Conte, de Boor - Elementary Numerical Analysis. An Algorithmic Approach (523140), страница 69
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Verify that the solution explodes forh = 0.003 and h = 0.01 while for h = 0.001 and h = 0.002 we obtain reasonable approximations to the exact solution y = e-x.8.13-2 For the systemy´ 1 =y2y´2 = -200y1 - 102y 2show that the eigenvalues of the coefficient matrix are -2 and -100 and hence that thegeneral solution is given byy 1 (x) = y(x) = Ae - 2 x + Be - 1 0 0 xUnder the conditions y(0) = 1, y´(0) = -2 which corresponds to y1 (0) = 1, y2 (0) = -2, the.exact solution is y(x) = e-2x Solvethis system from x = 0 to x = 1 using the trapezoidalmethod with a step h = 0.1 and compare your results with the exact solution.Previous Home NextCHAPTERNINEBOUNDARY-VALUE PROBLEMS INORDINARY DIFFERENTIAL EQUATIONSIn Chap.
8 we considered numerical methods for solving initial-valueproblems. In such problems all the initial conditions are given at a singlepoint. In this chapter we consider problems in which the conditions arespecified at more than one point. A simple example of a second-orderboundary-value problem isy´´(x) = y(x)y(0) = 0y(l) = 1An example of a fourth-order boundary-value problem isy i v (x) + ky(x) = qy(0) = y´(0) = 0(9.1)(9.2a)(9.2b)y(L) = y´´(L) = 0(9.2c)Here y may represent the deflection of a beam of length L which issubjected to a uniform load q. Condition (9.2b ) states that the end x = 0 isbuilt in, while (9.2c) states that the end x = L is simply supported. Weshall consider three methods for solving such problems: the method offinite differences and an adaptation of the methods of Chap. 8, which weshall call “shooting” methods, and the method of collocation.9.1 FINITE-DIFFERENCE METHODSWe assume that we have a linear differential equation of order greater thanone, with conditions specified at the end points of an interval [a,b].
Wedivide the interval [a,b] into N equal parts of width h. We set x0 = a,4069.1FINITE-DIFFERENCE METHODS407xN = b, and we definen = 1, 2, . . . , N - 1xn = x0 + nhas the interior mesh points. The corresponding values of y at these meshpoints are denoted byyn = y(x0 + nh)n = 0, l, . . . , NWe shall sometimes have to deal with points outside the interval [a,b].These will be called exterior mesh points, those to the left of x0 beingdenoted by x -1 = x 0 - h, x -2 = x 0 - 2h, etc., and those to the right ofx N being denoted by x N+1 = x N + h, x N+2 = x N + 2h, etc.
The corresponding values of y at the exterior mesh points are denoted in the obviousway as y-1 , y-2 , yN+1, yN+2, etc.To solve a boundary-value problem by the method of finite differences, every derivative appearing in the equation, as well as in theboundary conditions, is replaced by an appropriate difference approximation. Central differences are usually preferred because they lead to greateraccuracy. Some typical central-difference approximations are the following(see Chap. 7):(9.3)In each case the finite-difference representation is anapproximationto the respective derivative.
To illustrate the procedure, we consider thelinear second-order differential equationy´´(x) + f(x)y´ + g(x)y = q(x)(9.4)under the boundary conditionsy(x 0 ) = α(9.5)y(x N ) = β(9.6)The finite-difference approximation to (9.4) isn = 1, 2, . . . , N - 12Multiplying through by h , setting f(xn) = fn, etc., and grouping terms, we408BOUNDARY-VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONShaven = 1, 2, . .
. , N - 1(9.7)Since y0 and yN are specified by the conditions (9.5) and (9.6), (9.7) is alinear system of N - 1 equations in the N - 1 unknowns y n ( n =1, . . . , N - 1). Writing out (9.7) and replacing y0 by α and yN by β, thesystem takes the form(9.8)The coefficients in (9.8) can, of course, be computed since f(x), g(x), andq(x) are known functions of x. This linear system can now be solved byany of the methods discussed in Chap. 4. In matrix form we have A y = b,y = [y1,y2, . .
. , yN-1] T representing the vector of unknowns; b representing the vector of known quantities on the right-hand side of (9.8); and A,the matrix of coefficients. The matrix A in this case is tridiagonal and oforder N - 1. It has the special formThe system Ay = b can be solved directly using Algorithm 4.3 of Sec.4.2. We need only replace n by N - 1, identify x and y, and apply therecursion formulas of Algorithm 4.3.Returning to the boundary conditions, let us see how the system (9.8)is affected if in place of (9.5) we prescribe the following condition at9.1FINITE-DIFFERENCE METHODSy´(x 0 ) + γy(x0) = 0If we replace y´(x0) by a forward difference, we will have409(9.9)or on rearranging,(9.9a)y l + ( - 1 + γh)y0 = 0If we now write out (9.7) for n = 1 and then replace y0 by y1 /(1 - γh), wewill have(9.10)The first equation of (9.8) can now be replaced by (9.10).
