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Group 2 Bhadouria , Arjun Singh Glave , Theodore Dean Han, Zhe

Group 2 Bhadouria , Arjun Singh Glave , Theodore Dean Han, Zhe. Chapter 5. Laplace Transform Chapter 19. Wave Equation. Laplace Transform. Chapter 5. 5. Laplace Transform. 5.1. Introduction & Definition 5.2. Calculation of the Transform 5.3. Properties of the Transform

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Group 2 Bhadouria , Arjun Singh Glave , Theodore Dean Han, Zhe

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  1. Group 2Bhadouria, Arjun SinghGlave, Theodore DeanHan, Zhe Chapter 5. Laplace Transform Chapter 19. Wave Equation

  2. Laplace Transform Chapter 5

  3. 5. Laplace Transform • 5.1. Introduction & Definition • 5.2. Calculation of the Transform • 5.3. Properties of the Transform • 5.4. Application to the Solution of Differential Equations • 5.5. Discontinuous Forcing Functions; Heaviside Step Function • 5.6. Impulsive Forcing Functions; Dirac Impulse Function • 5.7. Additional Properties

  4. 5.1. Introduction & Definition • The Laplace transform is a widely used integral transform. Denoted , it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s). The Laplace transform has many important applications throughout the sciences. • The Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant systems. In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. http://en.wikipedia.org/wiki/Laplace_transform

  5. Basic idea http://faculty.mercer.edu/olivier_pd/documents/Ch2LaplaceTransforms.ppt

  6. Given a known function K(t,s), an integral transform of a function f is a relation of the form • The Laplace Transform of f is defined as where the kernel function is K(s,t) = e-st, a=0, b= . F (s) is the symbol for the Laplace transform, L is the Laplace transform operator, and f(t) is some function of time, t.

  7. How to solve problems Frequency Domain Time Domain Laplace transform Solve algebraic equation Inverse Laplace transform Douglas Wilhelm Harder, University of Waterloo.

  8. 5.2. Calculation of the Transform Since the Laplace Transform is defined by an improper integral, thus it must be checked whether the transform F(s) of a given function f(t) exists, that is whether the integral converges.

  9. EXAMPLE 2.1. Consider the following improper integral. We can evaluate this integral as follows: Note that if s = 0, then est= 1. Thus the following two cases hold:

  10. EXAMPLE 2.2. Consider the following improper integral. We can evaluate this integral using integration by parts: Since this limit diverges, so does the original integral.

  11. Exponential order Suppose that f is a function for which the following hold: (1) f is piecewise continuous on [0, b] for all b > 0. (2) | f(t) | Kect when t T, for constants c, K, M,with K, M > 0. A function f that satisfies the conditions specified above is said to to have exponential orderas t  EXAMPLE 2.3. Is f(t)=exp(4t)cost of exponential order? Solution: Yes. Therefore, K=10, c=4, and T=10 for instance EXAMPLE 2.4. Is f(t)=ln(t) of exponential order? Solution: Yes. Therefore, K=100, c=1, and T=10 for instance

  12. Piecewise Continuous If an interval [a, b] can be partitioned by a finite number of points a = t0 < t1 < … < tn = b such that Then the function f is piecewise continuous Or we can say f is piecewise continuous on [a, b] if it is continuous there except for a finite number of jump discontinuities. Picture from Paul's Online Math Notes

  13. EXAMPLE 2.5. Piecewise continuous function Consider the following piecewise-defined function f. • (a) From this definition of f, and from the graph of f below, we see that f is piecewise continuous on [0, 3]. • (b) From this definition of f, and from the graph of f below, we see that f is NOT piecewise continuous on [0, 3]. http://ebookbrowse.com/ch06-laplace-transform-ppt-d116265990

  14. THEOREM 2.1 Existence of the Laplace Transform Let f(t) satisfy these conditions: • f(t) is piecewise continuous on for every A>0, • f(t) is of exponential order as That is to say (1) f is piecewise continuous on [0, b] for all b > 0. (2) | f(t) | Kect when t T, for constants c, K, M,with K, M > 0. Then the Laplace Transform of f exists for s > c.

  15. Inverse Laplace transform operator By definition, the inverse Laplace transform operator, L-1, converts an s-domain function back to the corresponding time domain function: Importantly, both L and L-1 are linear operators. Thus,

  16. Examples of Calculation 2.6. 2.7 2.8

  17. Laplace transform table

  18. 5.3. Properties of the Transform • A various types of problems that can be treated with the Laplace transform include ordinary and partial differential equations as well as integral equations. THEOREM 3.0 The transform of an expression that is multiplied by a constant is the constant multiplied by the transform. That is:

  19. THEOREM 3.1 Linearity of the Transform • Suppose u and v are functions whose Laplace transforms exist for s > a1 and s > a2, respectively. • Then, for s greater than the maximum of a1 and a2, the Laplace transform of au(t) + bv(t) exists. That is, With Therefore

  20. EXAMPLE 3.1. f (t) = 5e-2t - 3sin(4t) for t  0. by linearity of the Laplace transform, and using results of Laplace transform table, the Laplace transform F(s) of f is:

  21. THEOREM 3.2 Linearity of the Inverse Transform For any U(s) and V(s) such that the inverse transforms For any constants a,b. Basic idea : Consider a general expression, Expand a complex expression for F(s) into simpler terms, each of which appears in the Laplace Transform table. Then you can take the L-1 of both sides of the equation to obtain f(t).

