How Do You Solve This Complex Laplace Inverse Problem?

  • Context: Graduate 
  • Thread starter Thread starter expi123
  • Start date Start date
  • Tags Tags
    Inverse Laplace
Click For Summary

Discussion Overview

The discussion centers around solving a complex Laplace inverse problem, specifically focusing on the inverse of the equation s+1/(s^3 + s + 1). Participants explore methods for factoring the denominator, finding roots, and relating the Laplace transform back to a differential equation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant struggles with factoring the cubic denominator and seeks hints for solving the inverse Laplace problem.
  • Another participant notes that the cubic function has one real root and two complex roots, suggesting a factorization approach but expresses uncertainty about how to proceed.
  • A different participant proposes using Cardano's method to find the exact roots and provides numerical approximations for the roots.
  • One suggestion involves factoring out the real root and performing partial fraction expansion on the denominator.
  • Another participant shifts the discussion to a different Laplace transform problem, asking how to relate the result back to the original differential equation.
  • A response questions the necessity of reconstructing the differential equation and notes that the highest power of s indicates a maximum order of three for the differential equation.
  • Further discussion raises the challenge of approximating the differential equation due to missing information about the original driving function and initial conditions.

Areas of Agreement / Disagreement

Participants express differing views on the methods for solving the Laplace inverse problem, with no consensus on the best approach or the feasibility of reconstructing the original differential equation.

Contextual Notes

Participants highlight limitations such as the inability to factor the cubic polynomial easily, the dependence on initial conditions, and the challenges posed by the lack of information regarding the original driving function.

Who May Find This Useful

This discussion may be of interest to students and practitioners dealing with Laplace transforms, differential equations, and those exploring methods for solving complex mathematical problems in engineering and physics contexts.

expi123
Messages
2
Reaction score
0
I am trying to do the inverse of the foll. eqn. to no avail:

s+1/(s^3 + s + 1)

I cannot factorise the denominator.
My only alternative was to use the definition of Laplace and try to integrate the equation. Still I could not. Can anyone provide some hints?
 
Physics news on Phys.org
That function in the denominator is cubic so it atleast has one real root. After graphing it, I can see that it has 1 real root. Hence, it has two complex roots. My guess would be to write that polynomial as a product of one real factor and two complex factors. I have no idea how to do that though.

It is a tough one though! I hope someone else in this forum is able to help you.
 
Last edited:
Actually, you can use the Cardano's method to find the exact roots.

Numerically, though, the roots are:
.341 + 1.16j; .341-1.16j; and -.682

Now you can treat these complex numbers just like any other real numbers and do the partial fraction expansion.
 
Factor out the real root, then preform partial fractions on the denominator, split up your expression into two, and solve.

Regards,

Nenad
 
Thanks, I just could not find a standard method to factorise the denominator.
 
Hi, I need some help!

I performed inverse laplace on (s^2+5s+6)/(s^3-s^2+2) and got 3/5*e^t*cos(t)+29/5*e^t*sin(t)+2/5*e^-t. My question is how do you take it back to the differential equation and how do you tell what type of differential equation it is?
 
hmm why would you want to?

if you used a laplace transform than it must have been a linear differential equation, combined with an initial value problem.

also because the highest power of s is s^3 I'd say that it couldn't have been more than third order. (although there might be special circumstances that I'm missing.)

I don't believe there is enough information to easily reconstruct the differential equation.
 
Is there a way to approximate the differential equation?
 
none that I know of, part of your problem is that you don't know what the original driving function was, or the initial conditions.

because a laplace transform depends on both of these things its impossible to determine what the differential equation was.

but once again why would you want to reconstruct the differential equation?
 

Similar threads

Graduate Inverse Laplace transform of a piecewise defined function
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad Solving PDE with Laplace Transforms & Inverse Lookup
  • · Replies 1 ·
Replies
1
Views
2K
MHB Emad's question via email about Inverse Laplace Transform
  • · Replies 1 ·
Replies
1
Views
5K
MHB Solve Laplace equation on unit disk
  • · Replies 33 ·
2
Replies
33
Views
5K
Engineering How would I solve this using Laplace transformation?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Taking the inverse laplace of this?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Inverse laplace transform for unique diffusion type problem
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Inverse Laplace transform with unit step function
  • · Replies 3 ·
Replies
3
Views
5K
Graduate Solving a Laplace Transform Problem: Where Am I Going Wrong?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Inverse Laplace Transformation of arctan (s/2)
  • · Replies 3 ·
Replies
3
Views
8K