what is the difference between analytical and numerical procedure of equation solving?

Question

equation meaning partial or odrinary differential equations.



thnx

Answer 1

An analytical procedure produces a formula for the solution. For example, solving a PDE using separation of variables, or solving a first-order linear ODE using an integrating factor. It's intended to produce an exact solution, if one is known.

A numerical procedure for an ODE or PDE (my research specialty :) produces an approximate solution, typically represented as a set of values defined on a grid (which may include points with coordinates in both space and time).

For example, suppose you have the ODE dy/dt = Ay, y(0)=y0, where A is a matrix and y0 is a given vector. An analytical procedure would involve computing the eigenvalues and eigenvectors of A, and computing exp(At)y0. A numerical procedure would mean using a method such as Euler's method to compute y(t_n+1) = y(t_n) + dt*A*y(t_n), for some chosen time step dt.

Answer 2

Analytical soln provides you with a general function which can be evaluted for any value within the domain.

The numerical soln fids out the exact or approx value of the funcn at a particular point.