When working with modelingtoolkit in Julia, it is common to encounter situations where you need to simplify the equations of an ODE system. In this article, we will explore three different ways to achieve this custom simplification.
Option 1: Using the simplify function
The first option is to use the simplify function provided by the ModelingToolkit package. This function takes an expression as input and applies a set of simplification rules to simplify it. Here is an example:
using ModelingToolkit
@variables x y
@parameters a b
eqs = [a*x + b*y, a*x^2 + b*y^2]
simplified_eqs = simplify.(eqs)
In this example, we have two equations stored in the eqs array. We apply the simplify function to each equation using the dot syntax to obtain the simplified equations in the simplified_eqs array.
Option 2: Defining custom simplification rules
If the simplify function does not provide the desired level of simplification, we can define custom simplification rules using the @rule macro provided by ModelingToolkit. Here is an example:
using ModelingToolkit
@variables x y
@parameters a b
eqs = [a*x + b*y, a*x^2 + b*y^2]
@rule simplify_rule(x^2) = x
simplified_eqs = simplify.(eqs)
In this example, we define a custom simplification rule using the @rule macro. The rule states that any expression of the form x^2 should be simplified to x. We then apply the simplify function to the equations, and the custom rule is automatically applied.
Option 3: Using symbolic manipulation packages
If neither the simplify function nor custom simplification rules provide the desired level of simplification, we can turn to symbolic manipulation packages such as SymPy.jl or SymEngine.jl. These packages offer more advanced simplification capabilities. Here is an example using SymPy.jl:
using ModelingToolkit
using SymPy
@variables x y
@parameters a b
eqs = [a*x + b*y, a*x^2 + b*y^2]
simplified_eqs = map(simplify, eqs)
In this example, we first import the SymPy package. We then use the map function to apply the simplify function from SymPy to each equation in the eqs array. This allows us to leverage the advanced simplification capabilities provided by SymPy.
After exploring these three options, it is clear that the best choice depends on the specific requirements of your problem. If the simplify function provides the desired level of simplification, it is the simplest and most straightforward option. However, if you need more control over the simplification process, defining custom simplification rules or using symbolic manipulation packages can be more suitable.
