Necessary Conditions to Solve Fractional Order Wave Equations Using Traditional Laplace Transforms

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Andrew W. Wharmby ◽  
Ronald L. Bagley
Keyword(s):  
Wave Equation ◽  
Dielectric Permittivity ◽  
Fractional Order ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Solid Materials ◽  
Viscoelastic Wave Equation ◽  
Viscoelastic Wave

AbstractMaxwell’s equations modified for dielectric permittivity lead to a modified wave equation similar to the viscoelastic wave equation for solid materials. In the process of the modification, certain terms arise that do not have traditional Laplace transforms. This paper addresses the necessary conditions needed to remove these problematic terms from the analysis.

A constant fractional-order viscoelastic wave equation and its numerical simulation scheme

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. T39-T48 ◽  
Author(s):  
Ning Wang ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Muming Xia ◽  
Shucheng Wang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Seismic Wave Propagation ◽  
Variable Order ◽  
New Wave ◽  
Viscoelastic Wave Equation ◽  
Viscoelastic Wave ◽  
Order Scheme ◽  
Simulation Error ◽  
Spatially Varying

Efficient modeling schemes currently exist to handle the spatially variable-order fractional Laplacians in the fractional Laplacian viscoacoustic wave equation. The simplest approach is to change the spatially variable-order fractional Laplacians into a linear combination of several constant fractional-order Laplacians. We generalize the constant fractional-order scheme to a spatially variable fractional-order viscoelastic wave equation and develop an almost-equivalent constant fractional-order viscoelastic wave equation. Our constant fractional-order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order; thus, our scheme simulates seismic wave propagation in viscoelastic media with sharp [Formula: see text] contrasts well. The fast Fourier transform is used in the approximation of the fractional Laplacians, which improves the spectral accuracy in space. Several simulation examples are performed to verify that the numerical solution of a homogeneous [Formula: see text] model obtained by solving our constant fractional-order viscoelastic wave equation agrees well with that obtained by solving the original viscoelastic wave equation. The numerical simulations for spatially varying [Formula: see text] models obtained by the new wave equation are more straightforward than those currently in use and match the reference solutions obtained by accurate, but inefficient, methods. This match of simulation results verifies the accuracy of our viscoelastic wave equation.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

2021 ◽  
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Jidong Yang ◽  
Zhenchun Li ◽  
Mukiibi Ssewannyaga Ivan
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Frequency Domain ◽  
Quality Factor ◽  
Fractional Order ◽  
Time Domain ◽  
Variable Order ◽  
Viscoelastic Wave Equation ◽  
Viscoelastic Wave ◽  
New Time

ABSTRACT Time-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

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