Laplace transform is an operational tool for solving constant conceits linear differential equations. The process of solution consists of three main steps:
The given “hard" problem is transformed into a simple" equation.
This simple equation is solved by purely algebraic manipulations.
The solution of the simple equation is transformed back to obtain the solution of the given problem.
Suppose that f is a real- or complex-valued function of the (time) variable t > 0 and (s) is a real or complex parameter. We define the Laplace transform of f as:
The symbol L is the Laplace transformation, which acts on functions f = f (t) and generates a new function, F(s) = L f (t) whenever the limit exists (as a finite number). When it does, the integral is said to converge. If the limit does not exist, the integral is said to diverge and there is no Laplace transform defined for f .
The key motivation for learning about Laplace transforms is that the process of solving
an ODE is simplified to an algebraic problem (and transformations). This type of
mathematics that converts problems of calculus to algebraic problems is known as
operational calculus. The Laplace transform method has two main advantages :
I. Problems are solved more directly: Initial value problems are solved without first
determining a general solution. Nonhomogenous ODEs are solved without first solving
the corresponding homogeneous ODE.
II. More importantly, the use of the unit step function
and Dirac’s delta make the method particularly powerful for problems with
inputs (driving forces) that have discontinuities or represent short impulses or complicated
periodic functions
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
الرجوع الى لوحة التحكم
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