# Simplifying output involving heaviside discontinuity functions

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Aleem Andrew on 14 Jan 2021
Answered: Amish on 14 May 2024
I wrote a program to find the discontinuity function of a beam. The output for the shear function was
f = (12*heaviside(x - 5/2) - 12*heaviside(x - 7/2))*x^2 + (84*heaviside(x - 7/2)...
- 60*heaviside(x - 5/2) - 24*heaviside(x - 1))*x + 75*heaviside(x - 5/2) - 12*heaviside(x - 1)...
- 109*heaviside(x - 7/2) + 46*heaviside(x)
How can the output be simplified so that there are no terms such as heaviside(x - 7/2)*x^2 where the heaviside function is shifted to the right by 7/2 but the function the heaviside function is multiplied by is not shifted by the same amount? For example, the above function is equivalent to
f = 46*heaviside(x)- 36*heaviside(x-1) - 24*heaviside(x-1).*(x-1) + 12*(heaviside(x-2.5).*(x-2.5).^2...
- heaviside(x-3.5).*(x-3.5).^2)+38*heaviside(x-3.5).*(x-3.5)

Amish on 14 May 2024
Hi Aleem,
I see that you want to simplify the output so that there are no heavyside functions.
In order to do so, you will need to express each term in a way that aligns the multiplication of Heaviside functions with their respective polynomial terms, ensuring that the shifts in the Heaviside functions correspond directly to the shifts in the polynomial expressions they multiply.
Therefore, to simplify such expressions, you can use the "Symbolic Math Toolbox" which has the "simplify" function. However, you should keep in mind that the symbolic computation can be computationally intensive, and the ability of the "simplify function to reduce an expression to a form that matches your exact needs might vary. You may need some sort of manual manipulation based on mathematical insight to achieve the desired form.
Here is a general snippet of how to use the simplify function:
syms x
% Define the original function using Heaviside functions
f = (12*heaviside(x - 5/2) - 12*heaviside(x - 7/2))*x^2 + ...
(84*heaviside(x - 7/2) - 60*heaviside(x - 5/2) - 24*heaviside(x - 1))*x + ...
75*heaviside(x - 5/2) - 12*heaviside(x - 1) - 109*heaviside(x - 7/2) + 46*heaviside(x);
f_simplified = simplify(f);
f_collected = collect(f_simplified, heaviside(x));
% Display the simplified function and the collected form
disp(f_simplified)
disp(f_collected)
Hope this helps!