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The return must contain 1 or 2 elements in the following order:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "df_dyarray_like with shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "(n, n, m)" }, { "__type": "Text", "__tag": 4046, "value": ", where an element " }, { "__type": "InlineCode", "__tag": 4051, "value": "(i, j, q)" }, { "__type": "Text", "__tag": 4046, "value": " equals to " }, { "__type": "InlineCode", "__tag": 4051, "value": "d f_i(x_q, y_q, p) / d (y_q)_j" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "df_dparray_like with shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "(n, k, m)" }, { "__type": "Text", "__tag": 4046, "value": ", where an element " }, { "__type": "InlineCode", "__tag": 4051, "value": "(i, j, q)" }, { "__type": "Text", "__tag": 4046, "value": " equals to " }, { "__type": "InlineCode", "__tag": 4051, "value": "d f_i(x_q, y_q, p) / d p_j" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Here q numbers nodes at which x and y are defined, whereas i and j number vector components. 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If we define " }, { "__type": "InlineCode", "__tag": 4051, "value": "r = y' - f(x, y)" }, { "__type": "Text", "__tag": 4046, "value": ", where y is the found solution, then the solver tries to achieve on each mesh interval " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm(r / (1 + abs(f)) < tol" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm" }, { "__type": "Text", "__tag": 4046, "value": " is estimated in a root mean squared sense (using a numerical quadrature formula). Default is 1e-3." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "max_nodes", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Maximum allowed number of the mesh nodes. If exceeded, the algorithm terminates. 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Equals to " }, { "__type": "ParamRef", "__tag": 4071, "name": "tol" }, { "__type": "Text", "__tag": 4046, "value": " by default. Up to 10 iterations are allowed to achieve this tolerance." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function numerically solves a first order system of ODEs subject to two-point boundary conditions " } ] }, { "__type": "Code", "__tag": 4050, "value": "dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b\nbc(y(a), y(b), p) = 0", "execution_status": null, "out": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Here x is a 1-D independent variable, y(x) is an n-D vector-valued function and p is a k-D 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 an (n + k)-D function." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The last singular term on 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 " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": " for the explanation how this term is handled when solving BVPs numerically." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "4" }, { "__type": "Text", "__tag": 4046, "value": "), 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)." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "build-install/usr/lib/python3.14/site-packages/scipy/integrate/_bvp.py", "item_line": 711, "item_type": "function", "aliases": [ "scipy.integrate.solve_bvp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "In the first example, we solve Bratu's problem::\n\n y'' + k * exp(y) = 0\n y(0) = y(1) = 0\n\nfor k = 1.\n\nWe rewrite the equation as a first-order system and implement its\nright-hand side evaluation::\n\n y1' = y2\n y2' = -exp(y1)\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\ndef fun(x, y):\n return np.vstack((y[1], -np.exp(y[0])))\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nImplement evaluation of the boundary condition residuals:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def bc(ya, yb):\n return np.array([ya[0], yb[0]])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nDefine the initial mesh with 5 nodes:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x = np.linspace(0, 1, 5)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nThis problem is known to have two solutions. To obtain both of them, we\nuse two different initial guesses for y. We denote them by subscripts\na and b.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "y_a = np.zeros((2, x.size))\ny_b = np.zeros((2, x.size))\ny_b[0] = 3\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nNow we are ready to run the solver.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.integrate import solve_bvp\nres_a = solve_bvp(fun, bc, x, y_a)\nres_b = solve_bvp(fun, bc, x, y_b)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nLet's plot the two found solutions. We take an advantage of having the\nsolution in a spline form to produce a smooth plot.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x_plot = np.linspace(0, 1, 100)\ny_plot_a = res_a.sol(x_plot)[0]\ny_plot_b = res_b.sol(x_plot)[0]\nimport matplotlib.pyplot as plt\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(x_plot, y_plot_a, label='y_a')\nplt.plot(x_plot, y_plot_b, label='y_b')\nplt.legend()\nplt.xlabel(\"x\")\nplt.ylabel(\"y\")\n", "execution_status": "failure", "out": "Text(0, 0.5, 'y')\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-3fde1c94c65e70d6.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nWe see that the two solutions have similar shape, but differ in scale\nsignificantly.\n\nIn the second example, we solve a simple Sturm-Liouville problem::\n\n y'' + k**2 * y = 0\n y(0) = y(1) = 0\n\nIt is known that a non-trivial solution y = A * sin(k * x) is possible for\nk = pi * n, where n is an integer. To establish the normalization constant\nA = 1 we add a boundary condition::\n\n y'(0) = k\n\nAgain, we rewrite our equation as a first-order system and implement its\nright-hand side evaluation::\n\n y1' = y2\n y2' = -k**2 * y1\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def fun(x, y, p):\n k = p[0]\n return np.vstack((y[1], -k**2 * y[0]))\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nNote that parameters p are passed as a vector (with one element in our\ncase).\n\nImplement the boundary conditions:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def bc(ya, yb, p):\n k = p[0]\n return np.array([ya[0], yb[0], ya[1] - k])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nSet up the initial mesh and guess for y. We aim to find the solution for\nk = 2 * pi, to achieve that we set values of y to approximately follow\nsin(2 * pi * x):\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x = np.linspace(0, 1, 5)\ny = np.zeros((2, x.size))\ny[0, 1] = 1\ny[0, 3] = -1\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nRun the solver with 6 as an initial guess for k.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sol = solve_bvp(fun, bc, x, y, p=[6])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nWe see that the found k is approximately correct:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sol.p[0]\n", "execution_status": "failure", "out": "np.float64(6.283294600464793)\n" }, { "__type": "Text", "__tag": 4046, "value": "\nAnd, finally, plot the solution to see the anticipated sinusoid:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x_plot = np.linspace(0, 1, 100)\ny_plot = sol.sol(x_plot)[0]\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(x_plot, y_plot)\nplt.xlabel(\"x\")\nplt.ylabel(\"y\")\n", "execution_status": "failure", "out": "Text(0, 0.5, 'y')\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-4a3b00d80fa14b63.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "fun", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "bc", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "x", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "y", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "p", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "S", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "fun_jac", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "bc_jac", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "tol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.001" }, { "__type": "SigParam", "__tag": 4030, "name": "max_nodes", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1000" }, { "__type": "SigParam", "__tag": 4030, "name": "verbose", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "bc_tol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "solve_bvp" }, "references": [ ".. [1] 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.", ".. [2] L.F. Shampine, P. H. Muir and H. Xu, \"A User-Friendly Fortran BVP", " Solver\", J. Numer. Anal., Ind. Appl. Math. (JNAIAM), Vol. 1,", " Number 2, pp. 201-217, 2006.", ".. [3] U. Ascher, R. Mattheij and R. Russell \"Numerical Solution of", " Boundary Value Problems for Ordinary Differential Equations\",", " Philidelphia, PA: Society for Industrial and Applied Mathematics,", " 1995.", " :doi:`10.1137/1.9781611971231`", ".. [4] `Cauchy-Riemann equations", " `_ on", " Wikipedia." ], "qa": "scipy.integrate._bvp:solve_bvp", "arbitrary": [], "local_refs": [ "S", "bc", "bc_jac", "bc_tol", "fun", "fun_jac", "max_nodes", "p", "result", "tol", "verbose", "x", "y" ] }