Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane–Emden type

Doha, E. H.; Abd-Elhameed, W. M.; Youssri, Y. H.

doi:10.1016/j.newast.2013.03.002

Cited by 95 publications

(65 citation statements)

References 20 publications

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“…Using equation (10) It can be seen from Table 1 and Figure 1 that the present method's errors are superior to those of others [6,17]. Figure 2 shows the effect of the residual correction technique.…”

Section: Example 1 Let Us Consider the Following Problemmentioning

confidence: 87%

“…Errors of the present method without the residual correction are better than those of the exponential approximation method [5]. Also, it is seen from Table 2 and Figure 4 that the results of our method are better than those of the Taylor method [6]. Example 3 Consider the following multi-pantograph equation [14]:…”

Section: Example 2 Let Us Consider the Following Differential Equatiomentioning

confidence: 95%

“…It is observed from Figure 3 that the errors decrease while the value of N increases. Hermite collocation method [6] Jacobi rational-Gauss method [17] Present method N=8 N=26 N=8 N=10 0.2 6.2E-9…”

Section: Example 1 Let Us Consider the Following Problemmentioning

confidence: 99%

“…Operational matrices of integration, derivation, and product for Bernstein, Jacobi, and Chebyshev * Correspondence: yuzbasi@akdeniz.edu.tr polynomials were presented in [2,6,10,17]. Our aim is to derive operational matrices of integration and product for Taylor polynomials and utilize them for solving pantograph type differential equations.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Taylor operation method for solutions of generalized pantograph type delay differential equations

Yüzbaşı¹,

Ismailov²

2018

Turk J Math

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In this paper, a new operational matrix method based on the Taylor polynomials is presented to solve generalized pantograph type delay differential equations. The method is based on operational matrices of integration and product for Taylor polynomials. These matrices are obtained by using the best approximation of function by the Taylor polynomials. The advantage of the method is that the method does not require collocation points. By using the proposed method, the generalized pantograph equation problem is reduced to a system of linear algebraic equations. The solving of this system gives the coefficients of our solution. Numerical examples are given to demonstrate the accuracy of the technique and the numerical results show that the error of the present method is superior to that of other methods.The residual correction technique is used to improve the accuracy of the approximate solution and estimation of absolute error. The estimation aspect of the residual method is useful when the exact solution of the considered equation is unknown and this feature of the method is shown in numerical examples.

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Section: Example 1 Let Us Consider the Following Problemmentioning

confidence: 87%

Section: Example 2 Let Us Consider the Following Differential Equatiomentioning

confidence: 95%

Section: Example 1 Let Us Consider the Following Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Taylor operation method for solutions of generalized pantograph type delay differential equations

Yüzbaşı¹,

Ismailov²

2018

Turk J Math

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“…(1) Boubakar polynomial expansion scheme [5], Legendre operational matrix of differenciation [6] and modified Legendre spectral method [7] -In 2013, different cases of these polytropic equations are studied by second kind Chebyshev operational matrix [8], by Haar wavelet [9] and by homotopy perturbation method (HPM) and fourier transform [10].…”

Section: Introductionmentioning

confidence: 99%

A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

Khan

Shaikh

2017

Journal of Artificial Intelligence and Soft Computing Research

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The actual motivation of this paper is to develop a functional link between artificial neural network (ANN) with Legendre polynomials and simulated annealing termed as Legendre simulated annealing neural network (LSANN). To demonstrate the applicability, it is employed to study the nonlinear Lane-Emden singular initial value problem that governs the polytropic and isothermal gas spheres. In LSANN, minimization of error is performed by simulated annealing method while Legendre polynomials are used in hidden layer to control the singularity problem. Many illustrative examples of Lane-Emden type are discussed and results are compared with the formerly used algorithms. As well as with accuracy of results and tranquil implementation it provides the numerical solution over the entire finite domain.

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Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations

Youssri

Abd-Elhameed

2020

Fractional Order Analysis

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Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane–Emden type

Cited by 95 publications

References 20 publications

A Taylor operation method for solutions of generalized pantograph type delay differential equations

A Taylor operation method for solutions of generalized pantograph type delay differential equations

A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations

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