Periodic, continuous and piecewise smooth function

In summary, the conversation is about a new member of physics forums seeking help with their research on applied mathematician problems. They are in need of an initial condition function that is continuous, periodic, and piecewise smooth, and are waiting for suggestions from other members. One member suggests using the function sin(x), but the original poster is looking for other options to compare with their numerical MATLAB simulation results. Another member suggests using cos(x) and also provides an alternative function f(x) that meets the criteria. The original poster is asked if they truly understand the meaning of a continuous, periodic, piecewise function.
  • #1
fractional
2
0
Dear friends,

I am a new member of physics forums, so this is may new message.
Already thanks to you for your helps to my question. I research some
applied mathematician problem's numerical solutions. There are initial-boundary value problems. I need an initial condition function which must be a periodic, continuous and
piecewise smooth type. Please vaiting for your suggestions.

ocean
 
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  • #2
sin(x) is a continuous smooth piecewise function.
 
  • #3
thanks for your suggestion, but I apply sin(x) to my problem, I need any other one to compare my numerical MATLAB simulation results. ıf you give me any other function, I can use for comparison.
thanks for your help.
 
  • #4
Cos(x) works.

You just want some continuous, periodic, piecewise function? Okay, what about f(x)= x if [itex]0\le x\le 1[/itex], 2- x if [itex]1\le x\le 2[/itex] and continued periodically with period 1?

My question is, do you really understand what "continous, periodic, piecewise function" means? If you do, then it should be easy to make up hundreds of examples.
 

1. What is the difference between a periodic, continuous, and piecewise smooth function?

A periodic function is one that repeats itself at regular intervals, where the value of the function at each interval is the same. A continuous function is one that has no sudden jumps or breaks in its graph, meaning that it can be drawn without lifting the pen from the paper. A piecewise smooth function is one that is composed of multiple smaller continuous functions, with each piece being smooth (having a continuous derivative).

2. Can a function be both periodic and continuous?

Yes, a function can be both periodic and continuous. For example, the sine and cosine functions are both periodic and continuous as they repeat themselves and have no sudden jumps or breaks in their graph.

3. Are all periodic functions also piecewise smooth?

No, not all periodic functions are piecewise smooth. For a function to be piecewise smooth, it must be composed of multiple smaller continuous functions, and not all periodic functions have this property. For example, the tangent function is periodic but not piecewise smooth.

4. How can I determine if a function is piecewise smooth?

To determine if a function is piecewise smooth, you can look for any breaks or sudden changes in the graph. If the function can be broken down into smaller continuous functions, then it is likely that it is piecewise smooth. Additionally, you can check if the function has a continuous derivative at all points, as this is a characteristic of a smooth function.

5. What are some real-world examples of piecewise smooth functions?

Piecewise smooth functions are commonly used in physics and engineering to model real-world phenomena. For example, the position of a moving object can be represented by a piecewise smooth function, with each piece representing a different phase of the object's motion. Another example is the temperature of a room over time, which may have different smooth sections as the heating or cooling systems turn on and off.

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