Global weak solutions to initial-boundary-value problems for the one-dimensional quasilinear wave equation with large data

1994 ◽  
Vol 126 (4) ◽  
pp. 333-368 ◽  
Author(s):  
Andreas Heidrich
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Large Data ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Global Weak Solutions ◽  
Quasilinear Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Initial Boundary Value

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

scholarly journals The pulsating orb: solving the wave equation outside a ball

2016 ◽  
Vol 472 (2189) ◽  
pp. 20160037 ◽  
Author(s):  
P. A. Martin
Keyword(s):  
Wave Equation ◽  
Laplace Transform ◽  
Continuum Mechanics ◽  
Weak Solutions ◽  
Acoustic Waves ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Compatibility Conditions ◽  
The Laplace Transform ◽  
Initial Boundary Value

Transient acoustic waves are generated by the oscillations of an object or are scattered by the object. This leads to initial-boundary value problems (IBVPs) for the wave equation. Basic properties of this equation are reviewed, with emphasis on characteristics, wavefronts and compatibility conditions. IBVPs are formulated and their properties reviewed, with emphasis on weak solutions and the constraints imposed by the underlying continuum mechanics. The use of the Laplace transform to treat the IBVPs is also reviewed, with emphasis on situations where the solution is discontinuous across wavefronts. All these notions are made explicit by solving simple IBVPs for a sphere in some detail.

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