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Journal articles on the topic 'Linear matrix equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 19 July 2025

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1

Allahviranloo, T., N. Mikaeilvand, and M. Barkhordary. "Fuzzy linear matrix equation." Fuzzy Optimization and Decision Making 8, no. 2 (2009): 165–77. http://dx.doi.org/10.1007/s10700-009-9058-1.

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2

Chiang, Chun-Yueh. "A Note on the⊤-Stein Matrix Equation." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/824641.

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This note is concerned with the linear matrix equationX=AX⊤B + C, where the operator(·)⊤denotes the transpose (⊤) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solutionX. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods,

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3

Sun, Yirong, Junyang An, and Xiaobin Guo. "Solving Complex Fuzzy Linear Matrix Equations." Mathematical Problems in Engineering 2021 (October 14, 2021): 1–11. http://dx.doi.org/10.1155/2021/9996566.

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In this paper, a kind of complex fuzzy linear matrix equation A X ˜ B = C ˜ , in which C ˜ is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.

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4

Shang, Dequan, and Xiaobin Guo. "Solving fuzzy linear matrix equation." Journal of Physics: Conference Series 1592 (August 2020): 012051. http://dx.doi.org/10.1088/1742-6596/1592/1/012051.

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5

HASANOV, VEJDI. "ITERATIVE METHODS FOR SOLVING A LINEAR MATRIX EQUATION." Annual of Konstantin Preslavsky University of Shumen, Faculty of mathematics and informatics XXV C (December 3, 2024): 3–14. https://doi.org/10.46687/vzhp2021.

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In this paper we study iterative methods for solving a linear matrix equation. The considered type of equations appear when applying Newton’s method for nonlinear matrix equations. The convergence of the studied methods was investigated. Theoretical results are illustrated by numerical examples.

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6

Zubov, N. E., and V. N. Ryabchenko. "Solution of a Linear Nondegenerate Matrix Equation Based on the Zero Divisor." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 5 (98) (October 2021): 49–59. http://dx.doi.org/10.18698/1812-3368-2021-5-49-59.

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New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the righ

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7

Su, Youfeng, and Guoliang Chen. "Iterative methods for solving linear matrix equation and linear matrix system." International Journal of Computer Mathematics 87, no. 4 (2010): 763–74. http://dx.doi.org/10.1080/00207160802195977.

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8

Yüzbaşi, Şuayip. "A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations." International Journal of Biomathematics 10, no. 07 (2017): 1750091. http://dx.doi.org/10.1142/s1793524517500917.

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In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improvement of solutions is presented by using the residual function. Applications of populati

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9

Titova, Tatiana. "Canonical transformations of linear Hamiltonian systems." E3S Web of Conferences 592 (2024): 04009. http://dx.doi.org/10.1051/e3sconf/202459204009.

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In this paper we consider the linear Hamiltonian systems of differential equations. We explore the normalization of a non-singular Hamiltonian matrix. We solve a system of matrix equations to find the generating function of the canonical transformation. In various cases we obtain the solution of the system of matrix equations. We get the solution of the algebraic matrix Riccati equation under certain conditions. Some properties of the Hamiltonian matrix have been proven. We get the normal form of a non-singular Hamiltonian matrix of order 4. We obtain the new method of normalization of the qua

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10

Li, Kefeng, and Chao Zhang. "The Solutions of Second-Order Linear Matrix Equations on Time Scales." Discrete Dynamics in Nature and Society 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/976914.

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This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.

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11

Dzhaliuk, N. S., and V. M. Petrychkovych. "The structure of solutions of the matrix linear unilateral polynomial equation with two variables." Carpathian Mathematical Publications 9, no. 1 (2017): 48–56. http://dx.doi.org/10.15330/cmp.9.1.48-56.

