scholarly journals Simulation of microwave circuits and laser structures including PML by means of FIT

2005 ◽  
Vol 2 ◽  
pp. 107-112
Author(s):  
G. Hebermehl ◽  
J. Schefter ◽  
R. Schlundt ◽  
Th. Tischler ◽  
H. Zscheile ◽  
...  
Keyword(s):  
Electric Field ◽  
Boundary Value Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Microwave Circuits ◽  
Algebraic Equations ◽  
Fine Grid ◽  
Preconditioning Techniques ◽  
Linear Algebraic Equations

Abstract. Field-oriented methods which describe the physical properties of microwave circuits and optical structures are an indispensable tool to avoid costly and time-consuming redesign cycles. Commonly the electromagnetic characteristics of the structures are described by the scattering matrix which is extracted from the orthogonal decomposition of the electric field. The electric field is the solution of an eigenvalue and a boundary value problem for Maxwell’s equations in the frequency domain. We discretize the equations with staggered orthogonal grids using the Finite Integration Technique (FIT). Maxwellian grid equations are formulated for staggered nonequidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells. The interesting modes of smallest attenuation are found solving a sequence of eigenvalue problems of modified matrices. To reduce the execution time for high-dimensional problems a coarse and a fine grid is used. The calculations are carried out, using two levels of parallelization. The discretized boundary value problem, a large-scale system of linear algebraic equations with different right-hand sides, is solved by a block Krylov subspace method with various preconditioning techniques. Special attention is paid to the Perfectly Matched Layer boundary condition (PML) which causes non physical modes and a significantly increased number of iterations in the iterative methods.

scholarly journals A PROBLEM WITH PARAMETER FOR THE INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (1) ◽  
pp. 34-54
Author(s):  
Elmira A. Bakirova ◽  
Anar T. Assanova ◽  
Zhazira M. Kadirbayeva
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Loaded Differential Equation ◽  
Linear Algebraic Equations

The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.

scholarly journals ON A METHOD FOR SOLVING A LINEAR BOUNDARY VALUE PROBLEM FOR A SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS CONTAINING THE PARAMETER

2021 ◽  
Vol 73 (1) ◽  
pp. 23-31
Author(s):  
N.B. Iskakova ◽  
◽  
G.S. Alihanova ◽  
А.K. Duisen ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Degenerate Kernel ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations ◽  
The Given

In the present work for a limited period, we consider the system of integro-differential equations of containing the parameter. The kernel of the integral term is assumed to be degenerate, and as additional conditions for finding the values of the parameter and the solution of the given integro-differential equation, the values of the solution at the initial and final points of the given segment are given. The boundary value problem under consideration is investigated by D.S. Dzhumabaev's parametrization method. Based on the parameterization method, additional parameters are introduced. For a fixed value of the desired parameter, the solvability of the special Cauchy problem for a system of integro-differential equations with a degenerate kernel is established. Using the fundamental matrix of the differential part of the integro-differential equation and assuming the solvability of the special Cauchy problem, the original boundary value problem is reduced to a system of linear algebraic equations with respect to the introduced additional parameters. The existence of a solution to this system ensures the solvability of the problem under study. An algorithm for finding the solution of the initial problem based on the construction and solutions of a system of linear algebraic equations is proposed.

scholarly journals AN ALGORITHM FOR SOLVING A CONTROL PROBLEM FOR A DIFFERENTIAL EQUATION WITH A PARAMETER

2018 ◽  
pp. 25-32
Author(s):  
Dzhumabaev D.S. ◽  
Bakirova E.A. ◽  
Kadirbayeva Zh.M.
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

On a finite interval, a control problem for a linear ordinary differential equations with a parameter is considered. By partitioning the interval and introducing additional parameters, considered problem is reduced to the equivalent multipoint boundary value problem with parameters. To find the parameters introduced, the continuity conditions of the solution at the interior points of partition and boundary condition are used. For the fixed values of the parameters, the Cauchy problems for ordinary differential equations are solved. By substituting the Cauchy problem’s solutions into the boundary condition and the continuity conditions of the solution, a system of linear algebraic equations with respect to parameters is constructed. The solvability of this system ensures the existence of a solution to the original control problem. The system of linear algebraic equations is composed by the solutions of the matrix and vector Cauchy problems for ordinary differential equations on the subintervals. A numerical method for solving the origin control problem is offered based on the Runge-Kutta method of the 4-th order for solving the Cauchy problem for ordinary differential equations. Key words: boundary value problem with parameter, differential equation, solvability, algorithm.

