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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 ...
More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. [1] All Gauss–Legendre methods are A-stable. [2] The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is:
The method is based on the theory of orthogonal collocation where the collocation points (i.e., the points at which the optimal control problem is discretized) are the Legendre–Gauss (LG) points. The approach used in the GPM is to use a Lagrange polynomial approximation for the state that includes coefficients for the initial state plus the ...
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...
which is an explicit formula for the Fourier coefficients a j,k. With periodic boundary conditions, the Poisson equation possesses a solution only if b 0,0 = 0. Therefore, we can freely choose a 0,0 which will be equal to the mean of the resolution. This corresponds to choosing the integration constant.
To accomplish this, a fractional differentiation matrix is derived at the Chebyshev Gauss–Lobatto collocation points by using the discrete orthogonal relationship of the Chebyshev polynomials. Then, using two proposed discretization operators for matrix functions results in an explicit form of solution for a system of linear FDEs with ...
The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s. [1] However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s, [2] followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications.
One possibility to "fix" the formula is use the stochastic collocation method and to project the corresponding implied, ill-posed, model on a polynomial of an arbitrage-free variables, e.g. normal. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free. [4]