Laplace Transform

Laplace Transform

by Souad Ayadi -

Questions pour la discussion :

  1. Quels types de problèmes trouvez-vous plus faciles à résoudre avec la transformée de Laplace qu’avec les méthodes classiques ?

  2. Comment la transformée de Laplace facilite-t-elle le traitement des conditions initiales dans les équations différentielles ?

  3. Pouvez-vous donner un exemple concret dans la physique ou l’ingénierie où la transformée de Laplace simplifie la résolution d’un système dynamique ?

  4. Discutez des limitations de la transformée de Laplace. Y a-t-il des situations où elle n’est pas appropriée ?

Laplace Transform

by SMAIL ISLAM -
1. Laplace makes linear differential equations and systems with jumps easier.


2. It handles initial conditions automatically in the transformed equation.


3. Example: solving RC/RLC circuit responses in electronics.


4. Limits: not great for nonlinear systems or when the function doesn’t have a valid transform.

Laplace Transform

by KAWTHER KLALDI -
1. Problems Best Solved with Laplace Transform:

· Linear differential equations with constant coefficients.
· Analyzing electrical circuits, especially with multiple currents.
· Systems of differential equations in dynamics and control.

2. Simplifies Initial Conditions:

· It converts derivatives into an algebraic form that automatically includes the initial conditions, saving steps in the solution process.

3. Example:

· Mass-spring system: Converting the differential equation directly into an algebraic equation makes finding the solution much easier.

4. Main Limitations:

· Only works for linear systems.
· Not useful for time-varying systems or nonlinear systems.

Laplace Transform

by NIHAD HENNICHI -
1. Types of Problems:
_​Discontinuous inputs and Systems of equations.
2.Initial Conditions
​Unlike classical methods that find a general solution first and solve for constants later, Laplace:
​Incorporate initial conditions directly into the algebraic step.
​Uses the property: \mathcal{L}\{f'(t)\} = sF(s) -

Laplace Transform

by MERKIDENE MAROUA -
1 :For linear differential equations with constant coefficients
For dynamic systems (electrical, mechanical, control systems)
For discontinuous or piecewise inputs ➡️ It converts differential equations into algebraic equations
2:Initial conditions appear directly in the transformed equation
No need to compute integration constants ➡️ Initial conditions are handled automatically
3 :RLC electrical circuit ➡️ The differential equation becomes an algebraic equation in the �-domain, making the solution ea
4 :Not suitable for nonlinear equations
Less effective for variable coefficients
Not ideal for spatial boundary-value problems ➡️ The Laplace transform is not universal