Solving Integral Equation w/ Laplace Transform - Abdullah

In summary, the conversation discusses the use of Laplace Transform in solving a convolution integral and finding the function $f(t)$ that satisfies the given equation. The Laplace Transform is a powerful tool used in various fields of science and engineering, and understanding it is crucial in solving real-world problems.
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We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $.

In this case, $\displaystyle g\left( t - u \right) = \sin{ \left[ 4\left( t - u \right) \right] } \implies g\left( t \right) = \sin{ \left( 4\,t \right) } \implies G\left( s \right) = \frac{4}{s^2 + 16} $.

Taking the Laplace Transform of the equation gives

$\displaystyle \begin{align*} F\left( s \right) &= 5 \left( \frac{1}{s} \right) + 12 \left( \frac{2}{s^3} \right) + 4\,F\left( s \right) \left( \frac{4}{s^2 + 16} \right) \\
F\left( s \right) &= \frac{5}{s} + \frac{24}{s^3} + \left( \frac{16}{s^2 + 16} \right) F\left( s \right) \\
F\left( s \right) - \left( \frac{16}{s^2 + 16} \right) F\left( s \right) &= \frac{5}{s} + \frac{24}{s^3} \\
\left( 1 - \frac{16}{s^2 + 16} \right) F\left( s \right) &= \frac{5}{s} + \frac{24}{s^3} \\
\left( \frac{s^2}{s^2 + 16} \right) F\left( s \right) &= \frac{5\,s^2 + 24}{s^3} \\
F\left( s \right) &= \frac{\left( s^2 + 16 \right) \left( 5\,s^2 + 24 \right) }{s^5} \\
F\left( s \right) &= \frac{5\,s^4 + 104\,s^2 + 384}{s^5} \\
F\left( s \right) &= \frac{5}{s} + \frac{104}{s^3} + \frac{384}{s^5} \\
F\left( s \right) &= 5\left( \frac{1}{s} \right) + 52 \left( \frac{2}{s^3} \right) + 16 \left( \frac{4!}{s^5} \right) \\
\\
f\left( t \right) &= 5 + 52\,t^2 + 16\,t^4
\end{align*} $
 

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I would like to point out that the Laplace Transform is a powerful tool in solving differential equations and finding solutions to complex mathematical problems. In this case, the Laplace Transform of the convolution integral helps us to find the function $f(t)$ that satisfies the given equation. This method is widely used in various fields of science and engineering, such as signal processing, control systems, and quantum mechanics. It is important to understand and utilize the Laplace Transform in order to solve real-world problems and advance our understanding of the physical world.
 

FAQ: Solving Integral Equation w/ Laplace Transform - Abdullah

What is an integral equation?

An integral equation is a mathematical equation that involves an unknown function within an integral. It is typically used to describe relationships between a function and its integral.

What is the Laplace transform?

The Laplace transform is a mathematical tool used to solve differential equations. It transforms a function of time into a function of a complex variable, making it easier to solve certain types of equations.

How does the Laplace transform help in solving integral equations?

The Laplace transform can be used to convert an integral equation into an algebraic equation, which is often easier to solve. This is because the transform simplifies the equation and eliminates the need for integration.

What are the steps for solving an integral equation with the Laplace transform?

The general steps for solving an integral equation with the Laplace transform are as follows:

  1. Take the Laplace transform of both sides of the equation.
  2. Solve for the transformed function.
  3. Apply the inverse Laplace transform to find the solution in terms of the original function.

What are some applications of solving integral equations with the Laplace transform?

The Laplace transform is commonly used in engineering and physics to solve problems involving differential equations. It is also used in signal processing, control systems, and circuit analysis.

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