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We investigate the structure of solutions of the matrix linear polynomial equation $A(\lambda)X(\lambda)+B(\lambda)Y(\lambda)=C(\lambda),$ in particular, possible degrees of the solutions. The solving of this equation is reduced to the solving of the equivalent matrix polynomial equation with matrix coefficients in triangular forms with invariant factors on the main diagonals, to which the matrices $A (\lambda), B(\lambda)$ \ and \ $C(\lambda)$ are reduced by means of semiscalar equivalent transformations. On the basis of it, we have pointed out the bounds of the degrees of the matrix polynomi

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12

Ladzoryshyn, N. B., V. M. Petrychkovych, and H. V. Zelisko. "Matrix Diophantine equations over quadratic rings and their solutions." Carpathian Mathematical Publications 12, no. 2 (2020): 368–75. http://dx.doi.org/10.15330/cmp.12.2.368-375.

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The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In

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13

Guo, Xiaobin, and Dequan Shang. "Fuzzy Symmetric Solutions of Fuzzy Matrix Equations." Advances in Fuzzy Systems 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/318069.

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The fuzzy symmetric solution of fuzzy matrix equationAX˜=B˜, in whichAis a crispm×mnonsingular matrix andB˜is anm×nfuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method.

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14

Li, Jin. "Linear barycentric rational collocation method for solving biharmonic equation." Demonstratio Mathematica 55, no. 1 (2022): 587–603. http://dx.doi.org/10.1515/dema-2022-0151.

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Abstract Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

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15

Sugaya, Ryo, Tomohiro Sogabe, Tomoya Kemmochi, and Shao-Liang Zhang. "Variational Quantum Algorithm for Solving Second-Order Linear Differential Equations." Quantum Information & Computation 25, no. 3 (2025): 232–47. https://doi.org/10.2478/qic-2025-0012.

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Abstract Variational quantum algorithms (VQAs) have attracted attention as quantum algorithms employing on noisy intermediate-scale quantum devices. VQAs for solving the Poisson equation were recently proposed, as this method transforms the linear equation with a symmetric positive definite matrix obtained by discretizing the Poisson equation into a minimization problem. In this study, we propose a VQA for second-order linear differential equations including the Poisson equation on the basis of the referenced study, where the coefficient matrix obtained through discretization is a non-symmetri

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16

McLEOD, J. B., and C. B. WANG. "DISCRETE INTEGRABLE SYSTEMS ASSOCIATED WITH THE UNITARY MATRIX MODEL." Analysis and Applications 02, no. 02 (2004): 101–27. http://dx.doi.org/10.1142/s0219530504000047.

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The orthogonal polynomials on the unit circle associated with the unitary matrix model have various interesting properties, and have been studied in connection with different applications, such as double scaling, Riemann–Hilbert problems and integrable Fredholm operators. In this paper, we study the orthogonal polynomials on the unit circle with the weight function [Formula: see text]. We use the orthogonality of the polynomials to show that the orthogonal polynomials on the unit circle satisfy the linear problems associated with the discrete PainlevéII hierarchy, alternate discrete PainlevéII

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17

Zhang, Wantong. "Two Methods of Matrix Factorization and Their Applications." Journal of Physics: Conference Series 2386, no. 1 (2022): 012002. http://dx.doi.org/10.1088/1742-6596/2386/1/012002.

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Abstract Many problems in real life are summed up as mathematical models. Linear equation set is a more common approach among all mathematical models. Given particular scenarios and conditions, linear equation sets can solve problems such as resource allocation. This paper mainly introduces two basic methods of matrix factorization, explains how to simplify the solving process of linear equations by using elimination method, and applies the model into a real-life problem.

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18

Li, Jianfeng, Linxi Qu, Zhan Li, et al. "A Novel Zeroing Neural Network for Solving Time-Varying Quadratic Matrix Equations against Linear Noises." Mathematics 11, no. 2 (2023): 475. http://dx.doi.org/10.3390/math11020475.

