A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method

Dehghan, Mehdi; Aryanmehr, S.; Eslahchi, M. R.

doi:10.1016/j.cam.2012.11.013

Cited by 28 publications

(19 citation statements)

References 31 publications

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“…Here, we solve them with initial conditions by using the proposed method for different values of h and m , and we compare the results with those in Costabile and Napoli by using a Hermite collocation method (HCM), in Noor and Mohyud‐Din by using an efficient method (EM), and in Guo and Yan using a Legendre‐Gauss collocation method (LGCM). In Example , we solve a Lane‐Emden type differential equation with the presented method and compare our results with collocation and Birkhoff‐type interpolation methods. In Example , we solve a singularly‐perturbed differential equation with proposed method and compare the results with the Chebyshev‐Gauss method (CGM), a perturbation method (PM), the B‐spline collocation method (BCM), and Laguerre approach method (LAM) .…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In Costabile and Napoli, a general collocation method has been derived for the numerical solution of high even order differential equations with two‐point Hermite boundary conditions. Also, in Costabile and Napoli and Dehghan et al, similar ideas have been generalized. Note that the idea in Costabile and Napoli is not valid for differential equations with initial conditions, and it is not a multistep scheme.…”

Section: Introductionmentioning

confidence: 97%

“…With the rapid growth of science and industry in the 20th century, the numerical methods that solve differential equations attract much attentions in different new fields. [1][2][3][4][5] Some of the recent methods include differential transform methods, [6][7][8][9] spectral Galerkin methods, [10][11][12] wavelet methods, 9,[13][14][15][16][17][18] collocation methods, [19][20][21][22][23][24] Legendre methods, [25][26][27] and some other numerical methods for ordinary differential equations. [28][29][30][31][32][33][34][35][36] We consider the following model problem:…”

Section: Introductionmentioning

confidence: 99%

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A collocation method for differential equations with one‐point Taylor initial conditions

Ghabarpoor

Kajani

Allame

2019

Math Methods in App Sciences

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This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A collocation method for differential equations with one‐point Taylor initial conditions

Ghabarpoor

Kajani

Allame

2019

Math Methods in App Sciences

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“…We also show how the proposed class of methods includes the methods presented in previous works [12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Section: Some Particular Casesmentioning

confidence: 99%

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Costabile¹,

Gualtieri²,

Napoli³

2017

Dynamical Systems - Analytical and Computational Techniques

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In this chapter, the connection between general linear interpolation and initial, boundary and multipoint value problems is explained. First, a result of a theoretical nature is given, which highlights the relationship between the interpolation problem and the Fredholm integral equation for high-order differential problems. After observing that the given problem is equivalent to a Fredholm integral equation, this relation is used in order to determine a general procedure for the numerical solution of high-order differential problems by means of appropriate collocation methods based on the integration of the Fredholm integral equation. The classical analysis of the class of the obtained methods is carried out. Some particular cases are illustrated. Numerical examples are given in order to illustrate the efficiency of the method.

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“…• Legendre operational matrix of integration, Pandey et al [26]. • Birkhoff interpolation method, Dehghan et al [27].…”

Section: Introductionmentioning

confidence: 99%

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

Ahmad

Raja

Bilal

et al. 2016

Neural Comput & Applic

109

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In the present study, stochastic numerical computing approach is developed by applying artificial neural networks (ANNs) to compute the solution of Lane-Emden type boundary value problems arising in thermodynamic studies of the spherical gas cloud model. ANNs are used in an unsupervised manner to construct the energy function of the system model. Strength of efficient local optimization procedures based on active-set (AS), interior-point (IP) and sequential quadratic programming (SQP) algorithms is used to optimize the energy functions. The performance of all three design methodologies ANN-AS, ANN-IP and ANN-SQP is evaluated on different nonlinear singular systems. The effectiveness of the proposed schemes in terms of accuracy and convergence is established from the results of statistical indicators.

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A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method

Cited by 28 publications

References 31 publications

A collocation method for differential equations with one‐point Taylor initial conditions

A collocation method for differential equations with one‐point Taylor initial conditions

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

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