What exaclty is a differential operator?

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SUMMARY

A differential operator, denoted as "L", is a mathematical construct that forms a linear combination with a given function, represented as L*v. In the context of differential equations, the operator can be expressed as L = P d²/dx² + Q d/dx, where P and Q are coefficients. This operator acts on a function y to yield L y = 0, which is a shorthand representation of the corresponding differential equation. The discussion clarifies the relationship between differential operators and their application in numerical methods.

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Homework Statement


I have fallen behind on my Numerical Methods course and am starting to fail it. I need to know how to make a differential approximation and I'm reading through the materials but I have too little time and it doesn't even explain what a differential operator is. At first it is referred to as "L". It apparently forms a linear combination with the given function v. And L*v is apparently the sum of all pattern in a given lattice (I'm sorry if my English terminology is off). So anyway, later there is a "difference operator v[itex]x\bar{x}[/itex][itex]x\bar{x}[/itex]". Is that supposed to be the same as L, just in some special case?

Homework Equations


Can't think of anything else.

The Attempt at a Solution


I'm sorry, but there isn't one as I am very tired and have little time or energy to spare for this problem right now, mostly because I don't even know what is going on.Can anybody enlighten me? I know genereally what a derviative is - it's a rate in how fast a function changes. But this - not really a whole lot to go on.Edit: nah it's ok I'm moving forward a little, I got a little panicked. A little messed up living in the dorms and all. Mods may delete this thread. I don't know if I can but it looks like I can't
 
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Consider this differential equation: P y'' + Q y' = 0.

The L = P d^2/dx^2 + Q d/dx is the corresponding differential operator.

All that is missing is the function y. Then L y = 0 is shorthand for that differential equation.

See http://en.wikipedia.org/wiki/Differential_operator
 

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