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The solving of quadratic matrix equations is a fundamental issue which essentially exists in the optimal control domain. However, noises exerted on the coefficients of quadratic matrix equations may affect the accuracy of the solutions. In order to solve the time-varying quadratic matrix equation problem under linear noise, a new error-processing design formula is proposed, and a resultant novel zeroing neural network model is developed. The new design formula incorporates a second-order error-processing manner, and the double-integration-enhanced zeroing neural network (DIEZNN) model is furth

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19

Guo, Xiaobin, and Dequan Shang. "Approximate Solution of LR Fuzzy Sylvester Matrix Equations." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/752760.

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The fuzzy Sylvester matrix equationAX~+X~B=C~in whichA,Barem×mandn×ncrisp matrices, respectively, andC~is anm×nLR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Som

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20

Xi, Yimeng, Zhihong Liu, Ying Li, Ruyu Tao, and Tao Wang. "On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application." AIMS Mathematics 8, no. 11 (2023): 27901–23. http://dx.doi.org/10.3934/math.20231427.

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<abstract><p>In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_C} $-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, $ \mathcal{GH} $-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the le

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21

Ghamkhar, Madiha, Laiba Wajid, Khurrem Shahzad, et al. "Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach." Open Physics 20, no. 1 (2022): 850–58. http://dx.doi.org/10.1515/phys-2022-0182.

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Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a strin

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22

Dookhitram, Kumar, Roddy Lollchund, Rakesh Kumar Tripathi, and Muddun Bhuruth. "Fully fuzzy Sylvester matrix equation." Journal of Intelligent & Fuzzy Systems 28, no. 5 (2015): 2199–211. https://doi.org/10.3233/ifs-141502.

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Abstract The Sylvester equation arises in many application areas, for instance process and system control, and in the fuzzy setting, solution of this equation has been considered only in the case when the right-hand side matrix is a fuzzy matrix. This paper introduces the fully fuzzy Sylvester matrix equation A ˜ X ˜ - X ˜ B ˜ = C ˜ , where the fuzzy matrices A ˜ , B ˜ and C ˜ are of order n , m and n × m , respectively. The fuzzy matrix X ˜ is the sought after solution. A two-step scheme is developed for the solution of this system. The first step solves the 1-cut of the problem and the secon

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23

Wu, Feng. "Sherman-Morrison-Woodbury Formula for Linear Integrodifferential Equations." Mathematical Problems in Engineering 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/9418730.

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The well-known Sherman-Morrison-Woodbury formula is a powerful device for calculating the inverse of a square matrix. The paper finds that the Sherman-Morrison-Woodbury formula can be extended to the linear integrodifferential equation, which results in an unified scheme to decompose the linear integrodifferential equation into sets of differential equations and one integral equation. Two examples are presented to illustrate the Sherman-Morrison-Woodbury formula for the linear integrodifferential equation.

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24

Uskov, Vladimir I. "Properties of one higher order matrix-differential operator." Russian Universities Reports. Mathematics, no. 138 (2022): 175–82. http://dx.doi.org/10.20310/2686-9667-2022-27-138-175-182.

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The article considers a linear matrix-differential operator of the n-th order of the form A^n. For it and for the operator (A ̃^(-1) )^n, an analytical expression is derived, for which an operator analog of the Newton binomial is obtained. A lemma on the solution of a linear equation is given. It is used in the study of the abstract Cauchy problem for an algebro-differential equation in a Banach space with the cube of the operator A at the highest derivative. The operator A has the property of having 0 as a normal eigenvalue. Conditions for the existence and uniqueness of the solution are dete

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25

Jiang, Bo, Yongge Tian, and Ruixia Yuan. "On Relationships between a Linear Matrix Equation and Its Four Reduced Equations." Axioms 11, no. 9 (2022): 440. http://dx.doi.org/10.3390/axioms11090440.

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Given the linear matrix equation AXB=C, we partition it into the form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A1, A1, B1, and B2 to obtain four reduced linear matrix equations. In this situation, each of the four reduced equations involves just one of the four unknown submatrices X11, X12, X21, and X22, respectively. In this paper, we study the relationships between the general solution of AXB=C and the general solutions of the four reduced equations using some highly

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26

CAI, Q. D. "CONTINUOUS NEWTON METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATION." Modern Physics Letters B 24, no. 13 (2010): 1303–6. http://dx.doi.org/10.1142/s0217984910023487.

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Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parame

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27

Dzhaliuk, N. S., and V. M. Petrychkovych. "Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations." Matematychni Studii 61, no. 2 (2024): 115–22. http://dx.doi.org/10.30970/ms.61.2.115-122.

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We investigate the solutions of the Sylvester-type matrix polynomial equation $$A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda),$$ where\ $A(\lambda),$ \ $ B(\lambda),$\ and \ $C(\lambda)$ are the polynomial matrices with elements in a ring of polynomials \ $\mathcal{F}[\lambda],$ \ $\mathcal{F}$ is a field,\ $X(\lambda)$\ and \ $Y(\lambda)$ \ are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations $$G \left\|\begin{array}{c}\mathbf{x} \\ \mathbf{y} \end{array} \right\|=\mathbf{c}$$ over a field $\mathcal{F}.$ In this case, th

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28

Toutounian, Faezeh, Emran Tohidi, and Stanford Shateyi. "A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/823098.

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This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus

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29

Secer, Aydin, and Selvi Altun. "A new numerical approach for solving high-order linear and non-linear differantial equations." Thermal Science 22, Suppl. 1 (2018): 67–77. http://dx.doi.org/10.2298/tsci170612272s.

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In this paper, the Legendre wavelet operational matrix method has been introduced for solving high-order linear and non-linear multi-point: initial and boundary value problems. It has been suggested that the technique is rest upon practical application of the operational matrix and its derivatives. The differential equation is presented that it is converted to a system of algebraic equations via the properties of Legendre wavelet together with the operational matrix method. As a result of this study, the scheme has been tested on five linear and non-linear problems. The results have demonstrat

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Abdullayev, Akmaljon, Kholsaid Kholturayev, and Nigora Safarbayeva. "Exact method to solve of linear heat transfer problems." E3S Web of Conferences 264 (2021): 02059. http://dx.doi.org/10.1051/e3sconf/202126402059.

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When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equat

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31

Jaiprasert, Janthip, and Pattrawut Chansagiam. "Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions." Electronic Research Archive 32, no. 4 (2024): 2789–804. http://dx.doi.org/10.3934/era.2024126.

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<abstract><p>We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included

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32

Zhang, Er Yan, and Xiao Feng Zhu. "The Recursive Algorithms of Yule-Walker Equation in Generalized Stationary Prediction." Advanced Materials Research 756-759 (September 2013): 3070–73. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.3070.

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Toeplitz matrix arises in a remarkable variety of applications such as signal processing, time series analysis, image processing. Yule-Walker equation in generalized stationary prediction is linear algebraic equations that use Toeplitz matrix as coefficient matrix. Making better use of the structure of Toeplitz matrix, we present a recursive algorithm of linear algebraic equations from by using Toeplitz matrix as coefficient matrix , and also offer the proof of the recursive formula. The algorithm, making better use of the structure of Toeplitz matrices, effectively reduces calculation cost. F

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Kılıçman, Adem, and Wasan Ajeel Ahmood. "On matrix fractional differential equations." Advances in Mechanical Engineering 9, no. 1 (2017): 168781401668335. http://dx.doi.org/10.1177/1687814016683359.

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The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fracti

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Elmaci, Deniz, Nurcan Baykus, and Savasaneril . "THE LUCAS POLYNOMIAL SOLUTION OF LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS." Matrix Science Mathematic 6, no. 1 (2022): 21–25. http://dx.doi.org/10.26480/msmk.01.2022.21.25.

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In this study, linear Volterra-Fredholm integral equations are approximatively solved in terms of Lucas polynomials about any point in this study using a practical matrix approach. This technique uses collocation points and Lucas polynomials to transform the aforementioned linear Volterra-Fredholm integral problem into a matrix equation. Lucas coefficients are unknown in the system of linear algebraic equations. With the use of an error estimation, some illustrated examples are also provided. The outcomes demonstrate how effective and practical the suggested methodology is. Code was created in

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35

Ivashnev, L. I. "Methods of linear multiple regression in a matrix form." Izvestiya MGTU MAMI 9, no. 4-4 (2015): 35–41. http://dx.doi.org/10.17816/2074-0530-67011.

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The article contains a summary of three basic and two weighted linear multiple regression tech- niques in matrix form, together with the method of least squares of Gauss constitute a new tool re- gression analysis. The article contains a matrix formula that can be used to obtain equations of line- ar multiple regression and the basic weighted least-squares method to obtain regression equations without constant term and the method of obtaining the regression equations of general form. The article provides an example of use of matrix methods to obtain the coefficients of regression equa- tion of

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36

Guerarra, Sihem. "Orthogonality and unitary conditions for solutions to some consistent linear matrix equations." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 1 (2024): 2608–25. http://dx.doi.org/10.54021/seesv5n1-128.

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The nonlinear matrix equation (A1 + B1X1D1)(A2 + B2X2D2) = A, presents a matrix identity, where A, Ai, Bi, and Di are known matrices of suitable sizes and Xi are unknown matrices for i = 1,2, over the field of complex numbers ℂ. Several known simple methods in linear algebra can handle the orthogonality problem. In this paper, we select some linear matrix equations as illustrative examples to discuss some properties of pairs of identically dimensional consistent linear matrix equations, then apply specific matrix analytic tools, among them is the matrix rank method, the rank of a matrix is one

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37

Li, Jin. "Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation." AIMS Mathematics 8, no. 7 (2023): 16494–510. http://dx.doi.org/10.3934/math.2023843.

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<abstract><p>The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.</p></abstract>

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Li, Jin, and Yongling Cheng. "Barycentric rational interpolation method for solving KPP equation." Electronic Research Archive 31, no. 5 (2023): 3014–29. http://dx.doi.org/10.3934/era.2023152.

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<abstract><p>In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving

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Zhang, Jian-bing, Canyuan Gu, and Wen-Xiu Ma. "Generalized Matrix Exponential Solutions to the AKNS Hierarchy." Advances in Mathematical Physics 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/1375653.

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Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponen

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Mansimov, K. B., and R. O. Mastaliyev. "Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations." Bulletin of Irkutsk State University. Series Mathematics 36 (2021): 29–43. http://dx.doi.org/10.26516/1997-7670.2021.36.29.

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The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partia

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W., S. W. Daud, Ahmad N., and Malkawi G. "A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation." TELKOMNIKA Telecommunication, Computing, Electronics and Control 19, no. 2 (2021): pp. 583∼598. https://doi.org/10.12928/TELKOMNIKA.v19i2.18023.

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Matrix equations have its own important in the field of control system engineering particularly in the stability analysis of linear control systems and the reduction of nonlinear control system models. There are certain conditions where the classical matrix equation are not well equipped to handle the uncertainty problems such as during the process of stability analysis and reduction in control system engineering. In this study, an algorithm is developed for solving fully fuzzy matrix equation particularly for A~X~B~-X~ = C~, where the coefficients of the equation are in near-zero fuzzy number

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Ilmi, Ulul. "Linear Equation System Study on Electrical Circuits Using Matlab." Jurnal Elektro 3, no. 1 (2018): 1. http://dx.doi.org/10.30736/je.v3i1.210.

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In everyday life, especially in the electrical circuit, there are many usage of matrix. One use of a matrix is found in the system of linear equations. In the field of electrical circuits there are also problems involving systems of linear equations in matrix form. To solve the system of linear equations in matrix form, in addition to using elementary row operations, also used matlab.

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Bolat, Cennet, and Ahmet İpek. "On the Solutions of Some Linear Complex Quaternionic Equations." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/563181.

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Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

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Akram, Muhammad, Ghulam Muhammad, Tofigh Allahviranloo, and Nawab Hussain. "LU Decomposition method to solve bipolar fuzzy linear systems." Journal of Intelligent & Fuzzy Systems 39, no. 3 (2020): 3329–49. http://dx.doi.org/10.3233/jifs-201187.

