Linear Algebraic System Research Articles

The Modelling of Light Absorption and Reflection in a SiOx/Si Structure with Al Nanoparticles for Solar Cells

In this study, the light propagation in a structure consisting of SiOx on Si substrate with Al nanoparticles regularly placed in the SiOx layer is considered. Numerical modelling is performed by solving the Maxwell equations for the electromagnetic waves. In distinction from the well-known finite-difference time-domain (FDTD) simulation technique, we do not solve time-dependent wave equations here; rather, we propose a new numerical technique. This technique allows us to determine the stationary amplitudes of the electromagnetic oscillations directly from the linear algebraic equation system. The obtained results apply to silicon solar cells with an SiOx + Al top layer to maximise their efficiency. We found that 26 nm and 39 nm diameters of spherical Al nanoparticles are nearly optimal for a λ = 435.8 nm wavelength of the incident light. In addition, we evaluated the (nearly) optimal parameters of their placement in the SiOx layer. The results show the possibility of increasing the efficiency of solar cells by increasing the light absorption inside the active Si layer from ≈60% to ≈80%. Future perspectives on the proposed method and its possible applications are discussed.

Numerical solution of fractional integro-differential equations using ultraspherical polynomials

In this work we will discuss the fractional integro-differential (FIDE) equations of the Fredholm and Volterra type, where we begin by presenting some special concepts about the definition of the fractional integral and the fractional derivative of Riemann-Liouville and Caputo, while addressing the Gegenbauer (Ultraspherical) polynomials and its special differential equation, mentioning the primary polynomials, in addition to presenting some theories such as the Krasnoselskii fixed point theorem, which guarantees the existence and uniqueness of solutions to fractional integro-differential equations (FIDE). To find the approximate solution of fractional integro-differential equations, we relied on orthogonal polynomials of the ultraspherical type (Gegenbauer), which have a great advantage in solving integral equations and integro-differential equations. These polynomials, with the application of fractional Caputo derivatives with the series method, in addition to fixing the values and changing the degree of the polynomials used with fixing the values of , lead to the equation becoming a linear algebraic system of equations. By solving this linear system, the approximate solution is obtained in the form of a Gegenbauer (ultraspherical) series. The change in the degree of the polynomials used will inevitably show the effectiveness and speed of the method through the examples that will be presented, where the approximate solutions obtained are compared with the exact solutions.

GPU Accelerating Algorithms for Three-Layered Heat Conduction Simulations

In this paper, we consider the finite difference approximation for a one-dimensional mathematical model of heat conduction in a three-layered solid with interfacial conditions for temperature and heat flux between the layers. The finite difference scheme is unconditionally stable, convergent, and equivalent to the solution of two linear algebraic systems. We evaluate various methods for solving the involved linear systems by analyzing direct and iterative solvers, including GPU-accelerated approaches using CuPy and PyCUDA. We evaluate performance and scalability and contribute to advancing computational techniques for modeling complex physical processes accurately and efficiently.

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate.

A Beneficial Numerical Approach to Solve Systems of Linear Integro-Differential Equations

The system of linear Fredholm-Volterra integro-differential equations (FVIDEs) has been solved in this paper by an improved approximation method. Generalised Bernstein polynomials and collocation points have been used to construct the theory of the method. The aim of the technique is to reduce systems of integro-differential equations into an algebraic matrix equation, which corresponds to a linear algebraic equation system, by means of Bernstein polynomials. In order to analyse the applicability of the method, some illustrative examples have also been considered. It has been shown that the proposed method is faster and more effective than the others when comparing the numerical results.

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

Математическая модель в задаче идентификации формы летательного аппарата по диаграмме рассеяния звукового поля

This paper proposes a mathematical model to solve an inverse problem reconstructing the shape and size of possible models of an unmanned aerial vehicle based on its acoustic trail. To simplify the calculations, four two-dimensional simplest geometric shapes were chosen for numerical experiments - a circle, an ellipse, a square and a four-petal rose. The scattering diagrams of the sound field, as a solution to direct diffraction problem for each contour, were taken as the input data. A residual functional is constructed as a difference between the true data of the scattered sound field and the data obtained using the semi-analytical boundary integral equation method and the Fredholm integral equation of the second kind, in the direct diffraction problem. The minimization of this functional leads to the inverse diffraction problem. After discretizing the boundary curve, the problem is reduced to a linear algebraic equation system, from which a function is found that numerically describes the shape of the object.

Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.

Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.

Solution of Volterra Integral Equations of the Second Kind with Weakly Singular Kernels Using Legendre Polynomials

The shifted Legendre polynomials of the first kind are employed to solve second kind Volterra integral equations with weakly singular kernels. In this method the unknown function and the data function are approximated through three matrices, while the kernel function is approximated through five matrices. Regarding the kernel's two variables, it will be approximated twice, first with respect to variable x and second with respect to variable t. The singularity of the kernel is removed analytically. It is proved that the solution is equivalent to an algebraic linear system without applying the collocation method. Two numerical experiments are solved to illustrate the efficiency of the proposed method.

Interval Iterative Decreasing Dimension Method for Interval Linear Systems and Its Implementation to Analog Circuits

The iterative decreasing dimension method (IDDM) is an iterative method used to solve the linear algebraic system Ax=f. Such systems are important in modeling many problems in applied sciences. For a number of reasons, such as estimated measurements made for modeling, errors arising from floating point calculations, and approximation methods used for solutions, it becomes necessary to study intervals in the solutions of systems of linear equations. The objective of this paper is to utilize IDDM to achieve resolution in the interval linear system (ILS). During the calculations, the Kaucher space is considered an extended classical interval space. The solutions of Barth-Nuding and Hansen interval linear systems, which are commonly used in the literature to test the solutions of ILSs, are obtained with the interval iterative decreasing dimension method for interval linear systems (I-IDDM). Since IDDM is a variation method of Gaussian elimination, a comparative analysis of the results with the interval Gaussian elimination method (I-GEM) is performed. It has been demonstrated that our approach, I-IDDM, produces better outcomes than I-GEM. I-IDDM is also used to investigate the analog circuit problem, where interval analysis is crucial.

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method.

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.

Phase field modeling and numerical algorithm for two-phase dielectric fluid flows

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property leads to a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data.

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

Linear relaxation method with regularized energy reformulation for phase field models

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models.

Orthonormal discrete Legendre polynomials for stochastic distributed‐order time‐fractional fourth‐order delay sub‐diffusion equation

In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples.

A Highly Accurate Computational Approach to Solving the Diffusion Equation of a Fractional Order

This study aims to present and apply an effective algorithm for solving the TFDE (Time-Fractional Diffusion Equation). The Chebyshev cardinal polynomials and the operational matrix for fractional derivatives based on these bases are relied on as crucial tools to achieve this objective. By employing the pseudospectral method, the equation is transformed into an algebraic linear system. Consequently, solving this system using the GMRES method (Generalized Minimal Residual) results in obtaining the solution to the TFDE. The results obtained are very accurate, and in certain instances, the exact solution is achieved. By solving some numerical examples, the proposed method is shown to be effective and yield superior outcomes compared to existing methods for addressing this problem.

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.