7: Finite-Difference Equations
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One of the most common tasks in scientific computing is finding solutions to differential equations, because most physical theories are formulated using differential equations. In classical mechanics, for example, a mechanical system is described by a second-order differential equation in time (Newton's second law); and in classical electromagnetism, the electromagnetic fields are described by first-order partial differential equations in space and time (Maxwell's equations).
In order to describe continuous functions (and the differential equations that act on them), computational schemes usually adopt the strategy of discretization. Consider a general mathematical function of one real variable,
We define the values at these points as
If
As we shall see, discretization converts differential equations into discrete systems of equations, called finite-difference equations. These can then be solved using the standard methods of numerical linear algebra.