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Laplace Transform

Course Material

Gyeongsang National University

Dept. of Information & Communication Engineering

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Laplace Transform

• Linear differential equation (initial value problem) with constant coefficients can be solved by Laplace transform

Differential Equation Algebraic Equation Differential Equation Solution Algebraic Equation Solution Laplace transform Inverse Laplace transform

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6.1 Laplace transform. Linearity.

s-Shifting

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Definition of Laplace Transform

• 𝑓(𝑡) is a function of time

• For 𝑡 ≥ 0, the Laplace transform of 𝑓(𝑡) is defined by following equation:

• 𝐹 𝑠 is a function of 𝑠

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Examples of Laplace Transform

1) 𝑓 𝑡 = 1, 𝑡 ≥ 0

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Example

1) 𝑡

3

2) sin 4𝑡

3) 𝑒

−2𝑡

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Linearity

• Linearity

• Example) 1) 3 + 2𝑡

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Shifting Theorem

• s-shifting theorem

• Example) When the Laplace transform of 𝑓(𝑡) is 𝐹 𝑠 = 2𝑠+1

𝑠(𝑠+1), find

the Laplace transform of the following functions. 1) 𝑒−2𝑡𝑓(𝑡)

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6.2 Transforms of Derivatives and

Integrals. ODE

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Laplace Transform of Derivatives

• Laplace transform of derivatives

• Example) When the Laplace transform of y(𝑡) is Y(𝑠) and, y 0 = 3, y′ 0 = 1, find the Laplace transform of followings.

1) y′ 2) y′′

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Inverse Laplace Transform

• When 𝐹(𝑠) is the Laplace transform of 𝑓(𝑡), 𝑓(𝑡) is an inverse Laplace transform of 𝐹(𝑠), and expressed as follows:

• Example) 1) 2

𝑠3 𝑠

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Inverse Laplace Transform

• Finding an inverse Laplace transform using a partial fraction

– Expressed in partial fraction and then find an inverse Laplace transform of the partial fraction

• Example) 1) 4𝑠−1

𝑠2−𝑠

2) 6𝑠+8

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Solving Initial Value Problem

• Solve following initial value problem by using Laplace transform.

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Solving Initial Value Problem

• Solve following initial value problem by using Laplace transform.

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Solving Initial Value Problem

• Solve following initial value problem by using Laplace transform.

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Example

• Solve following initial value problem by using Laplace transform.

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참조

관련 문서

double BuoyancyDisplacementCondition(); // 부력-중량 평형 조건을 계산하는 함수 double CCRequirementCondition(); // 화물창 요구 조건을 계산하는 함수

Step 1 Setting up a mathematical model (a differential equation) of the physical process.. By the physical law : The initial condition : Step

Step 2 The subsidiary equation is solved by purely algebraic manipulations3. Step 3 The solution in Step 2 is transformed back, resulting in the solution of

Substitute the series with undetermined coefficients and its derivatives.. [Reference] 5.4 Bessel’s Equation.. 11.6 Orthogonal Series.. 11.6 Orthogonal Series..

Bessel Series ((극좌표에서의 극좌표에서의 라플라스 라플라스 연산자 연산자 원형박막 원형박막 푸리에 푸리에 베셀 베셀 급수 급수)) Bessel Series.

12.6 Heat Equation: Solution by Fourier Integrals and Transforms 12.7 Modeling: Membrane, Two-Dimensional Wave Equation..

• Many physical behavior can be expressed as differential equation (containing derivatives of unknown function)..

2.2 Homogeneous Linear ODEs with Constant Coefficients 2.3 Differential