<P> To be useful in applications, a boundary value problem should be well posed . This means that given the input to the problem there exists a unique solution, which depends continuously on the input . Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well - posed . </P> <P> Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle . </P> <P> 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). </P> <P> For example, if the independent variable is time over the domain (0, 1), a boundary value problem would specify values for y (t) (\ displaystyle y (t)) at both t = 0 (\ displaystyle t = 0) and t = 1 (\ displaystyle t = 1), whereas an initial value problem would specify a value of y (t) (\ displaystyle y (t)) and y ′ (t) (\ displaystyle y' (t)) at time t = 0 (\ displaystyle t = 0). </P>

What is initial value problem and boundary value problem
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