An efficient algorithm for solving generalized pantograph equations with linear functional argument

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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The Tensor Pade´-Type Approximant with Application in Computing Tensor Exponential Function

Journal of Function Spaces ◽

10.1155/2018/2835175 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Chuanqing Gu ◽

Yong Liu

Keyword(s):

Orthogonal Polynomials ◽

Generating Function ◽

Exponential Function ◽

Efficient Algorithm ◽

Linear Functional ◽

Function Form ◽

Numerical Examples ◽

First Time ◽

Formal Orthogonal Polynomials

Tensor exponential function is an important function that is widely used. In this paper, tensor Pade´-type approximant (TPTA) is defined by introducing a generalized linear functional for the first time. The expression of TPTA is provided with the generating function form. Moreover, by means of formal orthogonal polynomials, we propose an efficient algorithm for computing TPTA. As an application, the TPTA for computing the tensor exponential function is presented. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

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A taylor collocation method for solving high-order linear pantograph equations with linear functional argument

Numerical Methods for Partial Differential Equations ◽

10.1002/num.20600 ◽

2010 ◽

Vol 27 (6) ◽

pp. 1628-1638 ◽

Cited By ~ 10

Author(s):

Mustafa Gülsu ◽

Mehmet Sezer

Keyword(s):

Collocation Method ◽

Linear Functional ◽

High Order ◽

Order Linear ◽

Functional Argument

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Spectral Tau method for solving general fractional order differential equations with linear functional argument

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-019-0039-4 ◽

2019 ◽

Vol 27 (1) ◽

Cited By ~ 1

Author(s):

Kamal R. Raslan ◽

Mohamed A. Abd El salam ◽

Khalid K. Ali ◽

Emad M. Mohamed

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Linear Functional ◽

Numerical Technique ◽

Tau Method ◽

Fractional Order Differential Equations ◽

Functional Argument

Abstract In this paper, a numerical technique for solving new generalized fractional order differential equations with linear functional argument is presented. The spectral Tau method is extended to study this problem, where the derivatives are defined in the Caputo fractional sense. The proposed equation with its functional argument represents a general form of delay and advanced differential equations with fractional order derivatives. The obtained results show that the proposed method is very effective and convenient.

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A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2005.12.015 ◽

2007 ◽

Vol 200 (1) ◽

pp. 217-225 ◽

Cited By ~ 49

Author(s):

Mehmet Sezer ◽

Ayşegül Akyüz-Daşcıogˇlu

Keyword(s):

Numerical Solution ◽

Linear Functional ◽

Taylor Method ◽

Functional Argument

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Computational Micrograph Registration with Sieve Processes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164660 ◽

1996 ◽

Vol 54 ◽

pp. 440-441

Author(s):

P.J. Phillips ◽

J. Huang ◽

S. M. Dunn

Keyword(s):

Planar Graph ◽

Efficient Algorithm ◽

Spatial Organization ◽

Spatial Arrangement ◽

Stereo Image ◽

Image Model ◽

Geometric Invariants ◽

Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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