Asymptotic expansion of the green function of an internal-wave equation fort→∞

1996 ◽  
Vol 37 (4) ◽  
pp. 606-614 ◽  
Author(s):  
V. A. Borovikov
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Green Function ◽  
Internal Wave ◽  
The Green Function

scholarly journals Relations between elliptic modular graphs

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Anirban Basu
Keyword(s):  
Asymptotic Expansion ◽  
Green Function ◽  
Type Ii ◽  
Superstring Theory ◽  
Genus Two ◽  
The Green Function

Abstract We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non-separating node of genus two string invariants that appear in the integrand of the D8ℛ4 interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti-holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given only by the Green function and not its derivatives. This results in a reduction in the number of basis elements in the space of elliptic modular graphs.

scholarly journals Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 464 ◽  
Author(s):  
Arsen Pskhu
Keyword(s):  
Wave Equation ◽  
Green Function ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Cylindrical Domain ◽  
Fractional Differentiation ◽  
Layer Potential ◽  
Diffusion Wave ◽  
Jump Relation ◽  
The Green Function

We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives.

scholarly journals The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 230
Author(s):  
Entsar A. Abdel-Rehim
Keyword(s):  
Wave Equation ◽  
Green Function ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Fractional Operator ◽  
Fractional Wave Equation ◽  
Constant Coefficients ◽  
The Green Function ◽  
Time Fractional Wave Equation ◽  
Fokker Planck

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

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