Direct And Indirect Optical Solutions To Linear Algebraic Equations: Error Source Modeling

1983 ◽  
Author(s):  
David Casasent ◽  
Anjan Ghosh ◽  
Charles P. Neuman
Keyword(s):  
Algebraic Equations ◽  
Error Source ◽  
Source Modeling ◽  
Linear Algebraic Equations

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1063
Author(s):  
Vladimir Mityushev ◽  
Zhanat Zhunussova
Keyword(s):  
Effective Conductivity ◽  
Dimensional Space ◽  
Elementary Function ◽  
Voronoi Tessellation ◽  
Random Packing ◽  
Periodicity Cell ◽  
Algebraic Equations ◽  
Optimal Packing ◽  
Linear Algebraic Equations ◽  
Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

The laminar boundary-layer equation: a method of solution by means of an automatic computer

1955 ◽  
Vol 51 (2) ◽  
pp. 320-332 ◽  
Author(s):  
D. C. F. Leigh
Keyword(s):  
Boundary Layer ◽  
Asymptotic Solution ◽  
Laminar Boundary Layer ◽  
Iterative Process ◽  
Boundary Layer Equation ◽  
Algebraic Equations ◽  
Automatic Computer ◽  
Linear Algebraic Equations ◽  
Method Of Solution ◽  
Incompressible Boundary

ABSTRACTA method, very suitable for use with an automatic computer, of solving the Hartree-Womersley approximation to the incompressible boundary-layer equation is developed. It is based on an iterative process and the Choleski method of solving a simultaneous set of linear algebraic equations. The programming of this method for an automatic computer is discussed. Tables of a solution of the boundary-layer equation in a region upstream of the separation point are given. In the upstream neighbourhood of separation this solution is compared with Goldstein's asymptotic solution and the agreement is good.

Solution of Levi Civita’s Problem by Infinite Matrix Inversion

1973 ◽  
Vol 40 (1) ◽  
pp. 31-36 ◽  
Author(s):  
M. Bentwich
Keyword(s):  
Flow Pattern ◽  
Flow Direction ◽  
Leading Edge ◽  
Arbitrary Cross Section ◽  
Matrix Inversion ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Flow Past ◽  
Linear Algebraic Equations ◽  
Wet Surface

The author proposes a new method by which one can solve for the two-dimensional irrotational fully cavitating flow past a cylinder of arbitrary cross section. Unlike the available solutions, it is in the form of two expansions each valid in part of the complex potential plane w = Φ + iΨ. The a priori unknown coefficients in the two expansions are linked by infinitely many linear algebraic equations. By inverting the associated matrix and utilizing the boundary condition, that represent the geometry of the wet surface, the coefficients in the expansions are evaluated and the solution is completed. Cases in which the wet surface is circular, the pressure along the free streamlines is constant, and the entire flow pattern is symmetric with respect to flow direction at infinity are considered in detail. Also, the well-known solution for the flow past a flat plate is compared to that obtained by the method of matrix inversion. Judging from these results, the convergence of the series appears to be very rapid. The author finally discusses the applicability of the method to cases in which the obstacle has a sharp leading edge, the pressure in the cavity is not uniform, or the flow pattern is not symmetric.

Analysis of Dynamic Systems With Periodically Varying Parameters via Chebyshev Polynomials

1991 ◽  
Author(s):  
S. C. Sinha ◽  
Der-Ho Wu ◽  
Vikas Juneja ◽  
Paul Joseph
Keyword(s):  
Dynamic Systems ◽  
Chebyshev Polynomials ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Varying Parameters ◽  
Approximate Analytical Solutions ◽  
Linear Algebraic Equations ◽  
Mathieu’S Equation ◽  
Runge Kutta

Abstract In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three blade helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton and Gear algorithms. The results obtained in the both examples indicate that the suggested approach extremely accurate and is by far the most efficient one.

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