All otherequations of (9.8) will remain unchanged, and the resulting system canagain be solved, using Algorithm 4.3. We note, however, that (9.9a ) is onlyanapproximation to the boundary condition (9.9) (see Sec. 7.1).The accuracy of the solution will then also be of order h. To obtain asolution which is everywhere of order h2, we replace (9.9) by the approximationor on rearranging,(9.11)y 1 - y - 1 + 2h γy0 = 0Since we have introduced an exterior point y-1, we must now consider y0as well as yl, y2, . . . , yN-1 as unknowns. Since we now have N unknowns,we must have N equations. We can obtain an additional equation bytaking n = 0 in (9.7).
If we then eliminate y-1 using (9.11), we will have forthe first two equationsThe remaining equations will be the same as those appearing in (9.8). Thesystem is still tridiagonal but now of order N. It can again be solvedexplicitly with the aid of Algorithm 4.3.The accuracy attainable with finite-difference methods will clearlydepend upon the fineness of the mesh and upon the order of the finitedifference approximation. As the mesh is refined, the number of equations410BOUNDARY-VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONSto be solved increases. As a result, the amount of computer time requiredmay become excessive, and good accuracy may be difficult to achieve. Theuse of higher-order approximations will yield greater accuracy for the samemesh size but results in considerable complication, especially near the endpoints of the interval where the exterior values will not be known.In practice, it is advisable to solve the linear system for severaldifferent values of h.
A comparison of the solutions at the same meshpoints will then indicate the accuracy being obtained. In addition, theextrapolation process, described in Sec. 7.5, can usually be applied to yieldfurther improvement. As adapted to the solution of finite-difference systems, extrapolation to the limit proceeds as follows. Let yh (x) denote theapproximate solution at one of the mesh points x of the boundary-valueproblem based on N = (b - a)/h subdivisions of the interval [a,b]. Letyh/2(x) be the approximate solution of the same problem based on 2N =(b - a)/(h/2) subdivisions of the interval [a,b].
At the N - 1 pointsx1 = a + h, x2 = a + 2h, . . . , xN-1 = a + (N - 1)h, we now have twoapproximations, yh (xn ) and yh/2 (xn ). Applying extrapolation to these, weobtainn = 1, 2, . . . , N - 1This extrapolation will usually produce a significant improvement in theapproximation.Example 9.1 Solve the boundary-value problem (9.1), using finite-difference methods.Taking f(x) = 0, g(x) = -1, q(x) = 0, and setting y 0 = 0, y N = 1 in (9.8), weobtain the system=0(-2 - h 2 )y 1 + y 2y n - 1 + ( - 2 - h 2 )yn + yn+1 = 0= -1y N - 2 + ( - 2 - h 2 )y N - 1n = 2, 3, . .
. , N - 2This is a system of N - 1 equations in the N - 1 unknowns: y 1 ,y 2 ,. . . ,y N-1 . Thissystem was solved on the IBM 7090 with h = 0.1 and h = 0.05, using a subroutine basedon Algorithm 4.3. The results are given on page 411. The fourth column gives theextrapolated values at intervals of 0.1 obtained from the formulaThe values in the last column are obtained from the exact solution to the problem,These results show that for h = 0.1 the solution is correct to three to foursignificant figures and for h = 0.05 to four to five significant figures, while the extrapolated solution is correct to about seven significant figures. To obtain seven significantfigures of accuracy without extrapolation would require a subdivision of the interval[0, 1] into approximately 100 mesh points (h = 0.01).9.1FINITE-DIFFERENCE METHODS411COMPUTER RESULTS FOR EXAMPLE 9.1XNYN(H = 0.05)YN(H = 0.10)YN(1)Y(XN)00.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.000.04256502.08523646.12812098.17132582.21495896.25912950.30394787.34952610.39597815.44342014.49197068.54175115.59288599.64550304.69973386.75571401.81358345.87348684.935573951000.04256363.08523369.12811689.17132045.21495239.25912183.30393920.34951659.39596794.44340942.49195965.54174004.59287506.64549258.69972418.75570543.81357635.87348163.935571071.08524469.08523372.17134184.17132048.25915240.25912187.34955449.34951663.4434521344340946.54178427.54174010.6455342564549263.75573958.75570550.87350228.8734816611EXERCISES9.1-l Solve by difference methods the boundary-value problemy (0) = 0y (1) = 1Take h = 1/4, and solve the resulting system, using a pocket calculator.Answer: yl = 0.2943, y2 = 0.5702, y3 = 0.8 104.Compare this solution with the exact solution y = (sin x)/(sin 1).9.1-2 Solve the boundary-value problem (9.1) with the condition y (0) = 0 replaced by thecondition y´(0) + y(0) = 0, using a mesh h = 0.1.9.1-3 Write anfinite-difference system for approximating the solution of theboundary-value problemy´´ + xy´ + y = 2xy(0) = 1y (1) = 0Let h = 0.1, and write the system in matrix form.
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