DR S. & S.S. GHANDHY ENGINEENRING COLLEGE SUBJECT:- ADVANCE ENGINEERING MATHEMATICS SUBJECT CODE : Topic : Laplace Transform

History of the Transform  Euler began looking at integrals as solutions to differential equations in the mid 1700’s:  Lagrange took this a step further while working on probability density functions and looked at forms of the following equation:  Finally, in 1785, Laplace began using a transformation to solve equations of finite differences which eventually lead to the current transform

LAPLACE TRANSFORM o LAPLACE TRANSFORM CAN BE USE TO FIND OUT CONTINUOUS SIGNAL WHICH HAS UNDEFINED VARIATION o NORMALLY THE FUNCTION E.G. VELOCITIES AND ACCELERATION OF PARTICLES OR POINTS ON BODIES WHICH CAN EASILY FIND USING SIMPLE ODE

DEFINATION:- The Laplace Transform The Laplace Transform of a function, f(t), is defined as;

LAPLACETRANSFORMS PROPERTY f 1 (t)  f 2 (t) a f(t) e at f(t) f(t - T) f(t/a) F 1 (s) ± F 2 (s) a F(s) F(s-a) e Ts F(as) a F(as) Linearity multiplication shift Real shift Scaling 4. Laplace transforms

Properties of Laplace Transforms  Linearity  Shift  Multiplication by t n  Integration  Differentiation

Properties: Linearity Example :Proof :

Properties: Time Shift Example :Proof : let

Properties: S-plane (frequency) shift Example :Proof :

Properties: Multiplication by t n Example : Proof :

The “D” Operator 1. Differentiation shorthand 2. Integration shorthand if then if

Properties: Integrals Example : Proof : let If t=0, g(t)=0 forso slower than

Properties: Derivatives (this is the big one) Example :Proof : let

Inverse Laplace Transforms Background: To find the inverse Laplace transform we use transform pairs along with partial fraction expansion: F(s) can be written as; F(S)=P(S)/Q(S) Where P(s) & Q(s) are polynomials in the Laplace variable, s. We assume the order of Q(s) P(s), in order to be in proper form. If F(s) is not in proper form we use long division and divide Q(s) into P(s) until we get a remaining ratio of polynomials that are in proper form.

Inverse Laplace Transforms Background: There are three cases to consider in doing the partial fraction expansion of F(s). Case 1: F(s) has all non repeated simple roots. Case 2: F(s) has complex poles: Case 3: F(s) has repeated poles. (expanded)

Definition  Definition -- Partial fractions are several fractions whose sum equals a given fraction  Purpose -- Working with transforms requires breaking complex fractions into simpler fractions to allow use of tables of transforms

Partial Fraction Expansions  Expand into a term for each factor in the denominator.  Recombine RHS  Equate terms in s and constant terms. Solve.  Each term is in a form so that inverse Laplace transforms can be applied.

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