# Multi-term FODE

By specifying different orders in the equation, we can handle multi-terms FODE now!

Let's see if we have an initial value problem with multiple terms derivative containing both fractional and integer, we can use the **PIEX** algorithm to solve the equation.

All we need to do is passing the parameters and orders of the fractional ordinary differential equation to the API `solve`

as two arrays.

When we are solving the multi-terms FODE problem, please note we should keep the parameters array and orders array have the same length.

## Detailed Usage

We need to first define our multi-terms FODE problem:

`julia> prob = MultiTermsFODEProblem(parameters, orders, rightfun, u0, tspan)`

Let's see if we have an equation like:

\[2y''(t)+4D^{1.5}y(t)=1\]

To solve the multi-terms fractional order equation, you can use the code:

```
rightside = 1
prob = MultiTermsFODEProblem([2, 4], [2, 1.5], rightside, [0, 0], (0, 30))
sol = solve(prob, 0.01, PIEX())
```

Bingo! The result `sol`

is the numerical solution of this equation!!!!

## Example

We have an initial problem:

\[y'''(t)+\frac{1}{16} {^C_0D^{2.5}_ty(t)}+\frac{4}{5}y''(t)+\frac{3}{2}y'(t)+\frac{1}{25}{^C_0D^{0.5}_ty(t)}+\frac{6}{5}y(t)=\frac{172}{125}\cos(\frac{4t}{5})\]

\[y(0)=0,\ y'(0)=0,\ y''(0)=0\]

Model this problem and solve the equation:

```
using FractionalDiffEq, Plots
tspan = (0, 30); h = 0.01
rightfun(x, y) = 172/125*cos(4/5*x)
prob = MultiTermsFODEProblem([1, 1/16, 4/5, 3/2, 1/25, 6/5], [3, 2.5, 2, 1, 0.5, 0], rightfun, [0, 0, 0, 0, 0, 0], tspan)
sol = solve(prob, h, PIEX())
plot(sol, legend=:bottomright)
```

By solving the equation and plotting the result, we can see its solution here:

This example is an official example in the source code, please see the example folder