scholarly journals Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

2018 ◽  
Vol 18 (1) ◽  
pp. 21-32 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Volodymyr Makarov ◽  
Nataliia Romaniuk
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Exponential Convergence ◽  
Linear Boundary ◽  
Recursive Procedure ◽  
Content Type ◽  
Numerical Examples ◽  
Discrete Scheme ◽  
Symbolic Algorithm

AbstractA new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of m terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides super-exponential convergence rate as {m\to\infty}. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for {m=\infty} and a truncated symbolic algorithm (TSA) for a finite m. Numerical examples are presented to support the theory.

scholarly journals Sensitivity analysis of waveguide eigenvalue problems

2011 ◽  
Vol 9 ◽  
pp. 85-89 ◽  
Author(s):  
N. Burschäpers ◽  
S. Fiege ◽  
R. Schuhmann ◽  
A. Walther
Keyword(s):  
Sensitivity Analysis ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Wave Number ◽  
Design Parameters ◽  
Dielectric Waveguides ◽  
Integration Technique ◽  
Numerical Examples ◽  
Numerical Effort ◽  
Complex Solutions

Abstract. We analyze the sensitivity of dielectric waveguides with respect to design parameters such as permittivity values or simple geometric dependencies. Based on a discretization using the Finite Integration Technique the eigenvalue problem for the wave number is shown to be non-Hermitian with possibly complex solutions even in the lossless case. Nevertheless, the sensitivity can be obtained with negligible numerical effort. Numerical examples demonstrate the validity of the approach.

Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives

2016 ◽  
Vol 16 (4) ◽  
pp. 633-652 ◽  
Author(s):  
Ihor Demkiv ◽  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov
Keyword(s):  
Convergence Rate ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Exponential Convergence ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Linear Differential Operators ◽  
Principal Property ◽  
Linear Differential

AbstractA new algorithm for eigenvalue problems for linear differential operators with fractional derivatives is proposed and justified. The algorithm is based on the approximation (perturbation) of the coefficients of a part of the differential operator by piecewise constant functions where the eigenvalue problem for the last one is supposed to be simpler than the original one. Another milestone of the algorithm is the homotopy idea which results at the possibility for a given eigenpair number to compute recursively a sequence of the approximate eigenpairs. This sequence converges to the exact eigenpair with a super-exponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The proposed method possesses the following principal property: its convergence rate increases together with the index of the eigenpair. Numerical examples confirm the theory.

Efficient solution of the fuzzy eigenvalue problem in structural dynamics

2014 ◽  
Vol 31 (5) ◽  
pp. 864-878 ◽  
Author(s):  
Yuying Xia ◽  
M. Friswell
Keyword(s):  
Eigenvalue Problem ◽  
Perturbation Method ◽  
Degrees Of Freedom ◽  
Eigenvalue Problems ◽  
Structural Vibration ◽  
Design Problems ◽  
Content Type ◽  
Analysis And Design ◽  
Stiffness Matrices ◽  
Fuzzy Eigenvalues

Purpose – Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as fuzzy parameters. The requirement is to propagate the parameter uncertainty to the eigenvalues of the structure, specified as fuzzy eigenvalues. However, the usual approach is to transform the fuzzy problem into several interval eigenvalue problems by using the α-cuts method. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem. Design/methodology/approach – Based on the fundamental perturbation principle and vertex theory, an efficient perturbation method is proposed, that gives the exact extrema of the first-order deviation of the structural eigenvalue. The fuzzy eigenvalue approach has also been improved by reusing the interval analysis results from previous α-cuts. Findings – The proposed method was demonstrated on a simple cantilever beam with a pinned support, and produced very accurate fuzzy eigenvalues. The approach was also demonstrated on the model of a highway bridge with a large number of degrees of freedom. Originality/value – This proposed Vertex-Perturbation method is more efficient than the standard perturbation method, and more general than interval arithmetic methods requiring the non-negative decomposition of the mass and stiffness matrices. The new increment method produces highly accurate solutions, even when the membership function for the fuzzy eigenvalues is complex.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

A GM(1,N)-based economic cybernetics model for the high-tech industries in China

Kybernetes ◽  
2014 ◽  
Vol 43 (5) ◽  
pp. 672-685 ◽  
Author(s):  
Zheng-Xin Wang
Keyword(s):  
Differential Equation ◽  
Patent Application ◽  
Small Sample ◽  
Total Output ◽  
High Tech ◽  
Data Set ◽  
Content Type ◽  
Fixed Assets ◽  
Grey Differential Equation ◽  
Medical Equipments

Purpose – The purpose of this paper is to propose an economic cybernetics model based on the grey differential equation GM(1,N) for China's high-tech industries and provide the necessary support to assist high-tech industries management departments with their policy making. Design/methodology/approach – Based on the principle of grey differential equation GM(1,N), the grey differential equations of five high-tech industries in China are established using the net fixed assets, labor quantity and patent application quantity as cybernetics variables. After the discretization and first-order subtraction reduction to the simultaneous equation of the five grey models, a linear cybernetics model is resulted in. The structure parameters in the cybernetics system show explicit economic significance and can be identified through least square principle. At last, the actual data in 2004-2010 are introduced to empirically analyze the high-tech industrial system in China. Findings – The cybernetics system for China's high-tech industries are stable, observable, and controllable. On the whole, China's high-tech industries show higher output coefficients of the patent application quantity than those of net fixed assets and labor quantity. This suggests that China's industry development mainly depends on technological innovation rather than capital or labor inputs. It is expected that the total output value of China's high-tech industries will grow at an average annual rate of 15 percent in 2011-2015, with contributions of pharmaceuticals, aircraft and spacecraft, electronic and telecommunication equipments, computers and office equipments, medical equipments and meters by 21, 16, 13, 10, and 28 percent, respectively. In addition, pharmaceuticals, as well as medical equipments and meters, present upward proportions in the gross of Chinese high-tech industries significantly. Electronic and telecommunication equipments, plus computers and office equipments exhibit an obvious decreasing proportion. The proportion of the output value of aircraft and spacecraft is basically stable. Practical implications – Empirical analysis results are helpful for related management departments to formulate reasonable industrial policies to keep the sustained and stable development of the high-tech industries in China. Originality/value – Based on the grey differential equation GM(1,N), this research puts forward an economic cybernetics model for the high-tech industries in China. This model is applicable to the economic system with small sample data set.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

scholarly journals A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsaid ◽  
D. Hammad
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Gordon Equation ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Klein Gordon ◽  
Fractional Orders ◽  
Klein Gordon Equation

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

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