The Explicit Solution of the -Neumann Problem in a Non-Isotropic Siegel Domain

1997 ◽  
Vol 49 (6) ◽  
pp. 1299-1322 ◽  
Author(s):  
Jingzhi Tie
Keyword(s):  
Differential Equation ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Heisenberg Group ◽  
Neumann Problem ◽  
Explicit Solution ◽  
Laplacian Operator ◽  
Siegel Domain ◽  
The Neumann Problem ◽  
Image Position

AbstractIn this paper, we solve the-Neumann problem on (0, q) forms, 0 ≤ q ≤ n, in the strictly pseudoconvex non-isotropic Siegel domain:where aj> 0 for j = 1,2, . . . , n. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

Existence de Solutions au Sens de Carathéodory Pour le Problème de Neumann y″ = f(t, y, y′)

1991 ◽  
Vol 43 (5) ◽  
pp. 998-1009 ◽  
Author(s):  
Zine E. A. Guennoun
Keyword(s):  
Differential Equation ◽  
Neumann Problem ◽  
Nonlinear Differential Equation ◽  
Existence Results ◽  
Carathéodory Function ◽  
The Neumann Problem ◽  
Image Position

AbstractWe announce some existence results and their consequences for the Neumann problem of the nonlinear differential equation: where f: [0, 1] x ℝ2 → ℝ is a Carathéodory function and can grow very rapidly in the y′ variable.

The fundamental solution for the axially symmetric wave equation II

1980 ◽  
Vol 87 (3) ◽  
pp. 515-521
Author(s):  
Albert E. Heins
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Fundamental Solution ◽  
Free Space ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Symmetric Wave ◽  
Alternate Forms ◽  
Image Position

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equationcould be written asif b > rorif r > b.

The Neumann problem for the generalized Bagley-Torvik fractional differential equation

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Svatoslav Staněk
Keyword(s):  
Differential Equation ◽  
Neumann Problem ◽  
Fractional Differential Equation ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
The Neumann Problem

AbstractWe discuss the existence and multiplicity of solutions to the generalized Bagley-Torvik fractional differential equation

Role of the fundamental solution in Hardy—Sobolev-type inequalities

2006 ◽  
Vol 136 (6) ◽  
pp. 1111-1130 ◽  
Author(s):  
Adimurthi ◽  
Anusha Sekar
Keyword(s):  
Fundamental Solution ◽  
Heisenberg Group ◽  
Sobolev Inequality ◽  
Fundamental Solutions ◽  
Sobolev Type ◽  
Best Constant ◽  
Optimal Estimates ◽  
Image Position

Let n ≥ 3, Ω ⊂ Rn be a domain with 0 ∈ Ω, then, for all the Hardy–Sobolev inequality says that and equality holds if and only if u = 0 and ((n − 2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy–Sobolev type inequalities on manifolds and also on the Heisenberg group.

scholarly journals A global branch of solutions to a semilinear equation on an unbounded interval

1985 ◽  
Vol 101 (3-4) ◽  
pp. 273-282 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Differential Equation ◽  
Neumann Problem ◽  
Order Differential Equation ◽  
Second Order ◽  
Semilinear Equation ◽  
Second Order Differential Equation ◽  
Unbounded Interval ◽  
The Neumann Problem ◽  
Global Branch

SynopsisFor a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.

Polynomial approximations to the solution of the heat equation

1973 ◽  
Vol 73 (1) ◽  
pp. 157-165 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Heat Equation ◽  
Least Squares ◽  
Linear Boundary ◽  
Chebyshev Series ◽  
Linear Boundary Condition ◽  
Image Position

AbstractAn approximation is found to the solution of the partial differential equationin the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.

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