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# scipy.integrate.solve_bvp

`scipy.integrate.``solve_bvp`(fun, bc, x, y, p=None, S=None, fun_jac=None, bc_jac=None, tol=0.001, max_nodes=1000, verbose=0)

  Solve a boundary-value problem for a system of ODEs.

  This function numerically solves a first order system of ODEs subject to two-point boundary conditions:

      dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b
      bc(y(a), y(b), p) = 0

  Here x is a 1-dimensional independent variable, y(x) is a n-dimensional vector-valued function and p is a k-dimensional vector of unknown parameters which is to be found along with y(x). For the problem to be determined there must be n + k boundary conditions, i.e. bc must be (n + k)-dimensional function.

  The last singular term in the right-hand side of the system is optional. It is defined by an n-by-n matrix S, such that the solution must satisfy S y(a) = 0. This condition will be forced during iterations, so it must not contradict boundary conditions. See [R47] for the explanation how this term is handled when solving BVPs numerically.

  Problems in a complex domain can be solved as well. In this case y and p are considered to be complex, and f and bc are assumed to be complex-valued functions, but x stays real. Note that f and bc must be complex differentiable (satisfy Cauchy-Riemann equations [R49]), otherwise you should rewrite your problem for real and imaginary parts separately. To solve a problem in a complex domain, pass an initial guess for y with a complex data type (see below).

  Parameters: | fun : callable Right-hand side of the system. The calling signature is `fun(x, y)`, or `fun(x, y, p)` if parameters are present. All arguments are ndarray: `x` with shape (m,), `y` with shape (n, m), meaning that `y[:, i]` corresponds to `x[i]`, and `p` with shape (k,). The return value must be an array with shape (n, m) and with the same layout as `y`. bc : callable Function evaluating residuals of the boundary conditions. The calling signature is `bc(ya, yb)`, or `bc(ya, yb, p)` if parameters are present. All arguments are ndarray: `ya` and `yb` with shape (n,), and `p` with shape (k,). The return value must be an array with shape (n + k,). x : array_like, shape (m,) Initial mesh. Must be a strictly increasing sequence of real numbers with `x[0]=a` and `x[-1]=b`. y : array_like, shape (n, m) Initial guess for the function values at the mesh nodes, i-th column corresponds to `x[i]`. For problems in a complex domain pass y with a complex data type (even if the initial guess is purely real). p : array_like with shape (k,) or None, optional Initial guess for the unknown parameters. If None (default), it is assumed that the problem doesn’t depend on any parameters. S : array_like with shape (n, n) or None Matrix defining the singular term. If None (default), the problem is solved without the singular term. fun_jac : callable or None, optional Function computing derivatives of f with respect to y and p. The calling signature is `fun_jac(x, y)`, or `fun_jac(x, y, p)` if parameters are present. The return must contain 1 or 2 elements in the following order: df_dy : array_like with shape (n, n, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d (y_q)_j. df_dp : array_like with shape (n, k, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d p_j. Here q numbers nodes at which x and y are defined, whereas i and j number vector components. If the problem is solved without unknown parameters df_dp should not be returned. If fun_jac is None (default), the derivatives will be estimated by the forward finite differences. bc_jac : callable or None, optional Function computing derivatives of bc with respect to ya, yb and p. The calling signature is `bc_jac(ya, yb)`, or `bc_jac(ya, yb, p)` if parameters are present. The return must contain 2 or 3 elements in the following order: dbc_dya : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d ya_j. dbc_dyb : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d yb_j. dbc_dp : array_like with shape (n, k) where an element (i, j) equals to d bc_i(ya, yb, p) / d p_j. If the problem is solved without unknown parameters dbc_dp should not be returned. If bc_jac is None (default), the derivatives will be estimated by the forward finite differences. tol : float, optional Desired tolerance of the solution. If we define `r = y' - f(x, y)` where y is the found solution, then the solver tries to achieve on each mesh interval `norm(r / (1 + abs(f)) < tol`, where `norm` is estimated in a root mean squared sense (using a numerical quadrature formula). Default is 1e-3. max_nodes : int, optional Maximum allowed number of the mesh nodes. If exceeded, the algorithm terminates. Default is 1000. verbose : {0, 1, 2}, optional Level of algorithm’s verbosity: 0 (default) : work silently. 1 : display a termination report. 2 : display progress during iterations.
  Returns: | Bunch object with the following fields defined: sol : PPoly Found solution for y as `scipy.interpolate.PPoly` instance, a C1 continuous cubic spline. p : ndarray or None, shape (k,) Found parameters. None, if the parameters were not present in the problem. x : ndarray, shape (m,) Nodes of the final mesh. y : ndarray, shape (n, m) Solution values at the mesh nodes. yp : ndarray, shape (n, m) Solution derivatives at the mesh nodes. rms_residuals : ndarray, shape (m - 1,) RMS values of the relative residuals over each mesh interval (see the description of tol parameter). niter : int Number of completed iterations. status : int Reason for algorithm termination: 0: The algorithm converged to the desired accuracy. 1: The maximum number of mesh nodes is exceeded. 2: A singular Jacobian encountered when solving the collocation system. message : string Verbal description of the termination reason. success : bool True if the algorithm converged to the desired accuracy (`status=0`).

