii) the importance of stepsize control, and (iii) the desirability of updating the functionx(t) according to Scheme (a) rather than Scheme (b).