I am totally confused between ODEs which I am familiar with, and differential algebraic equations (DAE) and Algebraic Differential Equations (ADE). Are they the same but just different names or what is the key difference between them (their nature and solution methods).Thanks and best regards
ODEs vs DAE vs ADE?
ode;numerical modelling;dae
At least one difference is that in a system of ODEs, all the equations are differential, e.g.: $$\dot{x}=f(x,y)\\\dot{y}=g(x,y)$$whereas the definition of DAEs that I'm familiar with includes some non-differential (i.e. algebraic) equations in the set, e.g.:$$\dot{x}=h(x,y)\\y=l(x,y)$$where $l$ is non-trival, and its solution can't be easily substituted into the first equation to simplify. These get more complex when there are more algebraic terms. DAEs are more challenging numerically; the challenges they entail are similar to but sometimes more severe than those attending stiff problems.  A very thorough explanation of DAEs and how to solve them numerically can be found in volume II of the text by Hairer and Wanner.