  Notes

  This function implements a 4-th order collocation algorithm with the control of residuals similar to [R46]. A collocation system is solved by a damped Newton method with an affine-invariant criterion function as described in [R48].

  Note that in [R46] integral residuals are defined without normalization by interval lengths. So their definition is different by a multiplier of h**0.5 (h is an interval length) from the definition used here.

  New in version 0.18.0.

  References

  [R46] | (1, 2, 3) J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001.

  [R47] | (1, 2) L.F. Shampine, P. H. Muir and H. Xu, “A User-Friendly Fortran BVP Solver”.

  [R48] | (1, 2) U. Ascher, R. Mattheij and R. Russell “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”.

  [R49] | (1, 2) Cauchy-Riemann equations on Wikipedia.

  Examples

  In the first example we solve Bratu’s problem:

      y'' + k * exp(y) = 0
      y(0) = y(1) = 0

  for k = 1.

  We rewrite the equation as a first order system and implement its right-hand side evaluation:

      y1' = y2
      y2' = -exp(y1)

      >>> def fun(x, y):
      ...     return np.vstack((y[1], -np.exp(y[0])))

  Implement evaluation of the boundary condition residuals:

      >>> def bc(ya, yb):
      ...     return np.array([ya[0], yb[0]])

  Define the initial mesh with 5 nodes:

      >>> x = np.linspace(0, 1, 5)

  This problem is known to have two solutions. To obtain both of them we use two different initial guesses for y. We denote them by subscripts a and b.

      >>> y_a = np.zeros((2, x.size))
      >>> y_b = np.zeros((2, x.size))
      >>> y_b[0] = 3

  Now we are ready to run the solver.

      >>> from scipy.integrate import solve_bvp
      >>> res_a = solve_bvp(fun, bc, x, y_a)
      >>> res_b = solve_bvp(fun, bc, x, y_b)

  Let’s plot the two found solutions. We take an advantage of having the solution in a spline form to produce a smooth plot.

      >>> x_plot = np.linspace(0, 1, 100)
      >>> y_plot_a = res_a.sol(x_plot)[0]
      >>> y_plot_b = res_b.sol(x_plot)[0]
      >>> import matplotlib.pyplot as plt
      >>> plt.plot(x_plot, y_plot_a, label='y_a')
      >>> plt.plot(x_plot, y_plot_b, label='y_b')
      >>> plt.legend()
      >>> plt.xlabel("x")
      >>> plt.ylabel("y")
      >>> plt.show()

  (Source code)

  We see that the two solutions have similar shape, but differ in scale significantly.

  In the second example we solve a simple Sturm-Liouville problem:

      y'' + k**2 * y = 0
      y(0) = y(1) = 0

  It is known that a non-trivial solution y = A * sin(k * x) is possible for k = pi * n, where n is an integer. To establish the normalization constant A = 1 we add a boundary condition:

      y'(0) = k

  Again we rewrite our equation as a first order system and implement its right-hand side evaluation:

      y1' = y2
      y2' = -k**2 * y1

      >>> def fun(x, y, p):
      ...     k = p[0]
      ...     return np.vstack((y[1], -k**2 * y[0]))

  Note that parameters p are passed as a vector (with one element in our case).

  Implement the boundary conditions:

      >>> def bc(ya, yb, p):
      ...     k = p[0]
      ...     return np.array([ya[0], yb[0], ya[1] - k])

  Setup the initial mesh and guess for y. We aim to find the solution for k = 2 * pi, to achieve that we set values of y to approximately follow sin(2 * pi * x):

      >>> x = np.linspace(0, 1, 5)
      >>> y = np.zeros((2, x.size))
      >>> y[0, 1] = 1
      >>> y[0, 3] = -1

  Run the solver with 6 as an initial guess for k.

      >>> sol = solve_bvp(fun, bc, x, y, p=[6])

  We see that the found k is approximately correct:

      >>> sol.p[0]
      6.28329460046

  And finally plot the solution to see the anticipated sinusoid:

      >>> x_plot = np.linspace(0, 1, 100)
      >>> y_plot = sol.sol(x_plot)[0]
      >>> plt.plot(x_plot, y_plot)
      >>> plt.xlabel("x")
      >>> plt.ylabel("y")
      >>> plt.show()