Search results
Results From The WOW.Com Content Network
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 .
The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation). [5] Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may ...
If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series. If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs.
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no analytical solution exists. Then one uses root-finding algorithms, such as Newton's method, to find the numerical solution. Crank-Nicolson method. With the Crank-Nicolson method
Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data ...
Suppose that we want to solve the differential equation ′ = (,). The trapezoidal rule is given by the formula + = + ((,) + (+, +)), where = + is the step size. [1]This is an implicit method: the value + appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear.
Predictor–corrector methods for solving ODEs [ edit ] When considering the numerical solution of ordinary differential equations (ODEs) , a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step.