scholarly journals Accelerated spectral refinement Part II: Cluster of eigenvalues

2000 ◽  
Vol 42 (2) ◽  
pp. 224-243
Author(s):  
Rafikul Alam ◽  
Rekha P. Kulkarni ◽  
Balmohan V. Limaye
Keyword(s):  
Integral Operator ◽  
Simple Eigenvalue ◽  
Algebraic Multiplicity ◽  
Numerical Examples ◽  
Multiple Eigenvalues

AbstractThe framework for accelerated spectral refinement for a simple eigenvalue developed in Part I of this paper is employed to treat the general case of a cluster of eigenvalues whose total algebraic multiplicity is finite. Numerical examples concerning the largest and the second largest multiple eigenvalues of an integral operator are given.

scholarly journals ON CALCULATION OF MULTIPLE EIGENVALUES OF THE LINEAR OPERATOR-FUNCTION BY REDUCTION PSEUDO-PERTURBATION METHODS

2013 ◽  
Vol 14 (2) ◽  
Author(s):  
Rakhimov Davran Ganievich
Keyword(s):  
Linear Operator ◽  
Operator Function ◽  
Perturbation Methods ◽  
Simple Eigenvalue ◽  
Algebraic Multiplicity ◽  
Generalized Eigenvectors ◽  
Multiple Eigenvalues ◽  
Multiplicity Of Eigenvalue

ABSTRACT: In this article, methods of theory of bifurcations, applies to the problem of retaining of the approximately given multiple eigenvalues and their generalized eigenvectors. This approach allows the reduction of algebraic multiplicity of eigenvalue to one and transfers the problem to the similar one but with simple eigenvalue. The method of the false perturbation is used to construct iterative processes. ABSTRAK: Dalam artikel ini, kaedah teori bifurkasi diaplikasikan terhadap masalah untuk mengekalkan penghampiran pelbagai nilai eigen dan vektor eigen yang teritlak. Kaedah ini membenarkan pengurangan kegandaan aljabar nilai eigen kepada satu dan memindahkan permasalahan kepada yang hampir serupa tetapi dengan nilai eigen yang lebih mudah. Pengkaedahan usikan palsu digunakan untuk proses pelelaran.

scholarly journals Accelerated spectral refinement Part I: simple eigenvalue

2000 ◽  
Vol 41 (4) ◽  
pp. 487-507 ◽  
Author(s):  
Rafikul Alam ◽  
Rekha P. Kulkarni ◽  
Balmohan V. Limaye
Keyword(s):  
Integral Operator ◽  
General Framework ◽  
Rates Of Convergence ◽  
Higher Order ◽  
Simple Eigenvalue ◽  
Numerical Examples

AbstractA general framework is developed for constructing higher order spectral refinement schemes for a simple eigenvalue. Well-known techniques for ordinary spectral refinement are carried over to higher order spectral refinement yielding faster rates of convergence. Numerical examples are given by considering an integral operator.

The approximate solution of nonlinear Fredholm implicit integro-differential equation in the complex plane

2021 ◽  
Author(s):  
Samir Lemita ◽  
Sami Touati ◽  
Kheireddine Derbal
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Numerical Process

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases

2019 ◽  
Vol 12 (04) ◽  
pp. 1950055 ◽  
Author(s):  
Majid Erfanian ◽  
Hamed Zeidabadi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Haar Wavelet ◽  
Banach Fixed Point Theorem ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Rationalized Haar Wavelet

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

scholarly journals An efficient algorithm to solve damped forced oscillator problems by Bernoulli operational matrix of integration

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Mithilesh Singh ◽  
Seema Sharma ◽  
Sunil Rawan
Keyword(s):  
Exact Solution ◽  
Integral Operator ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Numerical Examples ◽  
Orthonormal Polynomials ◽  
Fractal Medium ◽  
Forced Oscillator ◽  
Operational Matrix Of Integration ◽  
Asymptotic Perturbation

AbstractAn asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives was obtained in Yang and Srivastava (Commun Nonlinear Sci Numer Simul 29(1–3):499–504, 2015). In this paper, we obtain the numerical solution of damped forced oscillator problems by employing the operational matrix of integration of Bernoulli orthonormal polynomials. The operational matrix of integration is determined with the help of the integral operator on Bernoulli orthonormal polynomials. Numerical examples of two different problems of spring are given to delineate the performance and perfection of this approach and compared the results with the exact solution.

A Generalized Stationary Algorithm for Resonant Tunneling: Multi-Mode Approximation and High Dimension

2011 ◽  
Vol 10 (4) ◽  
pp. 882-898 ◽  
Author(s):  
Naoufel Ben Abdallah ◽  
Hao Wu
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
High Dimension ◽  
Resonant Tunneling ◽  
Schrodinger Equation ◽  
High Dimensional ◽  
Numerical Examples ◽  
Multiple Eigenvalues ◽  
Multi Mode

AbstractThe multi-mode approximation is presented to compute the interior wave function of Schrödinger equation. This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered. The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

Conservative solute transport with periodic velocity and sinusoidal source concentration in semi-infinite porous media: An analytical solution

2014 ◽  
Vol 6 (1) ◽  
pp. 1024-1031
Author(s):  
R R Yadav ◽  
Gulrana Gulrana ◽  
Dilip Kumar Jaiswal
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Laplace Transformation ◽  
Seepage Velocity ◽  
Transformation Technique ◽  
Numerical Examples ◽  
One Dimensional ◽  
Porous Domain ◽  
Conservative Solute Transport ◽  
Source Concentration

The present paper has been focused mainly towards understanding of the various parameters affecting the transport of conservative solutes in horizontally semi-infinite porous media. A model is presented for simulating one-dimensional transport of solute considering the porous medium to be homogeneous, isotropic and adsorbing nature under the influence of periodic seepage velocity. Initially the porous domain is not solute free. The solute is initially introduced from a sinusoidal point source. The transport equation is solved analytically by using Laplace Transformation Technique. Alternate as an illustration; solutions for the present problem are illustrated by numerical examples and graphs.

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

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