calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown...

boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution...

or initial point. Since the parameter s {\displaystyle s} is usually time, Cauchy conditions can also be called initial value conditions or initial value...

variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations...

boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions...

given function of x and t. Initial value problem on (−∞,∞) { u t = k u x x ( x , t ) ∈ R × ( 0 , ∞ ) u ( x , 0 ) = g ( x ) Initial condition {\displaystyle...

on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition)...

gauge of electromagnetism. One method to solve the initial-value problem (with the initial values as posed above) is to take advantage of a special property...

cosmologists to question how the initial density came to be so closely fine-tuned to this 'special' value. The problem was first mentioned by Robert Dicke...

Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem...

methods for solving initial value problems. This problem, in which an ordinary differential equation is given together with an initial condition, plays a...

solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution...

possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example...

value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [a,b], comparing the value this...

Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: d y d t = f ( t , y ) , y ( t 0 ) = y 0...

solution. To verify this prediction, recall the solution of the initial value problem u t t = u x x + u y y , u ( 0 , x , y ) = p ( x , y ) , u t ( 0...

boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in...

methods. The procedure for calculating the numerical solution to the initial value problem: y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle...

A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant...

problems, and must be a property of the differential system itself. Such systems are thus known as stiff systems. Consider the initial value problem The...

{\displaystyle \delta } is Dirac's delta function; the solution of the initial-value problem L y = f {\displaystyle Ly=f} is the convolution ( G ∗ f {\displaystyle...

( t , t 0 , y 0 ) {\displaystyle y(t)=y(t,t_{0},y_{0})} to the initial value problem y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle...

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order: f n ( x ) d n y d x n...

which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is...

method) states about existence and uniqueness of the solution to an initial value problem for the first order explicit ordinary differential equation. This...

coset. In this context, solving an initial value problem is interpreted as lying in the hyperplane given by the initial conditions. Stewart, James (2008)...

{\displaystyle \psi (t)} which is the solution to the inhomogeneous initial value problem d d t ψ ( t ) = f ( ψ ( t ) , ϕ ( t − τ ) ) , {\displaystyle {\frac...

6}A_{1}} We can determine A0 and A1 if there are initial conditions, i.e. if we have an initial value problem. So we have A 4 = 1 4 A 2 = ( 1 4 ) ( − 1 2 )...

theorem which guarantees the existence of solutions to certain initial value problems. Peano first published the theorem in 1886 with an incorrect proof...

theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall...