{
  "id": "enwiki-001353341524",
  "slug": "collocation-method",
  "title": "Collocation method",
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  "lead": "In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points.",
  "body": "In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points. y'(t) = f(t,y(t)), \\quad y(t_0)=y_0, is to be solved over the interval [t_0,t_0+h] . Choose c_k from 0 ≤ c 1 2 n ≤ 1. The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition p(t_0) = y_0 , and the differential equation p'(t_k) = f(t_k,p(t_k)) at all collocation points t_k = t_0 + c_k h for k = 1, \\ldots, n . This gives n + 1 conditions, which matches the n + 1 parameters needed to specify a polynomial of degree n. All these collocation methods are in fact implicit Runge–Kutta methods. The coefficients c k in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods. Pick, as an example, the two collocation points c 1 = 0 and c 2 = 1 (so n = 2). The collocation conditions are \\begin{align} p(t_0) &= y_0, \\\\ p'(t_0) &= f(t_0, p(t_0)), \\\\ p'(t_0{+}h) &= f(t_0{+}h, p(t_0{+}h)). \\end{align} There are three conditions, so p should be a polynomial of degree 2. Write p in the form p(t) = \\alpha (t-t_0)^2 + \\beta (t-t_0) + \\gamma \\, to simplify the computations. Then the collocation conditions can be solved to give the coefficients \\begin{align} \\alpha &= \\frac{1}{2h} \\Big( f(t_0{+}h, p(t_0{+}h)) - f(t_0, p(t_0)) \\Big), \\\\ \\beta &= f(t_0, p(t_0)), \\\\ \\gamma &= y_0. \\end{align} The collocation method is now given (implicitly) by y_1 = p(t_0 + h) = y_0 + \\frac{1}{2} h \\Big (f(t_0{+}h, y_1) + f(t_0,y_0) \\Big), \\, where y 1 = p(t 0  + h) is the approximate solution at t = t 1 = t 0  + h. This method is known as the \"trapezoidal rule\" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as y(t) = y(t_0) + \\int_{t_0}^t f(\\tau, y(\\tau)) \\, d\\tau, \\, and approximating the integral on the right-hand side by the trapezoidal rule for integrals. ==See also== Method of moments (electromagnetics) Boundary element method",
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    "title": "Collocation method",
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        "family": "Anticloud FZ LLE",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/collocation-method",
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