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The aim of this work is to solve the linear system of equations using LU decomposition method in bipolar fuzzy environment. We assume a special case when the coefficient matrix of the system is symmetric positive definite. We discuss this point in detail by giving some numerical examples. Moreover, we investigate m × n inconsistent bipolar fuzzy matrix equation and find the least square solution of the inconsistent bipolar fuzzy matrix using the generalized inverse matrix theory. The existence of the strong bipolar fuzzy least square solution of the inconsistent bipolar fuzzy matrix is discuss

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Reddy, M. Krishna. "Linear Algebra & Matrix in Mathematics: Understanding Importance." International Journal of Research and Review 11, no. 11 (2024): 592–97. https://doi.org/10.52403/ijrr.20241160.

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In linear algebra, we learn the basics of linear systems by studying matrices and vectors. A branch of mathematics known as linear algebra primarily studies vectors, vector spaces (sometimes called linear spaces), linear mappings (often called transformations), and systems of linear equations. Since vector spaces are fundamental to contemporary mathematics, abstract algebra and functional analysis both make extensive use of linear algebra. Analytic geometry provides a tangible example of linear algebra, while operator theory provides a generalization of the theory. It finds widespread use in t

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KIM, BUM-SHIN, SEONG-YEON YOO, and WOONG-SUN CHO. "DEVELOPMENT OF ANALYSIS PROGRAM FOR INCOMPRESSIBLE LOOPED FLOW NETWORK USING CONSTITUTIVE TOPOLOGIC MATRIX EQUATION." International Journal of Air-Conditioning and Refrigeration 20, no. 02 (2012): 1250005. http://dx.doi.org/10.1142/s2010132512500058.

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Flow network which consists of a number of flow paths and their junctions is popularly used in analysis of flow and pressure distribution for complicated pipe or channel flow structure. In order to analyze flow network, it is required to resolve mass conservation equation at each junction and energy conservation equation on all independent closed loops of the flow network. Topologic matrix which reflects characteristics of network connectivity simply transforms continuity equations into linear algebraic equations. However, typical solving procedure based on topologic matrix and linear analysis

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Garai, Hiranmoy, Lakshmi Kanta Dey, Sintunavarat Wutiphol, Sumit Som, and Sayandeepa Raha. "ON NEW EXISTENCE OF A UNIQUE COMMON SOLUTION TO A PAIR OF NON-LINEAR MATRIX EQUATIONS." Journal of Mathematical Analysis 15, no. 6 (2024): 12–29. https://doi.org/10.54379/jma-2024-6-2.

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The main goal of this article is to investigate the existence of a unique positive definite common solution to a pair of matrix equations. Our focus will be on two particular types of matrix equation systems. In order to achieve our target, we take the help of elegant properties of Thompson metric on the set of all n × n Hermitian positive definite matrices. To proceed with this, we first derive a common fixed point result for a pair of mappings utilizing a certain class of control functions in a metric space. Then we obtain sufficient conditions to ensure a unique positive definite common sol

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A.S., Iskhakov, and Skovpen S.M. "Exact Solution of a Linear Difference Equation in a Finite Number of Steps." Journal of Progressive Research in Mathematics 13, no. 2 (2018): 2259–62. https://doi.org/10.5281/zenodo.3974630.

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An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

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Simeonov, P. S., and D. D. Bainov. "Estimates for the Cauchy matrix of perturbed linear impulsive equation." International Journal of Mathematics and Mathematical Sciences 17, no. 4 (1994): 753–58. http://dx.doi.org/10.1155/s0161171294001055.

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Zhao, Mi, Huifang Li, Shengtao Cao, and Xiuli Du. "An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems." Engineering Computations 36, no. 1 (2018): 161–77. http://dx.doi.org/10.1108/ec-07-2018-0312.

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Purpose The purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems. Design/methodology/approach The algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order ac

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