scholarly journals Theory of differential offset continuation

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 718-732 ◽  
Author(s):  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Frequency Domain ◽  
Initial Value Problem ◽  
Data Transformation ◽  
Initial Value ◽  
Partial Differential ◽  
Reflection Data ◽  
Zero Offset

I introduce a partial differential equation to describe the process of prestack reflection data transformation in the offset, midpoint, and time coordinates. The equation is proved theoretically to provide correct kinematics and amplitudes on the transformed constant‐offset sections. Solving an initial‐value problem with the proposed equation leads to integral and frequency‐domain offset continuation operators, which reduce to the known forms of dip moveout operators in the case of continuation to zero offset.

scholarly journals Solution of a Class of Differential Equation with Variable Coefficients

2016 ◽  
Vol 8 (4) ◽  
pp. 140
Author(s):  
Huanhuan Xiong ◽  
Yuedan Jin ◽  
Xiangqing Zhao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Value Problem ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
Partial Differential

<p>In this paper, we obtain the formula of solution to the initial value problem for a hyperbolic partial differential equation with variable coefficient which is the modification of the famous D’ Alembert formula.</p>

scholarly journals On the initial value problem for a partial differential equation with operator coefficients

1980 ◽  
Vol 3 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Mahmoud M. El-Borai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Value Problem ◽  
Differential Operators ◽  
Bounded Operator ◽  
Partial Differential Operator ◽  
Initial Value ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
The Cauchy Problem

In the present work it is studied the initial value problem for an equation of the formL∂ku∂tk=∑j=1kLj∂k−ju∂tk−j,whereLis an elliptic partial differential operator and(Lj:j=1,…,k)is a family of partial differential operators with bounded operator coefficients in a suitable function space. It is found a suitable formula for solution. The correct formulation of the Cauchy problem for this equation is also studied.

scholarly journals THE MATHEMATICAL MODELLING OF THE NONLINEAR HEAT TRANSPORT IN THIN PLATE

1999 ◽  
Vol 4 (1) ◽  
pp. 44-50
Author(s):  
A. Buikis
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mathematical Modelling ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
First Order ◽  
Nonlinear Algebraic Equations

The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations.

Estimates for Solutions of Wave Equations with Vanishing Curvature

1985 ◽  
Vol 37 (6) ◽  
pp. 1176-1200 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Linear Combination ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

scholarly journals The Local Stability of Solutions for a Nonlinear Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Haibo Yan ◽  
Ls Yong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Equation ◽  
Local Stability ◽  
Nonlinear Partial Differential Equation ◽  
Strong Solutions ◽  
Stability Of Solutions ◽  
Initial Value ◽  
Partial Differential

The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the spaceL1(R)by assuming that the initial value only lies in the spaceL1(R)∩L∞(R).

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