12,154 research outputs found
Analytical-approximate solution of Abel integral equations
It is known that Abel integral equation has a solution in a closed form, with a removable singularity.The presence of Volterra integrals with weak singularity is not always integrable for continuous differentiable class of functions.In this work we propose an analytical approximate method for the solution of Abel integral equations.
We showed that the proposed method is exact for the known function in the cases of polynomials and irrational function of the form f(t)tα+1(a0+a1t+···+antn),0<α<1.For the derivation of the proposed method we expand the known function to the Taylor series around a singular points. Substituting this expansion into the solution of Abel equation we could remove the singularity. All evaluations of the integrals are calculated analytically. The obtained solution is a series that is uniformly convergence to the exact solution
The Schrodinger Equation as a Volterra Problem
The objective of the thesis is to treat the Schrodinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress,
the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution
of the integral equation. Similarly, the Schrodinger equation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of
the Rubinsteins and Kress to show for the Schrodinger equation similar results to those for the heat equation. The thesis proves that the Schrodinger equation with
a source function does indeed have a unique solution. The Poisson integral formula with the Schrodinger kernel is shown to hold in the Abel summable sense. The Green functions are introduced in order to obtain a representation for any function which satisfies the Schrodinger initial-boundary value problem. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact Green function as n goes to infinity. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces, and the Volterra and General Volterra theorems are proved and used in order to show that the Neumann series for the L^1 kernel, the L^infinity kernel, the Hilbert-Schmidt kernel, the unitary kernel, and the
WKB kernel converge to the exact Green function. In the WKB case, the solution of the Schrodinger equation is given in terms of classical paths; that is, the multiple scattering expansions are used to construct from, the action S, the quantum Green function. Then the interior Dirichlet problem is converted into a Volterra integral
problem, and it is shown that Volterra integral equation with the quantum surface kernel can be solved by the method of successive approximations
Nonlinear Abel type integral equation in modelling creep crack propagation
Copyright @ 2011 Birkhäuser BostonA nonlinear Abel-type equation is obtained in this paper to model creep crack time-dependent propagation in the infinite viscoelastic plane. A finite time when the integral equation solution becomes unbounded is obtained analytically as well as the equation parameters when solution blows up for all times. A modification to the Nyström method is introduced to numerically solve the equation and some computational results are presented
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
Cosmological Reionization by Stellar Sources
I use cosmological simulations that incorporate a physically motivated
approximation to three-dimensional radiative transfer that recovers correct
asymptotic ionization front propagation speeds for some cosmologically relevant
density distributions transfer to investigate the process of the reionization
of the universe by ionizing radiation from proto-galaxies. Reionization
proceeds in three stages and occupies a large redshift range from z~15 until
z~5. During the first, ``pre-overlap'' stage, HII regions gradually expand into
the low density IGM, leaving behind neutral high density protrusions. During
the second, ``overlap'' stage, that occurs in about 10% of the Hubble time, HII
regions merge and the ionizing background rises by a large factor. During the
third, ``post-overlap'' stage, remaining high density regions are being
gradually ionized as the required ionizing photons are being produced.
Residual fluctuations in the ionizing background reach significant (more than
10%) levels for the Lyman-alpha forest absorption systems with column densities
above 10^14 - 10^15 cm^-2 at z=3 to 4.Comment: Revised version accepted for publication in ApJ. Color versions of
Fig. 3a-h in GIF format, full (unbinned) versions of Fig. 5, 6, and 13, as
well as MPEG animations are available at
http://casa.colorado.edu/~gnedin/GALLERY/rei_p.htm
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