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Applications of Laplace Transforms

Applications of Laplace Transforms

Prepared By : Name : Ketaki Pattani Enroll. No. : 130210107039College : GEC, Bhavnagar

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.

Laplace transforms

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

Defination:

Diracs Delta FunctionMathematically impulsive forces are idealized by impulsive functions which is a discontinuous functions whose total value is concentrated at one point.The Impulse function having magnitude 1 is known as Dirac Delta function or Unit Impulse function.

Example:

Example:

11

A sawtooth functiontLaplace transforms are particularly effectiveon differential equations with forcing functionsthat are piecewise, like the Heaviside function,and other functions that turn on and off.XY

Top Hat functionTop Hat function is defined as follows:

H(t-a)-H(t-b) =1 ; a < = t < b =0 ; otherwise

Top Hat FunctionUsing this the Top Hat Function may be expressed as:F(t) = f1(t) [H(t) H(t-t1)] + f2(t)[H(t-t1) H(t-t2)] + f3(t)[H(t1-t2)] = f1(t)H(t) + [f2(t) f1(t)]H(t-t1) + [f3(t) f2(t)]H(t-t2)which is same as that of Heaviside Step Function.

Example of Top Hat Function: