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An initial value problem is a differential equation ′ = (, ()) with : where is an open set of , together with a point in the domain of (,),called the initial condition.. A solution to an initial value problem is a function that is a solution to the differential equation and satisfies
A linear matrix difference equation of the homogeneous (having no constant term) form + = has closed form solution = predicated on the vector of initial conditions on the individual variables that are stacked into the vector; is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being ...
When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions. Because there are no solvable equations for most three-body systems, the only way to predict the motions of the bodies is to estimate them using numerical methods. The three-body problem is a special case of the n-body problem.
A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest.
The Kepler problem and the simple harmonic oscillator problem are the two most fundamental problems in classical mechanics. They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem). [1]: 92
For the initial condition, ... % solve the initial value problem symbolically % for different initial conditions y1 = dsolve ... As example, the equation: ...
This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance ...
The problem has a solution; The solution is unique; The solution's behavior changes continuously with the initial conditions; Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are ...