اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSolving Abel integral equations by regularisation in Hilbert scales
84
0
0.0
(
0
)
تأليف
Cecile Della Valle
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a < 1 to arbitrary a > 0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.
قيم البحث
اقرأ أيضاً
A dispersive analysis of $etato 3pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be per
formed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $omegato 3pi$.
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $mathbb{R}$, we define fractional operators by means of a functional calculus using the Fou
rier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.
Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, $T^k$ and $T_t$, to be power convergent in the operator norm in a complex Banach space. These results cover also the case where $T
$ is unbounded and the corresponding Abel average is defined by means of the resolvent of $T$. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded $T$.
The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the eleg
ant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ODE of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e ~ 0, p = 1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii equation.
We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based method fo
r the Wiener-Hopf equation due to Henery, expressing it in terms of the Hilbert transform, and computing the latter in a more sophisticated way with sinc functions. We then generalise our method to the Fredholm equation reformulating it as two coupled Wiener-Hopf equations and solving them iteratively. We provide numerical tests and open-source code.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
