Singular first order perturbations of the heat equation

1984 ◽  
Vol 96 (3-4) ◽  
pp. 317-321
Author(s):  
Joel Avrin
Keyword(s):  
Heat Equation ◽  
Magnetic Vector ◽  
First Order ◽  
Vector Potentials ◽  
Semigroup Method

SynopsisWe exhibit dimension-independent conditions under which the formal operator A = −Δ + a.∇ + V can be defined on such that its closure Ā in L2(Rs, dx) is quasi-m-accretive. Here, a is real so that Ā is nonselfadjoint. the method of proof is a generalized version of the argument employed in the portion of the author's thesis where term a.∇ was originally considered. Specifically, we construct exp (−tĀ) as a limit of approximating semigroups. Since the thesis appeared, Kato has also dealt with the term a. ∇ his conditions on a and V are similar to, but more general than, the conditions that appear here; in addition, he considers magnetic vector potentials. Of interest here is the semigroup method itself, the conciseness of the arguments thereby produced, and a relaxed condition on div a.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

Schr�dinger operators with singular magnetic vector potentials

1981 ◽  
Vol 176 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Herbert Leinfelder ◽  
Christian G. Simader
Keyword(s):  
Magnetic Vector ◽  
Vector Potentials

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

Scattering by Spherically Symmetrical Objects

1972 ◽  
Vol 50 (8) ◽  
pp. 749-753 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Differential Equations ◽  
Scattered Field ◽  
Initial Phase ◽  
Radial Coordinate ◽  
Spherically Symmetric ◽  
First Order ◽  
Auxiliary Functions ◽  
Vector Potentials ◽  
Phase Functions ◽  
First Order Differential Equations

The solutions of vector potentials in the presence of spherically symmetric objects are expressed in terms of two auxiliary functions, related respectively to the phase and amplitude of the resulting field. It is shown that these auxiliary functions satisfy first-order differential equations of the radial coordinate, and the scattered field is described by the phase functions alone. Furthermore, the differential equations satisfied by the phase functions are found to be independent of the amplitude functions and are solved numerically by using the well-known initial phase shifts readily obtained from the boundary conditions.

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