Laplace and Fourier Transforms Essay Example for Free

Laplace and Fourier Transforms Essay Laplace and Fourier Transforms are operator which when applied on a function, lead to another function in a different variable. These transforms are very useful in solving many problems in different branches of engineering. What is essentially done is that an engineering problem is modeled as a mathematical equations and these equations are generally ordinary and / partial differential equations with boundary conditions. These equations are difficult to be solved by analytical methods, however, these equations can be converted into algebraic equations by using Laplace or Fourier Transforms and then it becomes easy to solve these equations. Once these subsidiary algebraic equations are solved, the solution of these algebraic equations is transformed back and thus the solution of the engineering problem is obtained. Thus it can be said that there are following three steps involved in solving differential equations with boundary conditions. (1) Transforming the differential equations with boundary conditions into simple algebraic equations (subsidiary equations). (2) Solution of the subsidiary algebraic equations by algebraic manipulations. (3) Transforming back the result(s) of subsidiary algebraic equations to obtain the solutions. Therefore, it can be seen that the problem of solving a differential equation is simplified into solving of algebraic equations by use of Laplace or Fourier transforms and needless to say that solving an algebraic equation is much simpler than solving a differential equation. Therefore, it is not unusual that Laplace and Fourier transforms find extensive application is solving engineering problems in mechanical as well as electrical domain where the driving force has discontinuities, is impulsive and is periodic function of complex shape. Besides, this method solves the problem directly. Initial value problems are solved without determining the general solution first. Also, nonhomogeneous equations are solved without solving the homogeneous equations first. These transformations are useful in solving not only the ordinary differential equations but in solving the partial differential equations as well. In this paper, the definition, properties and applications of Laplace and Fourier transforms is discussed in detail. Laplace Transform Let us consider a function f = f(t), which is defined for all t 0. When this function is multiplied by e-st and the product is integrated from t = 0 to t = ? and if this integral exists, then this integral will be a function of s, let us say it is F(s); then F(s) is Laplace transform of f(t).