Article
Mathematics, Applied
Tongxin Yan, Changfeng Ma
Summary: This work presents an iterative algorithm for solving a class of generalized coupled Sylvester-conjugate matrix equations over generalized Hamiltonian matrices. It is shown that a generalized Hamiltonian solution can be obtained within finite iteration steps in the absence of round-off errors if the equations are consistent. By choosing special initial matrices, the minimum-norm solution can be obtained, and numerical examples demonstrate the effectiveness of the iterative algorithm.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics
Huaxi Chen, Long Wang, Tingting Li
Summary: This paper presents a new necessary and sufficient condition for the solvability of the system of generalized Sylvester real quaternion matrix equations. Additionally, using purely algebraic technique, the solvability of the system of generalized Sylvester equations in a unital ring is considered.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2021)
Article
Mathematics, Applied
Yue Hao, Valeria Simoncini
Summary: This paper discusses the use of a matrix-oriented approach for numerically solving dense matrix equations, which leads to significantly lower computational costs and memory requirements. The method is illustrated in medium-sized and large-scale problems, showcasing its performance and efficiency. Additionally, a new explicit method for linear tensor equations is proposed, utilizing the discussed matrix equation procedure as a key building block.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Behnam Hashemi
Summary: The study presents sufficient conditions for the unique solvability of the matrix equation and also determines conditions for non-solvability in special cases. These conditions can be checked with a cubic complexity in terms of the size of the input matrices.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Davide Palitta
Summary: This study presents a novel solution strategy for addressing the discrete operator issues arising from the time-space discretization of evolutionary partial differential equations, efficiently solving problems with a large number of degrees of freedom while maintaining low storage demand.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Multidisciplinary Sciences
Lulin Xiong, Xin Tan, Shikun Zhong, Wei Cheng
Summary: This paper studies the supersymmetric quantum mechanics problems of the Schrodinger equation with a new kind of generalized trigonometric tangent superpotential. It provides a shape invariant relation for partner potential and explores the eigenvalues and eigenwave functions in different cases. It also discusses the potential algebra of such a superpotential and gives an outlook on the two-parameter shape-invariant potential.
Article
Mathematics, Applied
Durmus Albayrak
Summary: In this paper, various theorems and relationships are examined using a generalized Laplace-type integral transform. The harmonic oscillator, initial-boundary problems, and integral equations in non-resisting and resisting mediums are solved through this integral transform. Additionally, the well-known Basel problem series is obtained using a similar approach. Moreover, a numerical comparison between the classical and newly introduced integral transforms is conducted.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Tao Li, Qing-Wen Wang, Xin-Fang Zhang
Summary: This paper proposes a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations, and further derives a preconditioned modified conjugate residual method based on Kronecker product approximations. Theoretical analysis and numerical results demonstrate that our methods outperform the traditional conjugate gradient method in terms of convergence rate and computational efficiency.
Article
Mathematics, Interdisciplinary Applications
Bo Yu, Xiang Li, Ning Dong
Summary: The implicit difference approach discretizes a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with optimally determined iterative parameters is presented for solving the corresponding matrix equation. Numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems compared to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Zebin Chen, Xuesong Chen
Summary: In this note, the authors point out the insufficient value range of the iterative factor delta in a previous study and rederive the proof process to obtain the correct range. By introducing a new scaling method, the value range of delta is significantly increased.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics
Zhuo-Heng He
Summary: This paper establishes a different approach for solving a system of quaternion matrix equations, deriving new necessary and sufficient conditions for existence of a solution and showing equivalence with previous solvability conditions. The general solution to the system is provided when the solvability conditions are met, with applications discussed including general eta-Hermitian solution.
LINEAR & MULTILINEAR ALGEBRA
(2021)
Article
Mathematics, Applied
Jiri Neustupa, Minsuk Yang
Summary: The article assumes that Omega is either the whole space R-3, a half-space, or a smooth bounded or exterior domain in R-3; T > 0, and (u, b, p) is a suitable weak solution of the MHD equations in Omega x (0, T). The study shows that if the sum of the L-3-norms of u and b over an arbitrarily small ball B-rho(x(0)) is finite as t approaches t(0)-, then (x(0), t(0)) is a regular point of the solution (u, b, p).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Yanqiang Wu, Jipeng Cheng
Summary: A new generalization of the constrained modified KP hierarchy is presented in this paper, and two equivalent formulations are given, one based on the bilinear equations of the wave functions and tau functions, and the other based on the constraints on the tau functions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Zhongyun Liu, Fang Zhang, Carla Ferreira, Yulin Zhang
Summary: In this article, we propose an iterative method for solving large sparse continuous Sylvester equations. Theoretical analysis shows the convergence and upper bound of the proposed method. Computational comparison with alternative methods demonstrates its efficiency and reliability.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Wei-Kuo Chen, Dmitry Panchenko, Eliran Subag
Summary: In this study, we investigate the free energy of the mixed p-spin mean-field spin glass model using the TAP approach. We calculate the generalized TAP correction and establish the corresponding representation for the free energy. We introduce the concept of generalized TAP states and show their connection to the order parameter of the ancestor states in the Parisi ansatz.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
