Approximating solutions of nonlinear second-order boundary value problems: bvpsolve

### 10.3.7 Approximating solutions of nonlinear second-order boundary value problems: bvpsolve

The bvpsolve command finds an approximate solution of a boundary value problem

y″=f(x,y,y′), y(a)=α, y(b)=β

on an interval [a,b]. The procedure uses the method of nonlinear shooting which is based on Newton and Runge-Kutta methods. Values of y and its first derivative y′ are approximated at points xk=a+k δ, where δ=b−a/N and k=0,1,…,N. For the numeric tolerance (precision) threshold, the algorithm uses epsilon specified in the session settings in Xcas (see Section 3.5.7, item 9).

- bvpsolve takes three mandatory arguments and four optional arguments:
  - f(x,y,y′), an expression involving an independent variable x, a dependent variable y and y′.
  - [x=a..b,y], a list specifying the independent variable x, its range [a,b] and the sought function y
  - [α,β], a list of the boundary values of y.
  - Optionally, A, an initial guess for y′(a) (by default, (β−α)/(b−a)).
  - Optionally, N, an integer greater than or equal to 2 (by default 100).
  - output=type or Output=type, the type of the output, where type can be one of:
    - list
    - diff
    - piecewise
    - spline
    (By default list).
  - limit=M, a positive integer for a limit for the number of iterations before the procedure is stopped (by default there is no limit).
- bvpsolve(f(x,y,y′),[x=a..b,y],[α,β] ⟨,A,N, output=type,limit=M⟩) returns, for the different output types:
  - list, a list of pairs [xk,yk] where yk≈ y(xk),
  - diff, a list of lists [xk,yk,y′k], where y′k≈ y′(xk),
  - piecewise, a piecewise linear interpolation of the points (xk,yk).
  - spline, a piecewise spline interpolation of the points (xk,yk), based on the values y′k computed in the process.

Note that the shooting method is sensitive to roundoff errors and may fail to converge in some cases, especially when y is a rapidly increasing function. In the absence of convergence or if the maximum number of iterations is exceeded, bvpsolve returns undef. However, if the output type is list or piecewise and if N>2, a slower but more stable finite-difference method (which approximates only the function y) is tried first.

Sometimes setting an initial guess A for y′(a) to a suitable value may help the shooting algorithm to converge or to converge faster.

Examples.

- Solve the problem

  y″= |
  1
  8
  | (32+2 x3−y y′), 1≤ x≤ 3

  with boundary conditions y(1)=17 and y(3)=43/3. Use N=20, which gives an x-step of 0.01.
  Input:
  bvpsolve((32+2x^3-y*y’)/8,[x=1..3,y],[17,43/3],20)
  The output is shown in Table 10.1 (the middle two columns) alongside with the values y(xk) of the exact solution y=x2+16/x (the fourth column).

  > k | xk | yk | y(xk)
  > 0 | 1.0 | 17.0 | 17.0
  > 1 | 1.1 | 15.7554961579 | 15.7554545455
  > 2 | 1.2 | 14.7733911821 | 14.7733333333
  > 3 | 1.3 | 13.9977543159 | 13.9976923077
  > 4 | 1.4 | 13.388631813 | 13.3885714286
  > 5 | 1.5 | 12.9167227424 | 12.9166666667
  > 6 | 1.6 | 12.5600506483 | 12.56
  > 7 | 1.7 | 12.3018096101 | 12.3017647059
  > 8 | 1.8 | 12.1289281414 | 12.1288888889
  > 9 | 1.9 | 12.0310865274 | 12.0310526316
  > 10 | 2.0 | 12.0000289268 | 12.0
  > 11 | 2.1 | 12.0290719981 | 12.029047619
  > 12 | 2.2 | 12.1127475278 | 12.1127272727
  > 13 | 2.3 | 12.2465382803 | 12.2465217391
  > 14 | 2.4 | 12.4266798825 | 12.4266666667
  > 15 | 2.5 | 12.650010254 | 12.65
  > 16 | 2.6 | 12.9138537834 | 12.9138461538
  > 17 | 2.7 | 13.2159312426 | 13.2159259259
  > 18 | 2.8 | 13.5542890043 | 13.5542857143
  > 19 | 2.9 | 13.9272429048 | 13.9272413793
  > 20 | 3.0 | 14.3333333333 | 14.3333333333

  > Table 10.1: approximate and true values of the function y=x2+16/x on [1,3]

- Solve

  y″= |
  x2 (y′)2−9 y2+4 x6
  x5
  | , 1≤ x≤ 2,

  with the boundary conditions y(1)=0 and y(2)=ln256. Obtain the solution as a piecewise spline interpolation for N=10 and estimate the absolute error err of the approximation using the exact solution y=x3 lnx and the romberg command for numerical integration. You need to explicitly set an initial guess A for the value y′(1) because the algorithm fails to converge with the default guess A=ln256≈ 5.545. Therefore let A=1 instead.
  Input:

  f:=(x^2*diff(y(x),x)^2-9*y(x)^2+4*x^6)/x^5:;
  p:=bvpsolve(f,[x=1..2,y],[0,ln(256),1],10,output=spline):;
  err:=sqrt(romberg((p-x^3*ln(x))^2,x=1..2))

  Output:

  3.27751904973×10−6

  Note that, if the output type was set to list or piecewise, the solution would have been found even without specifying an initial guess for y′(1) because the algorithm would automatically apply the alternative finite-difference method, which converges.