scholarly journals Algorithm for using the boundary element method on the example of a mixed boundary value problem for the Poisson equation

2018 ◽  
Author(s):  
L. T. Boyko
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Normal Derivative ◽  
Algebraic Equations ◽  
Boundary Contour ◽  
Linear Algebraic Equations

The possibilities of the algorithm for applying the boundary element method to solving boundary value problems are discussed on the example of the two-dimensional Poisson differential equation. The algorithm does not change significantly when the type of boundary conditions changes: the Dirichlet problem, the Neumann problem, or a mixed boundary value problem. The idea of the algorithm is taken from the work of John T. Katsikadelis [1]. The algorithm is described in detail in the next sequence of actions. 1) The boundary- value problem for a two-dimensional finite domain is formulated. The desired function in the domain, its values, and its normal derivative on the boundary contour are connected by means of the second Green formula. 2) We pass from the boundary value problem for the Poisson equation to the boundary value problem for the Laplace equation. This simplifies the process of constructing an integral equation. We obtain the integral equation on the boundary contour using the boundary conditions. 3) In the integral equation, we divide the boundary contour into a finite number of boundary elements. The desired function and its normal derivative are considered constant values on each boundary element. We compose a system of linear algebraic equations considering these values. 4) We modify the system of linear algebraic equations taking into account the boundary conditions. After that, we solve it using the Gauss method. The computer program has been developed according to the developed algorithm. We used it in the learning process. The software implementation of the algorithm takes into account the capabilities of modern computer technology and modern needs of the educational process. The work of the program is shown in the test case. Further modification of the described algorithm is possible

scholarly journals NUMERICAL SOLUTION OF THE BOUNDARY PROBLEM FOR PARABOLIC EQUATION WITH NON-SELF-ADJOINT OPERATOR AND RELATED BOUNDARY CONDITIONS

2020 ◽  
pp. 7-11
Author(s):  
B. Dovgiy ◽  
L. Vakal ◽  
E. Vakal
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Diffusion Processes ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Second Order ◽  
Algebraic Equations ◽  
Sweep Method ◽  
Linear Algebraic Equations

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.

The irrotational flow of a perfect fluid past two spheres

1949 ◽  
Vol 45 (1) ◽  
pp. 81-87 ◽  
Author(s):  
I. N. Sneddon ◽  
J. Fulton
Keyword(s):  
Boundary Value Problem ◽  
Perfect Fluid ◽  
Infinite Series ◽  
Boundary Value ◽  
Laplace’S Equation ◽  
Algebraic Equations ◽  
Electrostatic Problem ◽  
Laplace's Equation ◽  
Linear Algebraic Equations ◽  
Infinite Set

1. The boundary value problem of Laplace's equation for two spheres is a classical one, and has been the subject of discussion by many mathematicians (1). The earliest attempt to solve a boundary value problem of this type is due to Poisson (2), but his analysis is applicable only to the electrostatic problem. The first of the methods which can be successfully applied to both electrostatic arid hydrodynamical problems was developed later by Lord Kelvin (3); this procedure, which is known as the ‘method of images’, was first applied to the problem of the motion of two spheres in a perfect fluid by Hicks (4). Another method of great generality, that of transforming Laplace's equation to bipolar coordinates and studying the solutions in these coordinates, was developed about the same time by Neumann (5) and much later by Jeffery (6). More recently a new method has been developed by Mitra (7) for the solution of the problem of two spheres in a potential field. It makes use of two sets of spherical polar coordinate systems; the solution is expressed in terms of infinite series whose coefficients satisfy an infinite set of linear algebraic equations. The chief interest of Mitra's method lies in the fact that he has found it possible to derive exact solutions of this infinite set of equations. All of these methods suffer from the disadvantage that the potential function is obtained in the form of an infinite series so that any numerical calculations are rendered cumbersome.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

scholarly journals A note on adaptivity in factorized approximate inverse preconditioning

2020 ◽  
Vol 28 (2) ◽  
pp. 149-159
Author(s):  
Jiří Kopal ◽  
Miroslav Rozložník ◽  
Miroslav Tůma
Keyword(s):  
Large Scale ◽  
Condition Numbers ◽  
Algebraic Equations ◽  
Approximate Factorization ◽  
Approximate Inverse ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
The Matrix ◽  
Preconditioned Iterative ◽  
Principal Submatrices

AbstractThe problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

scholarly journals Approximate method of solving one periodic optimal regulated boundary value problem

2019 ◽  
pp. 66-69
Author(s):  
I. Askerov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
General Problem ◽  
Asymptotic Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Nonlinear Algebraic Equations

In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

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