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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 ...
Guess values can be determined a number of ways. Guessing is one of them. If one is familiar with the type of problem, then this is an educated guess or guesstimate.Other techniques include linearization, solving simultaneous equations, reducing dimensions, treating the problem as a time series, converting the problem to a (hopefully) linear differential equation, and using mean values.
In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero. [1]
In computer programming, initialization or initialisation is the assignment of an initial value for a data object or variable. The manner in which initialization is performed depends on the programming language, as well as the type, storage class, etc., of an object to be initialized.
Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones.