The Analytical Method also referred to as a closed form solution, is a generic process of finding the solution to a problem using a series of logical steps and has proof, that can be followed and verified as correct. For example: if the quadratic formula is used to solve for X in a quadratic equation, this would be an analytical solution to the problem. The best solution for a given mathematical model is when you can use calculus, trigonometry, and other math techniques to write down the solution. In such cases, you know absolutely how the model will behave under any circumstances. This is called the ANALYTIC SOLUTION, because you used analysis to figure it out. But this is only feasible for simple models. For more complex models, the calculation becomes much too complex.
On the other hand..
For more complex models, you turn to numerical methods of solving the equations. For example: If, instead of using the quadratic formula, you try a lot of initial values of the variables X and Y, then this is a numerical solution. In case of a differential equation, that describes behavior over time, the numerical method starts with the initial values of the variables, and then uses the equations to figure out the changes in these variables over a very brief time period. Some problems simply do not have feasible analytical solutions and must be solved using approximation and numerical methods such as, complicated integrals, solution of Schrodinger equation to describe the quantum state of a molecule. A computer must be used to perform the thousands of repetitive calculations involved in a numerical algorithm. The result is a bunch of numbers, not an equation. This long list of numbers is then can be used to drive an animated simulation of the mathematical model.
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