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1.
2.
In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method. 相似文献
3.
Prolate spheroidal functions constitute a one-parameter (α) family of orthogonal functions in the interval. For α = 0, they
are the Legendre polynomials. For larger α, the prolate spheroidal functions oscillate more uniformly than the Legendre polynomials,
and provide more uniform resolution in the interval. The prolate spheroidal functions can be obtained by adding a zeroth-order
term to the Sturm–Liouville equation for the Legendre polynomials. Here, the Sturm–Liouville equation for orthogonal polynomials
in the triangle is modified in a similar fashion. The modification maintains the self-adjointness and symmetry properties
of the original Sturm–Liouville equation, so that the new eigenfunctions are orthogonal and give spectrally accurate approximations
of smooth functions with arbitrary boundary conditions in the triangle. The properties of the new eigenfunctions mimic those
in the interval. For larger α, the new eigenfunctions provide more uniform resolution in the triangle. 相似文献
4.
The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining
the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster
like O(1/N
2) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional
eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated
than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In
this work an alternative measure, ℓ, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville
equations that does not rely on knowledge of the roots. It is first shown that ℓ is a reasonable measure for evaluating the
eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown
that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional
result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing
the minimum distance between Fekete points to ℓ shows that ℓ is a reasonably good measure of the characteristic length scale
in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle
proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some
of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement. 相似文献
5.
考虑边值条件中含谱参数的一类Schrodinger算子逆谱的惟一性问题.由Sturm-Liouville问题逆谱理论中的惟一性定理及整函数的性质证明了基于一定条件下,特征值(包括重数)和一个相关的参数γ能惟一确定势函数. 相似文献
6.
Fatma Hıra 《Inverse Problems in Science & Engineering》2017,25(6):785-794
In this paper, we obtain a regularized trace formula for a Sturm–Liouville problem which has two points of discontinuity and also contains an eigenparameter in a boundary condition. 相似文献
7.
A scheme of the second order of accuracy for the Sturm—Liouville problem is constructed by the method of finite elements with
the use of a special basic system of compact functions. The convergence of the method of finite elements is proved. An exact
formula for estimating errors of eigenvalues is obtained.
Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 28–36, March–Apni, 2000. 相似文献
8.
This paper deals with the observability analysis of nonlinear tubular bioreactor models. Due to the lack of tools for the observability analysis of nonlinear infinite-dimensional systems, the analysis is performed on a linearized version of the model around some steady-state profile, in which coefficients can be functions of the spatial variable. The study starts from an example of tubular bioreactor that will serve as a case study in the present paper. It is shown that such linear models with coefficients dependent on the spatial variable are Sturm–Liouville systems and that the associated linear infinite-dimensional system dynamics are described by a Riesz-spectral operator that generates a C0 (strongly continuous)-semigroup. The observability analysis based on infinite-dimensional system theory shows that any finite number of dominant modes of the system can be made observable by an approximate point measurement. 相似文献
9.
This paper is concerned with the inverse problem for non-selfadjoint Sturm–Liouville operator with discontinuity conditions inside a finite interval. Firstly, we give the definitions of generalized weight numbers for this operator which may have the multiple spectrum and then investigate the connections between the generalized weight numbers and other spectral characteristics. Secondly, we obtain the generalized spectral data, which consists of the generalized weight numbers and the spectrum. Then the operator is determined uniquely by the method of spectral mappings. Finally, we give an algorithm for reconstructing the potential function and the coefficients of the boundary conditions and the coefficients of the discontinuity conditions. 相似文献
10.
In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation involving Jumarie’s modified Riemann–Liouville derivative have been derived. As a result, with the aid of symbolic computation, many types of new analytical solutions are obtained, which include new solitary wave, periodic wave and rational wave solutions. The graphical representations show that these gained solutions have abundant structures. 相似文献