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This paper focuses on the inverse extremal eigenvalue problem (IEEP) and a special inverse singular value problem (ISVP). First, a bordered tridiagonal matrix is constructed from the extremal eigenvalues of its leading principal submatrices and an eigenvector. Then, based on the previous construction, a Lefkovitch-type matrix is constructed from a particular set of singular values and a singular vector. Sufficient conditions are established for the existence of a symmetric bordered tridiagonal matrix, while the nonsymmetric case is also addressed. Finally, numerical examples illustrating these constructions derived from the main results are presented.

We study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems. We focus on the case when the given singular values are positive and multiple. This work extends the result of W. Ma (2022). We show that the new method is cubically convergent. Moreover, numerical experiments are given in the last section, which show that the proposed method is practical and efficient.

We present analytical models for variable cross-section absorber plate solar collectors, where the thermal conductivity, overall heat loss coefficient, and incident solar heat flux are power-law functions of temperature. The rectangular cross-section absorber plate (RE) serves as the base cross-section, while four reduced cross-section absorber plates namely the bigger edge convex (BE), conventional convex (Conv), smaller edge convex (SE), and triangular cross-section (TR) can be progressively derived by altering the thickness from the base cross-section. The governing equation for the rectangular cross-section is non-singular, while all the reduced cross-sections represent distinct singular value problems. The non-singular and singular value problems are solved using the classical Adomian decomposition method (ADM) and the modified Adomian decomposition method (MADM), respectively. For the present analysis, the average solar heat flux, along with the maximum and minimum declination angles, for a particular site in Santiniketan has been selected. Graphical plots of various absorber plate cross-sections, considering significant power index parameters, thermal conductivity, overall heat transfer coefficient, and solar heat flux based on the maximum and minimum declination angles, aspect ratio, and Biot number, are presented and physically interpreted. From the comparative studies, it has been observed that the mid point temperature of each cross-section of the absorber plate shows a progressively decreasing trend with increasing thickness. Furthermore, the study reveals that the efficiency of the reduced profiles lies between their decreasing tip thickness values.

In this paper, an efficient wavelet-based algorithm is introduced to investigate the approximate solutions for a few nonlinear singular initial value problems arising in astrophysics. Ultraspherical wavelet method (USWM) is utilized for solving the Lane-Emden type equations. The proposed method is utilized to convert the given nonlinear singular value differential equations into a system of algebraic equations using operational matrices of derivatives. Convergence analysis of the method is discussed. The obtained solutions are compared with LWM, CWM and exact solutions. A few numerical experiments are given to demonstrate the accuracy and efficiency of the proposed method. Satisfactory agreement with exact and other numerical solutions is observed. The efficiency of the proposed method is confirmed by means of computational CPU runtime. Moreover, the use of USWM is investigated to be simple, accurate and less computational cost.

The interaction of multiple parts with each other within a system according to certain intrinsic rules is a crucial natural phenomenon. The notion of entanglement and its decomposition of high-dimensional arrays is particularly intriguing since it opens a new way of thinking in data processing and communication, of which the applications will be broad and significant. Depending on how the internal parts engage with each other, there are different types of entanglements with distinct characteristics. This paper concerns the approximation over a multipartite system whose subsystems consist of symmetric rank-1 matrices that are entangled via the Kronecker tensor product. Such a structure resembles that arising in quantum mechanics where a mixed state is to be approximated by its nearest separable state, except that the discussion in this paper is limited to real-valued matrices. Unlike the conventional low-rank tensor approximations, the added twist due to the involvement of the Kronecker product destroys the multi-linearity, which makes the problem harder. As a first step, this paper explores the rank-1 multipartite approximation only. Reformulated as a nonlinear eigenvalue problem and a nonlinear singular value problem, respectively, the problem can be tackled numerically by power-like iterative methods and SVD-like iterative methods. The iteration in both classes of methods can be implemented cyclically or acyclically. Motivations, schemes, and convergence theory are discussed in this paper. Preliminary numerical experiments suggest these methods are effective and efficient when compared with some general-purpose optimization packages.

A new approach for numerical solving initial value problems for systems of second-order nonlinear ordinary differential equations with a singularity of the first kind at the start point x = 0 is proposed. By substitution of the independent variable x = e t , we reduce the original initial value problem on the interval [0,  a ] to the equivalent one on the interval ( - ∞ , ln a ] . For solving this initial value problem at the grid node t 0 of finite grid { t n ∈ ( - ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } , new fourth-order explicit Runge-Kutta-type methods have been constructed. For finding the solution in other nodes of the grid, we can apply any of the standard Runge-Kutta methods or linear multistep ones, using the solution at the point t 0 , calculated by the constructed in this article methods, as an initial condition. For the proposed approach, a new effective numerical algorithm with a given tolerance has been developed.

An interesting problem was raised in Vong et al. (SIAM J. Matrix Anal. Appl. 32:412–429, 2011 ): whether the Ulm-like method and its convergence result can be extended to the cases of multiple and zero singular values. In this paper, we study the convergence of a Ulm-like method for solving the square inverse singular value problem with multiple and zero singular values. Under the nonsingularity assumption in terms of the relative generalized Jacobian matrices, a convergence analysis for the multiple and zero case is provided and the quadratical convergence property is proved. Moreover, numerical experiments are given in the last section to demonstrate our theoretic results.

In this paper, we consider the existence of positive solutions for a singular tempered fractional equation with a p-Laplacian operator. By constructing a pair of suitable upper and lower solutions of the problem, some new results on the existence of positive solutions for the equation including singular and nonsingular cases are established. The asymptotic behavior of the solution is also derived, which falls in between two known curves. The interesting points of this paper are that the nonlinearity of the equation may be singular in time and space variables and the corresponding operator can have a singular kernel.

A high-order numerical scheme based on collocation of a quintic B-spline over finite element is proposed for the numerical solution of a class of nonlinear singular boundary value problems (SBVPs) arising in various physical models in engineering and applied sciences. Five illustrative examples are presented to illustrate the applicability and accuracy of the method. In order to justify the advantage of the proposed numerical scheme, the computed results are compared with the results obtained by two other fourth-order numerical methods, namely the finite difference method (Chawla et al. in BIT 28(1):88–97, 1988) and B-spline collocation method (Goh et al. in Comput Math Appl 64:115–120, 2012).

We consider the singular Dirichlet problem for the Monge-Ampère type equation d e t D 2 u = b ( x ) g ( − u ) ( 1 + | ∇ u | 2 ) q / 2 , u < 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a strictly convex and bounded smooth domain in ℝ n , q ∈ [0, n +1), g ∈ C ∞ (0, ∞) is positive and strictly decreasing in (0, ∞) with lim s → 0 + g ( s ) = ∞ , and b ∈ C ∞ (Ω) is positive in Ω. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g . Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.