Glossary

7.4 Step Functions and Discontinuous Forcing Functions

3 min read•august 6, 2024

Step functions and discontinuous forcing functions are game-changers in differential equations. They let us model sudden changes, like flipping a switch or applying a force out of nowhere. It's like adding a plot twist to our math story!

Laplace transforms make dealing with these jumpy functions a breeze. We can turn tricky discontinuous problems into smooth algebraic ones. It's like having a secret weapon for solving real-world problems with sudden changes.

Step Functions

Heaviside Step Function and Unit Step Function

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  • H(t)H(t) represents a discontinuous function that jumps from 0 to 1 at t=0t=0
    • Defined as H(t)=0H(t) = 0 for t<0t < 0 and H(t)=1H(t) = 1 for t≥0t \geq 0
    • Useful for modeling sudden changes or switches in a system (turning on a light switch)
  • u(t)u(t) is a shifted version of the Heaviside step function
    • Defined as u(t)=0u(t) = 0 for t<0t < 0 and u(t)=1u(t) = 1 for t≥0t \geq 0
    • Can be expressed in terms of the Heaviside step function: u(t)=H(t)u(t) = H(t)
    • Represents a unit change in a system at a specific time (applying a constant force at a given moment)

Piecewise Continuous Functions and Laplace Transforms

  • Piecewise continuous functions are functions that are continuous on a finite number of intervals but may have discontinuities at the endpoints of these intervals
    • Can be represented using step functions (Heaviside or unit step) to define different pieces of the function
    • Example: f(t)={0,t<0t,0≤t<11,t≥1f(t) = \begin{cases} 0, & t < 0 \\ t, & 0 \leq t < 1 \\ 1, & t \geq 1 \end{cases} can be written as f(t)=t[u(t)−u(t−1)]+u(t−1)f(t) = t[u(t) - u(t-1)] + u(t-1)
  • Laplace transform of step functions allows for solving differential equations with discontinuous forcing functions
    • Laplace transform of the Heaviside step function: L{H(t)}=1s\mathcal{L}\{H(t)\} = \frac{1}{s}
    • Laplace transform of the unit step function: L{u(t)}=1s\mathcal{L}\{u(t)\} = \frac{1}{s}
    • Laplace transform of a shifted unit step function: L{u(t−a)}=e−ass\mathcal{L}\{u(t-a)\} = \frac{e^{-as}}{s}, where aa is the shift amount

Discontinuous Forcing Functions

Discontinuous Forcing Functions and Dirac Delta Function

  • Discontinuous forcing functions are functions that have sudden changes or jumps in their values
    • Can be represented using step functions (Heaviside or unit step) or the
    • Example: a sudden impact force on a spring-mass system can be modeled using a step function or Dirac delta function
  • Dirac delta function δ(t)\delta(t) is a generalized function that represents an infinitely high, infinitely narrow spike at t=0t=0
    • Defined by its integral properties: ∫−∞∞δ(t)dt=1\int_{-\infty}^{\infty} \delta(t) dt = 1 and ∫−∞∞f(t)δ(t−a)dt=f(a)\int_{-\infty}^{\infty} f(t) \delta(t-a) dt = f(a)
    • Useful for modeling instantaneous changes or impulses in a system (a sharp blow to a structure)
    • Laplace transform of the Dirac delta function: L{δ(t)}=1\mathcal{L}\{\delta(t)\} = 1

Switching Circuits and Applications

  • Switching circuits are electrical circuits that can be modeled using discontinuous forcing functions
    • Example: a simple RC circuit with a switch that is closed at t=0t=0 can be modeled using a unit step function as the input voltage
    • The resulting current and voltage across the capacitor can be found using Laplace transforms and step function properties
  • Other applications of discontinuous forcing functions and step functions include:
    • (modeling sudden changes in input or disturbances)
    • (representing pulses or square waves)
    • Mechanical systems (modeling impact forces or sudden changes in applied forces)

Key Terms to Review (14)

