Thread: Computational methods

Let me try to explain the simple numerical method (in scientific computation) to deal with differential equations. Some of you might have lost touch with all this, so I shall consider going over the concepts. Please note that the concepts developed during the early stages of the voyage into the kernel will be used for solving problems in the upcoming days.



The most important point we need to note is that in the case of differential equations, it relates a function to its derivatives, so that we can compute the function itself. Take, for example: The general solution for this will be of the form 'a constant multiplied by e"t'. If you are sceptical, try differentiating the solution! The ordinary differential equations can be represented as shown below:

(The orders of the equations are different.) We also have partial differential equations, which differ from ODEs. And typical partial differential equations (PDE) can be classified as shown below: Out of these, there are homogeneous and non-homogeneous ones. Some equations (like Laplace equation) are homogeneous in nature: While others are non-homogeneous (like the Poisson equation):