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Laplace Transforms

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  1. Laplace Transforms Math 2306 Dr. Dillon SPSU Mathematics Department

  2. What Are Laplace Transforms?

  3. A Laplace transform is a type of integral transform. Plug one function in Get another function out The new function is in a different domain.

  4. When is the Laplace transform of Write

  5. A Laplace transform is an example of an improper integral : one of its limits is infinite. Define

  6. A Calculation Let This is called the unit step function or the Heaviside function. It’s handy for describing functions that turn on and off.

  7. 1 c t The Heaviside Function

  8. Remember that is zero until Calculating the Laplace transform of the Heaviside function is almost trivial. then it’s one.

  9. To What End Does One Use Laplace Transforms?

  10. We can use Laplace transforms to turn an initial value problem Solve for y(t) into an algebraic problem Solve for Y(s)

  11. Laplace transforms are particularly effective on differential equations with forcing functions that are piecewise, like the Heaviside function, and other functions that turn on and off. 1 1 t Asawtooth function

  12. Laplace transform I.V.P. Algebraic Eqn

  13. Then What?

  14. If you solve the algebraic equation and find the inverse Laplace transform of the solution, Y(s), you have the solution to the I.V.P.

  15. Inverse Laplace transform Algebraic Expression Soln. to IVP

  16. The inverse Laplace transform of is

  17. Thus is the solution to the I.V.P.

  18. How Do You Transform an Differential Equation?

  19. You need several nice properties of Laplace transforms that may not be readily apparent. First, Laplace transforms, and inverse transforms, are linear : for functions f(t), g(t), constant c, and transforms F(s), G(s).

  20. there is a very simple relationship between the Laplace transform of a given function and the Laplace transform of that function’s derivative. Second, These show when we apply differentiation by parts to the integral defining the transform.

  21. Now we know there are rules that let us determine the Laplace transform of an initial value problem, but...

  22. How Do You Find Inverse Laplace Transforms?

  23. First you must know that Laplace transforms are one-to-one on continuous functions. In symbols when f and g are continuous. That means that Laplace transforms are invertible.

  24. If then where Inverse Laplace Transforms

  25. An inverse Laplace transform is an improper contour integral, a creature from the world of complex variables. That’s why you don’t see them naked very often. You usually just see what they yield, the output. In practice, Laplace transforms and inverse Laplace transforms are obtained using tables and computer algebra systems.

  26. Why Use Such Dangerous Machines?

  27. Don’t use them... unless you really have to.

  28. When Might You Have To?

  29. When your forcing function is a piecewise, periodic function, like the sawtooth function... Or when your forcing function is an impulse, like an electrical surge.

  30. Impulse? An impulse is the effect of a force that acts over a very short time interval. A lightning strike creates an electrical impulse. The force of a major leaguer’s bat striking a baseball creates a mechanical impulse. Engineers and physicists use the Dirac delta function to model impulses.

  31. The Dirac Delta Function This so-called quasi-function was created by P.A.M. Dirac, the inventor of quantum mechanics. People use this thing all the time. You need to be familiar with it.

  32. The Laplace Transform of theDirac Delta Function

  33. Beware!

  34. Laplace transforms have limited appeal. You cannot use them to find general solutions to differential equations. You cannot use them on initial value problems with initial conditions different from Initial conditions at a point other than zero will not do.

  35. What Do We Expect You to Be Able to Do?

  36. Know the definition of the Laplace transform • Know the properties of the Laplace transform • Know that the inverse Laplace transform is an improper integral • Know when you should use a Laplace transform on a differential equation • Know when you should not use a Laplace transform on a differential equation

  37. Be able to solve IVPs using Laplace transforms… When Appropriate