Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator L\mathcal{L}. It can be seen in this paper that the auxiliary parameter (^h/_2p),\hbar, which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.     

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm-Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm-Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm-Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm-Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.     

Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.     

Boundary value method for inverse Sturm-Liouville problems

In this paper we present a method to recover symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. The linear multistep method coupled with suitable boundary conditions known as boundary value method (BVM) is the main tool to approximate the eigenvalues in each iteration step of the used Newton method. The BVM was extended to work for Neumann-Neumann boundary conditions. Moreover, a suitable approximation for the asymptotic correction of the eigenvalues is given. Numerical results demonstrate that the method is able to give good results for both symmetric and non-symmetric potentials.     

Matrix Riccati equations and matrix Sturm-Liouville problems

The asymptotic behavior of determinants of unitary solutions of matrix Riccati differential equations containing a large parameter is determined. The result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix Sturm-Liouville problems.     

A collocation-shooting method for solving fractional boundary value problems

In this paper, we discuss the numerical solution of special class of fractional boundary value problems of order 2. The method of solution is based on a conjugating collocation and spline analysis combined with shooting method. A theoretical analysis about the existence and uniqueness of exact solution for the present class is proven. Two examples involving Bagley–Torvik equation subject to boundary conditions are also presented; numerical results illustrate the accuracy of the present scheme.     

A space-time finite element method for fractional wave problems

Numerical Algorithms - In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ...     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

An improved Tau method for a class of Sturm-Liouville problems

We consider the numerical solution of Sturm-Liouville eigenvalue problems by Legendre-Gauss Tau method. The latter approximates the solution of differential equations as a finite sum of Legendre polynomials . We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials for some real or complex frequencies {ω_k}. With the introduction of such exponentials, Legendre-Gauss Tau method can detect the sharp variations exhibited by the highly indexed Sturm-Liouville eigenfunctions. The efficiency of our results is illustrated through numerical examples.     

The method of external excitation for solving generalized Sturm-Liouville problems

A new numerical technique for solving the generalized Sturm-Liouville problem , b_l[w(0),λ]=b_r[w(1),λ]=0 is presented. In particular, we consider the problems when the coefficient q(x,λ) or the boundary conditions depend on the spectral parameter λ in an arbitrary nonlinear manner. The method presented is based on mathematically modelling the physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the eigenvalues.The results of the numerical experiments justifying the method are presented.     

The shooting method for some nonlinear Sturm-Liouville boundary value problems

The shooting method is used to prove existence and uniqueness of the solution for a semilinear Sturm-Liouville boundary value problem (N). lies between two consecutive eigenvalues of the related linear problem, the shooting function turns out to be strongly monotone.     

A sequential method for a class of pseudoconcave fractional problems

The aim of the paper is to maximize a pseudoconcave function which is the sum of a linear and a linear fractional function subject to linear constraints. Theoretical properties of the problem are first established and then a sequential method based on a simplex-like procedure is suggested.      

Set membership parameter estimation of fractional models based on bounded frequency domain data

This paper deals with parameter estimation of fractional models based on frequency domain uncertain but bounded data. Fractional models parameters, including differentiation orders, are expressed as intervals. Real and complex interval analysis is then used to estimate the whole set of feasible parameters. Two methods based on interval constraints satisfaction problem (CSP) are compared to each other. The first one formulates a real-CSP based on an explicit decomposition of the frequency response in the real and the imaginary parts. The second one formulates a complex-CSP using the complex frequency response without any decomposition. To have a fair comparison, rectangular representation of complex intervals is considered. Contractors combined to bisection algorithms are used to solve both CSPs.     

A domain embedding method for mixed boundary value problems

We propose a domain embedding (fictitious domain) method for elliptic equations subject to mixed boundary conditions, and prove the sharp convergence rate. The theory provides a unified treatment for Dirichlet, Neumann, and Robin boundary conditions. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

分数次Orlicz极大算子在齐型空间中的局部加权端点估计

利用齐型空间中的覆盖引理及其有界区域的二进方体分解得到了分数次Orlicz极大算子在齐型空间（X,d,μ）中的有界区域Ω上的局部加权端点估计．该工作为分数次积分交换子[b,Iα】在欧式空间R^n中的有界区域上的加权端点弱型估计推广到齐型空间奠定了基础．     

A reproducing kernel method for solving nonlocal fractional boundary value problems

In our previous works, we proposed a reproducing kernel method for solving singular and nonsingular boundary value problems of integer order based on the reproducing kernel theory. In this letter, we shall expand the application of reproducing kernel theory to fractional differential equations and present an algorithm for solving nonlocal fractional boundary value problems. The results from numerical examples show that the present method is simple and effective.