Numerical Analysis

Finding the solution of a linear boundary problem is assisted by the linear structure of the equation and the use of two special initial value problems. Suppose that u(t) is the unique solution to the I.V.P. with . Furthermore, suppose that v(t) is the unique solution to the I.V.P. with . Then the linear combination . is a solution to with . Program (Linear Shooting Method) : To approximate the solution of the boundary value problem with over the interval [a,b] by using the Runge-Kutta method of order n=4. The method involves solving a two systems of equations over . First solve with , and . Then solve with , and . Finally, the desired solution x(t) is the linear combination . The subroutine Runge2D will be used to construct the two solutions , and .

Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems. Consider the linear equation (1) over [a,b] with . Form a partition of [ a, b ] using the points , where and for . The central-difference formulas discussed in Chapter 6 are used to approximate the derivatives (2) and (3) Use the notation for the terms on the right side of (2) and (3) and drop the two terms . Also, use the notations , , and this produces the difference equation which is used to compute numerical approximations to the differential equation (1). This is carried out by multiplying each side by and then collecting terms involving and arranging them in a system of linear equations: for , where and . This system has the familiar tridiagonal form.

One of the most important weighted residual methods was invented by the Russian mathematician Boris Grigoryevich Galerkin (February 20, 1871 - July 12, 1945). Galerkin's method selects the weight function functions in a special way: they are chosen from the basis functions, i.e. . It is required that the following equations hold true (6) for . To apply the method, all we need to do is solve these equations for the coefficients . Galerkin's Method for solving an I. V. P. Suppose we wish to solve the initial value problem (i) , with over the interval . We use the trial function (ii) . There are equations to solve for , i.e. (iii) for . Remark : For the solution of an I. V. P. we choose . Galerkin's Method for solving an a B. V. P. Suppose we wish to solve a boundary value problem over the interval , (I) , with We define and use the trial