Laplace Transform Example

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26th May 2017 Mathematics Reference this

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Abstract:

This paper describes the Laplace transform used in solving the differential equation and the comparison with the other usual methods of solving the differential equation. The method of Laplace transform has the advantage of directly giving the solution of differential equation with given boundary values without the necessity of first finding the general solution and then evaluating from it the arbitrary constants. Moreover the ready formulas of the Laplace reduce the problem of solving differential equations to mere algebraic manipulation.

Introduction:

Differential equation is an equation which involves differential coefficients or

differentials. It may be defined in a more refined way as an equation that defines a Relationship between a function and one or more derivatives of that function. Let y be some function of the independent variable t. Then following are some differential equations relating y to one or more of its derivatives.

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The equation states that the first derivative of the function y equals the product of and the function y itself. An additional, implicit statement in this differential equation is that the stated relationship holds only for all t for which both the function and its first derivative are defined. Some other differential equations:

Differential equations arise from many problems in oscillations of mechanical and electrical systems, bending of beams conduction of heat, velocity of chemical reactions etc., and as such play a very important role in all modern scientific and engineering studies. There are many ways of solving the

differential equation and the most effective way is to use the Laplace equation because it provides the easy path to solve the differential equation without involving any long process of finding out the complementary function and particular integral.

Solution of differential equation:

A solution of a differential equation is a relation between the variables which satisfy the given differential equation. A first order homogeneous differential equation involves only the first derivative of a function and the function itself, with constants only as multipliers. The equation is of the form

and can be solved by the substitutio

The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives

The general solution to a differential equation must satisfy both the homogeneous and non-homogeneous equations. It is the nature of the homogeneous solution that the equation gives a zero value. If you find a particular solution to the non-homogeneous equation, you can add the homogeneous solution to that solution and it will still be a solution since its net result will be to add zero. This does not mean that the homogeneous solution adds no meaning to the picture; the homogeneous part of the solution for a physical situation helps in the understanding of the physical system. A solution can be formed as the sum of the homogeneous and non-homogeneous solutions, and it will have a number of arbitrary (undetermined) constants. Such a solution is called the general solution to the differential equation. For application to a physical problem, the constants must be determined by forcing the solution to fit physical boundary conditions. Once a general solution is formed and then forced to fit the physical boundary conditions, one can be confident that it is the unique solution to the problem, as gauranteed by the uniqueness theorem.

Uniqueness theorem:

For the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem. This kind of approach is made possible by the fact that there is one and only one solution to the differential equation, i.e., the solution is unique.

Stated in terms of a first order differential equation, if the problem meets the condition such that f(x,y) and the derivative of y is continuous in a given rectangle of (x,y) values, then there is one and only one solution to the equation which will meet the boundary conditions.

Laplace in solving differential equation:

The Laplace transform method of solving differential equations yields particular solutions without the necessity of first finding the general solution and then evaluating the arbitrary constants. This method is in general shorter than the above mentioned methods and is specially used for solving the linear differential equation with constant coefficients.

Working procedure:

  • Take the Laplace transform of both sides of the differential equation using the formulas of Laplace and the given initial conditions.
  • Transpose the terms with minus sign to right.
  • Divide by the coefficient of y, getting y as a known function of s.
  • Resolve this function of s into partial fractions and take the inverse transform of both sides.

This gives y as a function of t which is the desired solution satisfying the given conditions.

Solving the algebraic equation in the mapped space

Back transformation of the solution into the original space.

Figure 1: Schema for solving differential equations using the Laplace transformation

Some of the examples which demonstrate the use of the Laplace in solving the differential equation are as follows:

Example no.1 Consider the differential equation

with the initial conditions .

Proceeding using the steps given above one has

Step 1:

Step 2:

Step 3: The complex function must be decomposed into partial fractions in order to use the tables of correspondences. This gives

By using the formulas of the inverse laplace transform we can convert these frequency domains back in the time domain and hence get the desired result as ,

Another example of the laplace involving trigonometric function is

We want to solve

with initial conditions f(0) = 0 and f ′(0)=0.

We note that

and we get

So this is equivalent to

We deduce

So we apply the Laplace inverse transform and get

Periodic functions:

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity.

A function f is said to be periodic if

for all values of x. The constant P is called the period, and is required to be nonzero. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods.

For example, the sine function is periodic with period 2π, since

for all values of x. This function repeats on intervals of length 2π (see the graph to the right).

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.A function that is not periodic is called aperiodic.

Laplace transform of periodic functions:

If function f(t) is periodic with period p > 0, so that f(t + p) = f(t), and f1(t) is one period (i.e. one cycle) of the function, then the Laplace of this periodic function is given by

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The basic concept of the formula is the Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by (1 − e-sp).Laplace transform of some of the common functions like the graph given below is given by

Fig no3:continous graphical function

From the graph, we see that the first period is given by:

and that the period p = 2.

Now

So

Hence, the Laplace transform of the periodic function, f(t) is given by:

Other continuous wave forms and there Laplace transforms are

This wave is an example of the full wave rectification which is obtained by the rectifier used in the electronic instruments.

Here,

and the period, p = π.

So the Laplace Transform of the periodic function is given by:

Conclusion:

The knowledge of Laplace transform has in recent years become an essential part of mathematical background required of engineers and scientists. This is because the transform method an easy and effective means for the solution of many problems arising in engineering. The method of laplace transformation is proving to be the most effective and easy way of solving differential equations and hence it is replacing other methods of solution of the differential equation. The most frequent function encompassed in electronics engineering is continuous function and most of the functions are in the time domain and we need to convert them in the frequency domain, this operation is performed excellently by the Laplace transform and hence its application is further enlarged using it in the solution of the continuous functions.

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