These methods are widely used in both academic and real world applications.

[PumkinMonster]

Can you introduce me some modern and mostly used computer based methods to solve differential equations ?

I know only Newton, KRupta but I am not sure if they are often used in academic or real world applications.

Thank you

 

[MasterX]

First of all Newton's method is used to solve nonlinear algebraic equation!

For differential equations, you can use:
Backward Euler,
Forward Euler,
Crank-Nikolson
Adams-Bashford
Finite Difference Method
Finite Elements Method
Finite Volume Method