Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions by Haifa Bin Jebreen

“…Using this matrix, the spectral element method reduces the desired equation to a system of algebraic equations. To find the solution, the generalized minimal residual method (GMRES method) and Newton’s method are used in linear and nonlinear forms of this system, respectively. …”
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Matrix Formulation of EISs of Graphs and Its Application to WSN Covering Problems by Yongyi Yan, Jumei Yue, He Deng

“…By introducing a characteristic value of a vertex subset of a graph and using the algebraic representation of pseudological functions, several necessary and sufficient conditions of matrix form are proposed to express the externally independent sets (EISs), minimum externally independent sets (MEISs), and kernels of graphs. …”
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Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films by Matthew Michal, Marina Chugunova, Roman Taranets

“…Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. …”
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Dimensional Enhancement via Supersymmetry by M. G. Faux, K. M. Iga, G. D. Landweber

“…We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. …”
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Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions by S. Akhlaghi, M. Tavassoli Kajani, M. Allame

“…Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. …”
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Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients by Li-hua Zhang

“…At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.…”
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Numerical Methods Based on the Hybrid Shifted Orthonormal Polynomials and Block-Pulse Functions for Solving a System of Fractional Differential Equations by Abdulqawi A. M. Rageh, Adel R. Hadhoud

“…Using these hybrid bases and the operational matrices method, the system of fractional differential equations is reduced to a system of algebraic equations. Error analysis is performed and some simulation examples are provided to demonstrate the efficacy of the proposed techniques. …”
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Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients by M.H. Heydari, D. Baleanu, M. Bayramu

“…The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.…”
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Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions by Bashar Zogheib, Emran Tohidi, Stanford Shateyi

“…., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. …”
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Prime Ideals and Three-generated Ideals with Large Regularity by McCullough, Jason

“…Relatedly, Caviglia et. al. proved the existence of a function $\Psi (e)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}(P) \le \Psi (\deg (P))$. …”
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Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations by S. M. Sayed, A. S. Mohamed, E. M. Abo-Eldahab, Y. H. Youssri

“…We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. …”
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Application of Radial Basis Function Method for Solving Nonlinear Integral Equations by Huaiqing Zhang, Yu Chen, Chunxian Guo, Zhihong Fu

“…The unknown function was firstly interpolated by MQ functions and then by forming the nonlinear algebraic equations by the collocation method. Finally, the coefficients of RBFs were determined by Newton’s iteration method and an approximate solution was obtained. …”
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Free Vibrations with Large Amplitude of Axially Loaded Beams on an Elastic Foundation Using the Adomian Modified Decomposition Method by Desmond Adair, Askar Ibrayev, Alima Tazabekova, Jong R. Kim

“…The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple mathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural frequencies and corresponding closed-form series solution of the mode shapes. …”
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Graph Invariants and Large Cycles: A Survey by Zh. G. Nikoghosyan

“…In this paper, we have collected 36 pure algebraic relations between basic (initial) graph invariants ensuring the existence of a certain type of large cycles. …”
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Efficient Convex Optimization of Reentry Trajectory via the Chebyshev Pseudospectral Method by Chun-Mei Yu, Dang-Jun Zhao, Ye Yang

“…The Chebyshev-Gauss Legend (CGL) node points are used to transcribe the original dynamic constraint into algebraic equality constraint; therefore, a nonlinear programming (NLP) problem is concave and time-consuming to be solved. …”
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BUCKLING OF BEAMS WITH A BOUNDARY ELEMENT TECHNIQUE by ABDERRAZEK MESSAOUDI, LASZLO PETER KISS

“…The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. …”
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Approximate Series Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology by Randhir Singh, Jitendra Kumar, Gnaneshwar Nelakanti

“…Unlike the classical ADM, the proposed method avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. …”
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A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta

“…The derivatives of these polynomials and their operational metrics of derivatives are established to tackle the problem using the typical collocation method to transform the nonlinear fifth-order Korteweg–de Vries equation governed by its underlying conditions into a system of nonlinear algebraic equations, thereby obtaining the approximate solutions. …”
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Projections in Moduli Spaces of the Kleinian Groups by Hala Alaqad, Jianhua Gong, Gaven Martin

“…In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.…”
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A Numerical Approach for the Analytical Solution of Multidimensional Wave Problems by Jinxing Liu, Jiahua Fang, Muhammad Nadeem, Yahya Alsayyad

“…This NIS does not require any discretization in the findings of results and derives the algebraic equations. Some numerical models and graphical representations are illustrated to verify the performance of this scheme.…”
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