Parabolic equations and It�'s stochastic equations with coefficients discontinuous in the time variable

1982 ◽  
Vol 31 (4) ◽  
pp. 278-283 ◽  
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Parabolic Equations ◽  
Stochastic Equations ◽  
Time Variable

scholarly journals Wavelet Method for Numerical Solution of Parabolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
A. H. Choudhury
Keyword(s):  
Parabolic Equations ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Time Variable ◽  
Basis Functions ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Parabolic Partial Differential Equations ◽  
Convection Diffusion ◽  
Diffusion And Convection

We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

scholarly journals A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

2020 ◽  
Vol 28 (3) ◽  
pp. 323-339 ◽  
Author(s):  
Phuong Mai Nguyen ◽  
Loc Hoang Nguyen
Keyword(s):  
Inverse Problem ◽  
Numerical Method ◽  
Parabolic Equations ◽  
Source Term ◽  
Nonlinear Coefficient ◽  
Time Variable ◽  
Inverse Source Problem ◽  
Coefficient Inverse Problem ◽  
Second Stage ◽  
Two Stages

AbstractTwo main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

scholarly journals Asymptotic properties of solutions of the fractional diffusion-wave equation

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Asymptotic Properties ◽  
Fractional Diffusion ◽  
Time Variable ◽  
Diffusion Wave

AbstractFor the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

scholarly journals Problem with two-point conditions for parabolic equation of second order on time

2014 ◽  
Vol 6 (2) ◽  
pp. 351-359 ◽  
Author(s):  
M.M. Symotyuk ◽  
I.R. Tymkiv
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Second Order ◽  
Time Variable ◽  
Small Denominators ◽  
Dirichlet Type ◽  
Linear Parabolic Equations

The  correctness of the problem with two-point conditions on time variable and Dirichlet-type conditions  on spetial coordinates for the linear  parabolic equations are established. The metric theorem about estimate from below of small denominators of the problem is proved.

Saturn's E, G and F rings: modulated by the plasma sheet

1984 ◽  
Vol 75 ◽  
pp. 597
Author(s):  
E. Grün ◽  
G.E. Morfill ◽  
T.V. Johnson ◽  
G.H. Schwehm
Keyword(s):  
Solar Wind ◽  
Direct Contact ◽  
Transport Processes ◽  
Time Variable ◽  
Dust Particles ◽  
Spatial Diffusion ◽  
Drag Forces ◽  
Orbit Perturbation ◽  
Time Variations ◽  
F Ring

ABSTRACTSaturn's broad E ring, the narrow G ring and the structured and apparently time variable F ring(s), contain many micron and sub-micron sized particles, which make up the “visible” component. These rings (or ring systems) are in direct contact with magnetospheric plasma. Fluctuations in the plasma density and/or mean energy, due to magnetospheric and solar wind processes, may induce stochastic charge variations on the dust particles, which in turn lead to an orbit perturbation and spatial diffusion. It is suggested that the extent of the E ring and the braided, kinky structure of certain portions of the F rings as well as possible time variations are a result of plasma induced electromagnetic perturbations and drag forces. The G ring, in this scenario, requires some form of shepherding and should be akin to the F ring in structure. Sputtering of micron-sized dust particles in the E ring by magnetospheric ions yields lifetimes of 102to 104years. This effect as well as the plasma induced transport processes require an active source for the E ring, probably Enceladus.

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