JADEs Issue #11: The Wave Equation Revisited
In this week's issue, we revisit the Wave Equation, one of the most popular if not the most popular linear hyperbolic partial differential equation (PDE).
PROBLEM
Definition
Here is the homogeneous Wave Equation with its prescribed initial and boundary conditions:
By definition, we get the following form of a solution with two cases, with the second one (1):
Source: A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover, New York (1990).
Example
So, here is a specific worked example with a = 4, with the prescribed boundary conditions:
We get two cases for the solution to the given Wave equation and its boundary conditions. The first is the following:
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And, in the second case, the solution becomes the following forms (2):
Definition Addendum
Now, if the boundary condition u = 0 at x = 0 above is instead or becomes u = h(t) at x = 0, then the forms (1) above after the Integral term ADDS this new term s.t. we have the following:
Example Addendum
Now, if the boundary condition u = 0 at x = 0 above is instead or becomes u = h(t) = t^3 at x = 0, then the solution form above in (2) shows the Integral term, then ADDS this new term s.t. we have the following:
ABOUT JADEs
The Journal of Applied Differential Equations (JADEs)?is a free Open Access (OA) journal, e-journal that covers linear and nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs) modeling many of today's complex problems.
While most journals take a very theory-driven analysis approach to PDEs and ODEs, JADEs explores problems in a very pragmatic, applied lab-like problem-solution approach with definitions and examples for data science, engineering, science, and more.
For more, you can contact the Editor/Publisher at?[email protected]
Thank you! See you next week.
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DrHuang.com - cto
2 年there is a simple solution https://server.drhuang.com/input/?guess=pdsolve%28ds%28y%2Ct%2C2%29%3Da%5E2*ds%28y%2Cx%2C2%29%29&inp=ds%28y%2Ct%2C2%29%3Da%5E2*ds%28y%2Cx%2C2%29&lang=null
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2 年Thanks for sharing Stefania Loredana Nita!
Physics and Calculus Teacher at Appomattox Regional Governor's School
2 年Steve A., MSc, PhD(hc), I love the Journal, and it has helped me refresh on DEs. I do have a question: perhaps it's me or notation, but it seems to me that the integral sign is missing on the equation in the 4th line below "solution 1." Or else there is an infinitesimal increment "d(something) left over. Could you please clarify your notation for me? Thanks.