1.3.4  Approximate Solution of Problems in Calculus of Variations It is hard to solve the Euler's boundary value problem in general.  Nevertheless, numerically, it appears to be much easier to solve the original problem directly. An Approximate Method of Solution Consider now, the problem defined by (1) and/or (2).  The interval [a,b] is divided into n equal subintervals, each of which has length , by the points x1, x2, ... , xn as shown in the following figure.   Mathematically, we have the grid of points a=xo<x1<x2<......................................xn=b. h=xi-xi-1 ,i=1,2,...................................,n and yi=y(xi) for i=0,1,2,....................,n. Then, we approximate the functional by the function Note that the function In is obtained from (2) by a simple rectangular integration approximation of (2) in which derivatives are replaced by forward differences. (One can obtain better approximations using higher order integration techniques like trapezoidal method, Simpson method, etc., .... , and/or, replacing the derivatives by central differences). Since  and , it follows that In is a function of only y1, y2, ... ,yn. To find an extremal of I n, then, consider the system of equations;   (4) A solution of (4) will constitute an approximation at  x1, x2, ... , xn-1  of a function y(x) which is a solution of the original problem.