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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
The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.
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 ...
The above equations are, in fact, the general functions sought, but they are in their general form (with unspecified values of A and B), whilst we want to actually find their exact forms and solutions. So now we consider the problem’s given initial conditions (the problem including given initial conditions is the so-called initial value problem).
Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists.
This problem, in which an ordinary differential equation is given together with an initial condition, plays a central role in all natural and engineering sciences and is also becoming increasingly important in the economic and social sciences, for example. Initial value problems are used to analyze, simulate or predict dynamic processes. [25]
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain.The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...