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LAPLACE TRANSFORMS

BY

B.SUGUNA SELVARANI

ASSISTANT PROFESSOR

DEPARTMENT OF MATHEMATICS

CPA COLLEGE,BODI

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Transformation

  • A transformation is an operation which converts an mathematical expression to a different but equivalent form.
  • For eg:

Logarithm – converts multiplication to addition and division to subtraction

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Laplace Transform

  • TRANSFORMS -- A MATHEMATICAL CONVERSION FROM ONE WAY OF THINKING TO ANOTHER TO MAKE A PROBLEM EASIER TO SOLVE

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transform

solution

in transform

way of

thinking

inverse

transform

problem in original way of thinking

solution in original way of thinking

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�The French Newton Piere-Simon L � Pierre simon de Laplace�

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  • French mathematician

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  • Developed mathematics in astronomy, physics, and statistics

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  • Began his work in calculus which led to the Laplace Transform

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  • Used this transformation in theory of probabilities.

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� Uses of Laplace Transforms

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  • Find solution to differential equation (both ordinary and partial)
  • Relationship to Fourier Transform allows easy way to characterize systems
  • Useful with multiple processes in system

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Laplace transforms- Definition

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  1. The Laplace transform ℒ, of a function f(t) for t > 0 is defined by the following integral over 0 to ∞
  2. :t is real, s is complex
  3. Note “transform”: f(t) → F(s), where t is integrated and s is variable
  4. Conversely F(s) → f(t), t is variable and s is integrated
  5. Assumes f(t) = 0 for all t < 0

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Conditions for existence

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Some results of Laplace Transforms

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Example

  •  

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THE INVERSE LAPLACE TRANSFORM

  • Wide variety of function can be transformed
  • Inverse Transform

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F(s) is the Laplace transform of f (t) then f (t) is the inverse Laplace transform of F (s).

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THE INVERSE LAPLACE TRANSFORM

if f (t) = 4 then:

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So, if

Then the inverse Laplace transform of F (s) is,

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PROPERTIES OF LAPLACE TRANSFORM

(1) The Laplace transform and its inverse are linear transforms. That is,

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The transform of a sum (or difference) of expressions is the sum

(or difference) of the transforms. That is,

(2) The transform of an expression that is multiplied by a constant is

the constant multiplied by the transform. That is,

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Thank you