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 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.

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 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>

Weyl transforms and a degenerate elliptic partial differential equation

2005 ◽  
Vol 461 (2064) ◽  
pp. 3863-3870 ◽  
Author(s):  
M.W Wong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Elliptic Partial Differential Equation ◽  
Hermite Functions ◽  
Partial Differential ◽  
Pseudo Differential Operators ◽  
Weyl Transforms ◽  
Degenerate Elliptic

We give a formula for the inverse of a degenerate elliptic partial differential operator P on related to the Heisenberg group. The formula is in terms of pseudo-differential operators of the Weyl type, i.e. Weyl transforms. The technique is to use the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for . Using the formula for the inverse, we give an estimate for the L p norm of the solution u of the partial differential equation Pu = f on in terms of the L 2 norm of f , 2≤ p ≤∞.

Boundary-Value Problems of a Degenerate Sobolev-Type Differential Equation

1977 ◽  
Vol 20 (2) ◽  
pp. 221-228 ◽  
Author(s):  
C. V. Pao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Sobolev Type ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Type Differential Equation

AbstractThe purpose of this paper is to study a degenerate Sobolev type partial differential equation in the form of Mut + Lu = f, where M and L are second order partial differential operators defined in a domain (0, T]×Ω in Rn+1. The degenerate property of the equation is in the sense that both M and L are not necessarily strongly elliptic and their coefficients may vanish or be negative in some part of the domain (0, T]×Ω. Two types of boundary conditions are investigated.

scholarly journals A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

1997 ◽  
Vol 145 ◽  
pp. 125-142
Author(s):  
Takeshi Mandai
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Negative Integer ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Real Analytic

Consider a partial differential operator(1.1) where K is a non-negative integer and aj,a are real-analytic in a neighborhood of (0, 0)

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