Initial-Value Problem for a Higher-Order Quasilinear Partial Differential Equation

2021 ◽  
Author(s):  
T. K. Yuldashev ◽  
K. Kh. Shabadikov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Value Problem ◽  
Higher Order ◽  
Initial Value ◽  
Partial Differential ◽  
Quasilinear Partial Differential Equation

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>

Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument

2020 ◽  
pp. 31-56
Author(s):  
T.K. Yuldashev ◽  
J.A. Artykova
Keyword(s):  
Differential Equation ◽  
Initial Value Problem ◽  
Higher Order ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Degenerate Kernel ◽  
Hyperbolic Operator ◽  
Initial Value ◽  
Partial Differential ◽  
Integro Differential Equation

In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.

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 Exponential asymptotics and Stokes lines in a partial differential equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2385-2421 ◽  
Author(s):  
S. Jonathan Chapman ◽  
David B Mortimer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Second Generation ◽  
Saddle Points ◽  
Asymptotic Series ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Stokes Line ◽  
Partial Differential ◽  
Stokes Lines

A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al . (Berk et al . 1982 J. Math. Phys. 23 , 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al . (Aoki et al . 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.

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