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In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form. Since the ratio of coefficients A k / A k − 1 {\displaystyle A_{k}/A_{k-1}} is a rational function , the power series can be written as a generalized hypergeometric series .
Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats. The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has + on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n×n matrix given by the power series = =!
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.