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 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 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.

Integration of Partial Differential Equation for Transient Linear Flow of Gas-Condensate Fluids in Porous Structures

1964 ◽  
Vol 4 (04) ◽  
pp. 291-306 ◽  
Author(s):  
C. Kenneth Eilerts
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Effective Permeability ◽  
Flow Characteristics ◽  
Compressibility Factor ◽  
Variable Coefficients ◽  
Back Pressure ◽  
Absolute Permeability ◽  
Partial Differential ◽  
Linear Flow

Abstract Finite difference equations were programmed and used to integrate the second-order, second-degree, partial differential equation with variable coefficients that represents the transient linear flow of gas-condensate fluids. Effect was given to the change with pressure of the compressibility factor, the viscosity, and the effective permeability and to change of the absolute permeability with distance. Integrations used as illustrations include recovery of fluid from a reservoir at a constant production rate followed by recovery at a declining rate calculated to maintain a constant pressure at the producing boundary. The time required to attain such a limiting pressure and the fraction of the reserve recovered in that time vary markedly with properties of the fluid represented by the coefficients. Fluid also is returned to the reservoir at a constant rate, until initial formation pressure is attained at the input boundary, and then at a calculated rate that will maintain but not exceed the limiting pressure. The computing programs were used to calculate the results that would be obtained in a series of back-pressure tests made at selected intervals of reservoir depletion. If effect is given to the variations in properties of the fluid that actually occur, then a series of back-pressure curves one for each stage of reserve depletion -- is required to indicate open-flow capacity and related flow characteristics dependably. The isochronal performance method for determining flow characteristics of a well was simulated by computation. Introduction The back-pressure test procedure is based on a derivation of the equation for steady-state radial flow of a gas, the properties of which are of necessity assumed to remain unchanged in applying the test results. The properties of most natural gases being recovered from reservoirs change as the reserve is depleted and pressures decline, and the results of an early back-pressure test may not be dependable for estimating the future delivery capacity of a well. The compressibility factor of a fluid under an initial pressure of 10,000 psia can change 45 per cent and the viscosity can change 70 per cent during the productive life of the reservoir. There are indications that the effective permeability to the flowing fluid can become 50 per cent of the original absolute permeability before enough liquid collects in the structure about a well as pressure declines to permit flow of liquid into the well. The advent of programmed electronic computing made practicable the integration of nonlinear, second-order, partial differential equations pertaining to flow in reservoirs. Aronofsky and Porter represented the compressibility factor and the viscosity by a linear relationship, and integrated the equation for radial flow of gas for pressures up to 1,200 psi. Bruce, Peaceman, Rachford and Rice integrated the partial differential equations for both linear and radial unsteady-state flow of ideal gas in porous media. Their published results were a substantial guide in this study of integration of the partial differential equation of linear flow with coefficients of the equation variable. The computing program was developed to treat effective permeability as being both distance-dependent and pressure-dependent. In this study all examples of effective permeability are assumptions designed primarily to aid in developing programs for giving effect to this and other variable coefficients. The accumulation of data for expressing the pressure dependency of the effective permeability is the objective of a concurrent investigation. SPEJ P. 291^

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