Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations

Yüzbaşı, Şuayip; Gök, Emrah; Sezer, Mehmet

doi:10.1002/mma.2801

Cited by 19 publications

(6 citation statements)

References 28 publications

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“…Equation (32) can be written in the following form: Journal of Applied Mathematics Walsh series method [26] DUSF series method [27] Hermite series method [3] Taylor series method [4] Laguerre matrix method [28] PIA ( …”

Section: Pia(1 1) Solutionmentioning

confidence: 99%

Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

Bahşı

Çevik

2015

Journal of Applied Mathematics

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The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.

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Section: Pia(1 1) Solutionmentioning

confidence: 99%

Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

Bahşı

Çevik

2015

Journal of Applied Mathematics

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“…as λ ( τ ) = φ ( τ ) + i ω ( τ ) such that φ ( τ 0 ) = 0, ω ( τ 0 ) = ω 0 . From , we obtain

τ_{j} = \frac{1}{ω_{0}} \arccos ((), \frac{a_{4} ω_{0}^{4} + (a_{1} a_{3} - a_{2} a_{4}) ω_{0}^{2} - a_{3} a_{5}}{a_{3}^{2} + a_{4}^{2} ω_{0}^{2}}) + \frac{2 jπ}{ω_{0}}, j = 0, 1, 2, \dots

and

τ_{0} = \frac{1}{ω_{0}} \arccos ((), \frac{a_{4} ω_{0}^{4} + (a_{1} a_{3} - a_{2} a_{4}) ω_{0}^{2} - a_{3} a_{5}}{a_{3}^{2} + a_{4}^{2} ω_{0}^{2}})

According to , we have a transversality condition:

{(∥, \frac{normald}{normald τ} Reλ (τ))}_{τ = τ_{0}} = {(|, \frac{normald}{normald τ} φ (τ))}_{τ = τ_{0}} > 0

When τ > τ 0 , the real part of λ ( τ ) becomes positive; hence, the steady state

…”

Section: Model Analysismentioning

confidence: 99%

A multi‐stage ‘infection’ model of stock investors' reaction to new product announcement signal

Hao

Wang

2016

Math Methods in App Sciences

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Drawing on viral dynamics theory, this paper presents a differential equations model with time delay to investigate the stock investor behavior driven by new product announcement (NPA) signal. Visually, we look upon investors in stock market as cells in vivo and the NPA signals as free virus. The potential investors will be ‘infected’ by the dissociative NPA signal and then make investment decisions. In order to better understand the ‘infection’ process, we extract and establish a multi‐stage process during which NPA signal is delivered and ‘infects’ the potential investors. A time‐delay effect is employed to reflect the evaluation stage at which potential investors comprehensively evaluate and decide whether to invest or not. In addition, we introduce a set of external and internal factors into the model, including information sensitivity and investor sentiment, and so on, which are pivotal for examining investor behavior. Equilibrium analysis and numerical simulations are employed to check out the properties of the model and highlight the practical application values of the model. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Utilizing diverse kinds of polynomials such as Legendre, Chebyshev, Bessel, Chelyshkov, Laguerre, and Vieta-Lucas functions is considered in literature. [14][15][16][17][18][19][20][21][22][23] Combinations of orthogonal functions with quasilinearization method (QLM) have been successfully applied to many important models in physical sciences, see, cf. previous studies.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that collocation‐based numerical approximations provide a promising tool to treat various initial and boundary value model problems in science and engineering. Utilizing diverse kinds of polynomials such as Legendre, Chebyshev, Bessel, Chelyshkov, Laguerre, and Vieta‐Lucas functions is considered in literature 14–23 . Combinations of orthogonal functions with quasilinearization method (QLM) have been successfully applied to many important models in physical sciences, see, cf.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

Abbasi

Izadi

Hosseini

2022

Math Methods in App Sciences

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This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted L2$$ {L}_2 $$ norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

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Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations

Cited by 19 publications

References 28 publications

Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

A multi‐stage ‘infection’ model of stock investors' reaction to new product announcement signal

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

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