Dear Sarmad, the two methods should give the same "qualitative" numerical values, considering the small differences due to numerical approximation.

Approximated-Analytical methods give a comprehension of the mathematical concepts used in equations resolution (e.g. the Picard method in ODE y'(x) = f(x, y(x)), which is the base for extension of ODE resolution in Functional Spaces, see H. Brezis, Functional Analysis and its Applications).

Numerical methods are often the only available technique in the case of complex domains and PDE (with or without mesh).

I think that Approximated-Analytical methods can be considered mathematical models, while Numerical methods are mathematical algorithms for numerical management of these models.

But each of them are the same dignity and importance in mathematical item of differential equations and relative problems.

Gianluca

Even though both the methods are approximate, according to me, approximate analytical methods give more comprehensive insight of the solution of the problem, which we cannot get from numerical solutions.

For example, we get certain behavior of the solution for certain parameters; and we cannot predict the reason behind such behavior if the solution is obtained using numerical techniques. Whereas, if we obtain the solution using approximate analytical methods, we can easily study the effect of every parameters on the solution just by looking at it.

However, though there are global techniques available, approximate analytical techniques generally are valid only within certain domain, whereas numerical techniques might be valid over larger domain. Another advantage of numerical techniques is their simplicity especially in complex problems/geometries. In such cases, analytical solutions are difficult to obtain.

Sometimes, I think we can make use of both the techniques to earn advantages of both. For example, we can solve the simpler version of the problem using approximate analytical methods to understand the basic nature of solution; and then we can use this solution as our initial guess to obtain the numerical solution of actual more complicated problem. This makes the numerical solution more efficient as well in terms of computational time.

Many thanks dear colleges for these useful responses.

Best regards.