Parabolic equations in motionless and moving media

2021 ◽  
Vol 149 (4) ◽  
pp. A93-A93
Author(s):  
Vladimir Ostashev ◽  
D. Keith Wilson
Keyword(s):  
Parabolic Equations ◽  
Moving Media

scholarly journals Extra-wide-angle parabolic equations in motionless and moving media

2021 ◽  
Author(s):  
Vladimir Ostashev ◽  
Michael Muhlestein ◽  
D. Wilson
Keyword(s):  
Refractive Index ◽  
Parabolic Equations ◽  
Sound Propagation ◽  
Perturbation Technique ◽  
Integral Form ◽  
Spectral Domain ◽  
Wide Angle ◽  
Starting Point ◽  
Spectral Algorithm ◽  
Moving Media

Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.

Extra-wide-angle parabolic equations in motionless and moving media

2019 ◽  
Vol 145 (2) ◽  
pp. 1031-1047 ◽  
Author(s):  
Vladimir E. Ostashev ◽  
Michael B. Muhlestein ◽  
D. Keith Wilson
Keyword(s):  
Parabolic Equations ◽  
Wide Angle ◽  
Moving Media

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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