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Trapezoidal rule (differential equations) In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method ...
Adaptive step size. In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method and to ensure stability properties such as A-stability.
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form. with and . in the vicinity of the regular singular point . One can divide by to obtain a differential equation of the form.
Use of Newton's method to compute square roots. Newton's method is one of many known methods of computing square roots. Given a positive number a, the problem of finding a number x such that x2 = a is equivalent to finding a root of the function f(x) = x2 − a. The Newton iteration defined by this function is given by.
Linear multistep method. Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution.
where a, b and λ are constants and 0 < λ < 1. This equation and some more general forms are named after the pantographs on trains.; Solving DDEs. DDEs are mostly solved in a stepwise fashion with a principle called the method of steps.
The finite element method ( FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential .
Solving deconstructed matrix ordinary differential equations. The process of solving the above equations and finding the required functions of this particular order and form consists of 3 main steps. Brief descriptions of each of these steps are listed below: Finding the eigenvalues; Finding the eigenvectors; Finding the needed functions
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