D'Lembert Method for Solving the Wave Equation with Boundary Conditions

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SUMMARY

The discussion focuses on applying the D'Lembert Method to solve the wave equation given specific boundary and initial conditions. The equation under consideration is \( u_{tt} = u_{xx} \) for \( x > 0 \) and \( t > 0 \), with boundary condition \( u(0,t) = 0 \) and initial conditions \( u(x,0) = x e^{-x^2} \) and \( u_t(x,0) = 0 \). Participants clarify the handling of the boundary condition at \( (0,0) \) and confirm that it can be ignored if the solution satisfies the other conditions. The method is validated through practical application.

PREREQUISITES
  • Understanding of wave equations, specifically \( u_{tt} = u_{xx} \)
  • Familiarity with the D'Lembert Method for solving partial differential equations
  • Knowledge of boundary and initial value problems in mathematical physics
  • Basic calculus and differential equations concepts
NEXT STEPS
  • Study the D'Lembert Method in greater detail, focusing on its application to various boundary conditions
  • Explore the implications of different initial conditions on wave equation solutions
  • Learn about other methods for solving wave equations, such as separation of variables
  • Investigate numerical methods for approximating solutions to wave equations
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Mathematicians, physicists, and engineering students interested in solving wave equations and understanding boundary value problems in mathematical modeling.

Markov2
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Solve

$\begin{aligned} & {{u}_{tt}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\
& u(0,t)=0,\text{ }t>0 \\
& u(x,0)=x{{e}^{-{{x}^{2}}}},\text{ }0<x<\infty \\
& {{u}_{t}}(x,0)=0.
\end{aligned}
$

The condition $u(0,t)$ is new to me, since I usually apply the method when only having $u(x,0)$ and $u_t(x,0),$ what to do in this case?
 
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The boundary and intial value conditions match at (0, 0) so I would just ignore it, then check to make sure my result satisfied that.
 
Okay, I'll apply it then and see how it works, thanks!
 

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