  22. EXAMPLE 3.2. Perform a partial fraction expansion (PFE) where coefficients and have to be determined. To find : Multiply both sides by s + 1 and let s = -1 To find : Multiply both sides by s + 4 and let s = -4 Therefore,

  23. THEOREM 3.3 Transform of the Derivative Let f(t) be continuous and f’(t) be piecewise continuous on 0≤t≤t。For every finite t。, and let f(t) be of exponential order as t  so that there are constants K, c, T such that | f(t) | Kect when t T. Then L{f’(t)} exists for all s>c. • This is a very important transform because derivatives appear in the ODEs we wish to solve. • Similarly, for higher order derivatives: where:

  24. Proof Deriving the Laplace transform of f (t) often requires integration by parts. However, this process can sometimes be avoided if the transform of the derivative is known: For example, if f (t ) = tthen f ‘(t ) = 1 and f (0) = 0 so that, since: That is: It has already been established that if: then: Now let so that: Therefore:

  25. For example, if: then: and Similarly: And so the pattern continues. EXAMPLE 3.3 Then substituting in: yields So:

  26. EXAMPLE 3.4 *Additional Section • COMMENT: Difference in The values are only different if f(t) is not continuous at t=0 Example of discontinuous function: u(t)

  27. EXAMPLE 3.5 Try to solve the differential equation: take the Laplace transform of both sides of the differential equation to yield: Resulting in: The right-hand side can be separated into its partial fractions to give: From the table of transforms it is then seen that: Thus,

  28. THEOREM 3.4 Laplace Convolution Theorem If both exist for s>c, then As Laplace convolution of f and g.

  29. Proof: We have, therefore

  30. EXAMPLE 3.6 If f(t)=exp(t), g(t)=t, then EXAMPLE 3.7 Schiff. Joel L. The Laplace transform: theory and applications. P92

  31. 5.4. Application to the Solution of Differential Equations • Laplace transforms play a key role in important engineering concepts and techniques. Examples: • Transfer functions • Frequency response • Control system design • Stability analysis • …

  32. Solution of ODEs by Laplace Transforms Procedure: • Take the L of both sides of the ODE. • Rearrange the resulting algebraic equation in the s domain to solve for the L of the output variable, e.g., F(s). • Perform a partial fraction expansion. • Use the L-1 to find f(t) from the expression for F(s).

  33. EXAMPLE 4.1. Solve • Equation with initial conditions • Laplace transform is linear • Apply derivative formula • Rearrange • Take the inverse

  34. EXAMPLE 4.2. resulted in the expression Take L-1 of both sides:

  35. EXAMPLE 4.3. Taking Laplace transforms of both sides of this equation gives: K.A. Stroud. Engineering Mathematics. P1107

  36. EXAMPLE 4.4. A mass m is suspended from the end of a vertical spring of constant k (force required to produce unit stretch). An external force f(t) acts on the mass as well as a resistive force proportional to the instantaneous velocity. Assuming that x is the displacement of the mass at time t and that the mass starts from rest at x=0, Set up a differential equation for the motion Find x at any time t Solution: (a)The resistive force is given by –βdx/dt. The restoring force is given by –kx. Then by Newton’s law,

  37. EXAMPLE 4.4. (b) Taking the Laplace transform of (1), using we obtain So that on using (2) Where There are three cases to be considered.

  38. EXAMPLE 4.4. • Case 1, R>0. In this case let We have Then we find from (3) • Case 2, R=0. In this case thus

  39. EXAMPLE 4.4. • Case 3. R<0, In this case let We have Schaum’s Outline of Theory and Problems of Advanced Mathematics for Engineers and Scientists. P115, Problem 4.46

  40. 5.5. Discontinuous Forcing Functions; Heaviside Step Function The unit step function is widely used. It is defined as: Because the step function is a special case of a “constant”, it follows

  41. More generally, It is possible to express various discontinuous functions in terms of the unit step function. Therefore, or

  42. EXAMPLE 5.1 Schiff. Joel L. The Laplace transform: theory and applications. P92

  43. EXAMPLE 5.2 Determine L{g(t)} for Schiff. Joel L. The Laplace transform: theory and applications. P92

  44. 5.6. Impulsive Forcing Functions; Dirac Impulse Function Pictorially, the unit impulse appears as follows: Mathematically: Picture from web.utk.edu

  45. We can prove that Where further thus is known as Dirac delta function, or unit impulse function

  46. The Laplace transform of a unit impulse: if we let f(t) = (t) and take the Laplace *Additional Section * Rectangular Pulse Function

  47. EXAMPLE 6.1 • Some useful properties of • Dirac Impulse Function

  48. 5.7. Additional Properties THEOREM 7.1S-plane (frequency) shift If exists for s>s0, then for any real constant a, for s+a>s0, or equivalently, Proof : EXAMPLE 7.1. EXAMPLE 7.2.

  49. THEOREM 7.2Time Shift If exists for s>s0, then for any real constant a>0, for s>s0, or, equivalently, Proof : EXAMPLE 7.3.

  50. THEOREM 7.3Multiplication by 1/s (Integrals) If exists for s>s0, then for s>max{0,s0}, or, equivalently, Proof : Notice: EXAMPLE 7.4.

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