Continuity at a point: Continuity at a point means that a function behaves predictably around that point, allowing you to approach it from either direction and still arrive at the same value. This concept ensures that small changes in input produce small changes in output, making it easier to analyze functions and their behavior. When discussing functions like step functions and discontinuous forcing functions, understanding continuity helps identify where these functions change abruptly and how those changes affect differential equations.
Control Systems: Control systems are mathematical models that describe how inputs are transformed into outputs through a series of processes, allowing for the regulation of dynamic systems. They are crucial in engineering and physics as they help manage the behavior of systems by applying various input signals, such as step functions or discontinuous forcing functions, to achieve desired outcomes. By analyzing these systems, one can predict responses to inputs and design systems that behave in a controlled manner.
Dirac Delta Function: The Dirac delta function is a mathematical construct that represents an idealized point mass or point charge, acting as a distribution rather than a conventional function. It is defined such that it is zero everywhere except at zero and integrates to one over the entire real line. This unique property makes it extremely useful for modeling impulsive forces or inputs in various mathematical contexts, especially in relation to step functions and convolution in differential equations.
First-order ode with discontinuity: A first-order ordinary differential equation (ODE) with discontinuity is an equation that includes terms or conditions that change abruptly at certain points, leading to solutions that may not be continuous across those points. This type of ODE is often used to model systems where inputs or conditions experience sudden changes, such as in step functions or discontinuous forcing functions.
Heaviside Step Function: The Heaviside step function is a mathematical function that represents a discontinuous change at a specific point, typically denoted as H(t). It is defined as 0 for negative values and 1 for positive values, making it a useful tool in modeling situations where a sudden change occurs, such as in piecewise continuous functions or discontinuous forcing functions in differential equations. The function serves as a foundation for understanding how systems respond to abrupt inputs over time.
Jump Discontinuity: A jump discontinuity occurs in a function when there is a sudden change in the value of the function at a certain point, resulting in distinct left-hand and right-hand limits that do not equal each other. This characteristic is particularly relevant when analyzing step functions and discontinuous forcing functions, as it highlights how a function can abruptly change its behavior, impacting the solutions of differential equations that rely on these functions.
Non-homogeneous linear ode: A non-homogeneous linear ordinary differential equation is an equation of the form $$L[y] = f(t)$$, where $$L$$ is a linear differential operator, $$y$$ is the unknown function, and $$f(t)$$ is a non-homogeneous term or forcing function that is not equal to zero. This type of equation can describe systems where external forces or influences are present, leading to solutions that are combinations of complementary functions and particular solutions based on the nature of the forcing function.
Piecewise continuous function: A piecewise continuous function is a type of function that is defined by different expressions or formulas in different intervals of its domain. While it may have discontinuities at certain points, it remains continuous within each individual interval. This characteristic makes piecewise continuous functions particularly useful for modeling real-world situations where behavior changes at specific thresholds, often seen in scenarios involving step functions and discontinuous forcing functions.
Piecewise graph: A piecewise graph is a type of graph that is defined by multiple sub-functions, each applicable to a specific interval of the input variable. These graphs are often used to represent functions that have different behaviors in different regions, making them useful for modeling situations where a single formula does not apply. The distinct sections can include linear segments, step functions, or other types of curves that connect or transition at specified points.
Signal processing: Signal processing is the technique of analyzing, modifying, and synthesizing signals such as sound, images, and scientific measurements. It plays a crucial role in various applications by improving the quality of information, enabling better transmission, and extracting valuable insights. In the context of circuits and electrical applications, signal processing helps in filtering and amplifying signals for clearer communication, while with step functions and discontinuous forcing functions, it aids in understanding how systems respond to sudden changes in inputs.
Solution continuity: Solution continuity refers to the property of solutions to differential equations being continuous functions over their domains. This concept is crucial when dealing with step functions and discontinuous forcing functions, as it highlights how solutions behave at points of discontinuity and ensures that small changes in initial conditions or forcing functions lead to small changes in the solution.
Stability analysis: Stability analysis is a mathematical technique used to determine the behavior of a system as it approaches equilibrium over time. It helps assess whether small perturbations in the system's initial conditions lead to significant changes in the long-term behavior, thereby indicating if the system is stable or unstable. This concept is crucial in various fields, allowing us to predict how systems respond to changes or disturbances.
Step Graph: A step graph is a piecewise constant function that visually represents abrupt changes in value, resembling a series of horizontal lines and vertical jumps. This type of graph is especially useful in modeling situations where a variable experiences sudden shifts rather than gradual changes, making it ideal for illustrating discontinuous forcing functions in differential equations.
Unit step function: The unit step function, often denoted as $u(t)$, is a piecewise function that is zero for negative values of time and one for positive values, effectively 'turning on' at $t=0$. It is crucial in engineering and mathematics for modeling sudden changes in systems, such as a switch being turned on or off. The unit step function serves as a foundation for understanding discontinuous forcing functions and plays a significant role in the context of Laplace transforms, which are used to analyze linear time-invariant systems.
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