Numerical solution of linear two‐point boundary problems via the fundamental‐matrix method

Abstract

SUMMARYA numerical method is presented for the solution of linear systems of differential equations with initialvalue or two-point boundary conditions. For y ' ( x ) = A(x)y(x) + f(x) the domain of interest [ a, 61 is divided into an appropriate number L of subintervals. The coefficient matrix A(x) is replaced by its value Ak at a poirir XA within the kth subinterval, thus replacing the original system by the L discretized systems yB(x) = Akyk(x) + fk(x), k = 1,2, ._., I,. The fundamental matrix solution @k(x, xk) over each subinterval is found by computing the eigenvalues and eigenvectors of each Ak. By matching the solutions yk(x) at the L -I equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different yk(x). The values of yl(a) and y~( b ) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method, ' the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.

Multiple scales analysis of chirped Bragg gratings

“…These equations constitute a two-point boundary problem that is solved numerically by using the fundamental matrix method [8]. Comparison is made with the characteristic matrix method [11].…”

“…In this paper, we study the influence of grating strength and chirp parameter on chirped grating response by using the method of multiple scales [7] and employ the fundamental matrix method [8] to solve the resulting coupled-mode equations numerically. Work on apodized unchirped reflection gratings [9] using these two methods showed that the coupled-amplitude equations of coupled-mode theory were insufficiently accurate in predicting the spectral response, especially when the strength of the periodic index perturbation ( ) increases beyond = 0.1.…”

Three-dimensional nonlinear vibrations of composite beams ? II. flapwise excitations

“…To solve the linear parts of equations (1) and (3) for the coupled vibration modes and frequencies, we make use of the state-space concept and the fundamental-matrix method [13]. To this end, we assume that v(s, t) = V(s)e zot and 7(s, t) = C(s)e i~t .…”

“…jlw21313, A64 -11313 A65 -jlw2 A46 -/~11 -/~213 /311 ' /~11 " (15) According to the fundamental-matrix method [13], the solution of equation (13) can be expressed in the general form Figures 2(a) and 3. The linear mode shapes and their corresponding frequencies for motions in the z direction can be obtained from equations (2), (4), and (5).…”

An Fe Shooting Method for Solving Infinite Domain Problems

“…2 The introduction of this matrix reduces the stiness matrix into a banded block triangular one, i.e. via the shooting method, the equilibrium conditions of each nodal point (or a group of nodal points) can be considered singly.…”