Partial differential equation

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In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000. Partial differential equations occur very widely in mathematically oriented...

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