13.9 - Examples
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Evaluating Convolution Integral
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Today, we will evaluate the convolution of the functions f(t) = t and g(t) = e^{-t}. Can anyone remind us what convolution means in this context?
Is it when we combine the two functions in some integral form?
Exactly! We define convolution as (f * g)(t) = ∫[0 to t] f(τ) g(t - τ) dτ. Let's apply this definition to our functions.
So, we need to integrate τ * e^{-(t-τ)}?
Correct! Can you set up the integral for me?
Sure, it will be ∫[0 to t] τ * e^{-(t-τ)} dτ.
Well done! Now, using integration by parts, let’s find the solution.
What happens when we compute this integral? What will it yield?
After performing integration by parts, we find that (f * g) results in (t - 1) + e^{-t}.
To summarize, convolution allows us to blend functions while retaining their unique characteristics.
Solving Differential Equations via Convolution
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Now, let's turn our attention to a differential equation: y'' + y = sin(t), with initial conditions y(0) = 0 and y'(0) = 0. How do we start solving this?
Should we take the Laplace transform to solve it?
Absolutely! By applying the Laplace transform, we can handle the equation more effectively. What do we obtain for Y(s)?
We get Y(s) = F(s)/(s^2 + 1) due to the initial conditions.
Great! Now, how can we apply convolution here?
By finding the inverse Laplace transform, we can use the convolution theorem to express y(t) as the convolution of f(t) with h(t).
Correct! This yields y(t) = t * sin(t) after we solve it using known transforms. Does anyone see how this connects back to our use of convolution?
It shows us how convolution helps solve differential equations effectively and generates the system response.
Exactly! In conclusion, convolution provides a powerful framework in both analysis and applications in engineering.
Introduction & Overview
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Quick Overview
Standard
The section contains two examples focused on evaluating convolution integrals and solving differential equations using the convolution theorem. The first example involves the convolution of linear and exponential functions, while the second relates to solving a second-order differential equation through convolution with an impulse response.
Detailed
Detailed Summary
In this section, we delve into two specific examples illuminating the concept and applications of convolution in practical scenarios:
- Example 1 explores the convolution of two functions: a linear function, represented as f(t) = t, and an exponential decay function, g(t) = e^{-t}. The solution process illustrates the direct application of the convolution integral formula, emphasizing integration by parts as a technique for evaluating the integral. The result showcases how convolution allows us to blend these functions into a new function, capturing their joint behavior.
- Example 2 transitions to a practical scenario involving a differential equation defined as y'' + y = sin(t) with initial conditions y(0) = 0 and y'(0) = 0. By transforming this differential equation using the Laplace transform, we find the expression for Y(s) and use the convolution concept to derive y(t). This step illustrates how convolution serves as an effective tool for solving differential equations in systems engineering.
The significance of these examples lies in their demonstration of applying the convolution theorem both in pure function analysis and in solving real-world engineering problems.
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Example 1: Convolution of Functions
Chapter 1 of 2
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Chapter Content
Evaluate the convolution (f ∗ g)(t), where:
f(t) = t, g(t) = e^{-t}
Solution:
\[
(f ∗ g)(t) = \int_0^t \tau \, e^{-(t - \tau)} d\tau = e^{-t} \int_0^t \tau e^{\tau} d\tau
\]
Using integration by parts:
\[
= e^{-t}\left[ (t - 1)e^{t} + 1 \right] = (t - 1) + e^{-t}
\]
Detailed Explanation
In this example, we are asked to evaluate the convolution of two functions: f(t) = t and g(t) = e^{-t}. The convolution is defined by the integral that combines these two functions. First, we express the convolution using the integral definition. Then, we simplify it using integration by parts, which helps us solve complex integrals more easily. The final solution shows how the combination of these functions behaves over time.
Examples & Analogies
Think of two different types of materials being mixed together — like flour and water to make dough. The way the final product depends on the characteristics of both ingredients can be likened to how the convolution describes the combined effect of the two functions over time.
Example 2: Solving a Differential Equation Using Convolution
Chapter 2 of 2
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Chapter Content
Solve y'' + y = sin(t), with y(0) = 0, y'(0) = 0, using convolution.
Solution:
Take Laplace Transform:
\[
\frac{1}{s^2}Y(s) + Y(s) = \frac{1}{s^2 + 1} \Rightarrow Y(s) = \frac{1}{(s^2 + 1)^2}
\]
Now find y(t) = L^{-1} \left\{ \frac{1}{(s^2 + 1)^2} \right\}.
Using known inverse transform:
\[
L^{-1}\left\{ \frac{1}{(s^2 + 1)^2} \right\} = t \sin(t)
\]
Thus, y(t) = t \sin(t)
Detailed Explanation
In this example, we solve a second-order linear differential equation using convolution techniques. We start by applying the Laplace Transform to our differential equation to simplify it into algebraic equations in the s-domain. We then solve for Y(s) and finally find its inverse to get the time-domain solution, y(t). Here, y(t) describes how the system behaves over time when acted on by the input sin(t). This shows how convolution is used to derive outputs from known inputs systematically.
Examples & Analogies
Imagine a swing (the system) that responds to someone pushing it (the input). The swing's motion over time depends not only on the force of the push but also on how it was already moving beforehand. By analyzing these elements using convolution, we can predict how the swing will move after several pushes over time.
Key Concepts
-
Convolution: The operation of combining functions to produce a new function that represents how the shape of one function is altered by the other.
-
Laplace Transform: A tool for converting functions into a different domain, making it easier to manipulate linear systems.
-
Differential Equations: Mathematical equations involving functions and their derivatives, crucial for describing dynamic systems.
Examples & Applications
Example 1: Evaluating the convolution of f(t) = t and g(t) = e^{-t} illustrates the direct computation of the integral.
Example 2: Solving the differential equation y'' + y = sin(t) using convolution demonstrates how to apply the Laplace transform effectively.
Memory Aids
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Rhymes
To convolve is divine, blend functions and align.
Stories
Once upon a time, two functions met at an integral party and created a new shape called convolution, which revealed their blended traits.
Memory Tools
Remember 'CIF' for Convolution: Combine Integrate Function.
Acronyms
Use 'LDC' to recall Laplace, Differential, Convolution.
Flash Cards
Glossary
- Convolution
A mathematical operation that combines two functions to produce a third function, reflecting the way one function modifies the shape of another.
- Laplace Transform
A technique used to transform a function of time into a function of a complex variable, facilitating the analysis of linear time-invariant systems.
- Integration by Parts
A mathematical method for integrating products of functions, based on the product rule of differentiation.
- Impulse Response
The output of a system when presented with a brief input signal, significant in defining system behavior.
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