## Exact Solutions

In mathematics and especially physics, an exact solution is a solution to a problem that encapsulates the whole mathematics or physics of the problem without using an approximation.

## Analytic solutions

Whenever we use analysis to find the solution to the set of equations and understanding the behavior of a simple mathematical model using basic math techniques, the solution will be analytic solution.

- Suppose you have a mathematical model andyou want to understand its behavior, you want to find a solution to the set of equations. The best way is you can use calculus, trigonometry and other math techniques to find the solutions.
- Now you know absolutely how the model will behave under any circumstances. This is called analytic solution, because you used analysis to figure it out. It is also referred to as closed form solution. But this tends to work only for simple models.

## Numerical Solutions

For more complex models, the math becomes too much complicated. Then we turn to numerical methods of solving the equations, such as Runge Kutta method.

For a differential equation that describes behavior over time, the numerical method starts with the initial values of the variables and then use equations to figure out the changes in these variables over a very brief time period.

- It is only a approximate solution, but it can be very good under certain circumstances.

Numerical model which uses numerical methods are:

- Finite differences.
- Finite volumes.
- Finite elements.
- Boundary elements.

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