scholarly journals Starlikeness, convexity and landau type theorem of the real kernel α-harmonic mappings

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2629-2644
Author(s):  
Bo-Yong Long ◽  
Qi-Han Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Type Theorem ◽  
Second Order ◽  
Harmonic Mappings ◽  
Partial Differential ◽  
Geometric Properties ◽  
The Real ◽  
Landau Type Theorem ◽  
Univalence Criteria

In [26], Olofsson introduced a kind of second order homogeneous partial differential equation. We call the solution of this equation real kernel ?-harmonic mappings. In this paper, we study some geometric properties of this real kernel ?-harmonic mappings. We give univalence criteria and sufficient coefficient conditions for real kernel ?-harmonic mappings that are fully starlike or fully convex of order ?, ? ? [0, 1). Furthermore, we establish a Landau type theorem for real kernel ?-harmonic mappings.

Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

Geophysics ◽  
2021 ◽  
pp. 1-53
Author(s):  
Jiangtao Hu ◽  
Jianliang Qian ◽  
Jian Song ◽  
Min Ouyang ◽  
Junxing Cao ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Ray Tracing ◽  
Seismic Waves ◽  
Real Space ◽  
Advection Equation ◽  
Partial Differential ◽  
The Real ◽  
Eikonal Function ◽  
Complex Valued

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.

XIII.—Partial Differential Equations and the Calculus of Variations

1927 ◽  
Vol 46 ◽  
pp. 126-135 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Calculus Of Variations ◽  
Order Equation ◽  
Second Order ◽  
Physical Problem ◽  
Partial Differential ◽  
Hamiltonian Principle

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

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