A Comparative Study of Two Legendre-Collocation Schemes Applied to Fractional Logistic Equation

Izadi, Mohammad

doi:10.1007/s40819-020-00823-4

Cited by 19 publications

(13 citation statements)

References 26 publications

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“…We now go beyond σ=1 and examine the behaviour of the solutions for 0<σ<1. Based on our experiments and previous numerical proofs [13,14], choosing α=σ yields the smallest magnitude of errors. Because of this, we use σ=0.25,0.5,0.75 and show the numerical results in figure 3 with J=8 and α=σ.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…In this respect, tremendous numerical and computational procedures have been investigated in the past towards solving FDEs. Let us mention only the variational iteration method and the Adomian decomposition method [5], the differential transform method [6], the local discontinuous Galerkin approaches [7–9], the spectral collocation methods based on Chebyshev [10–12], those based on Legendre [13,14], Bessel of the first kind [15], the novel Bessel functions [16,17] and the wavelet-based techniques [18,19].…”

Section: Introductionmentioning

confidence: 99%

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A novel matrix technique for multi-order pantograph differential equations of fractional order

Izadi

Srivástava

2021

Proc. R. Soc. A.

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The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted by S n ( x ) and called special polynomials , were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

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Section: Computational Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A novel matrix technique for multi-order pantograph differential equations of fractional order

Izadi

Srivástava

2021

Proc. R. Soc. A.

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“…The Legendre polynomials satisfy the following relations [ 17 ]

where

denotes the Kronecker delta. The first property shows that these set of orthogonal polynomials are orthogonal with respect to weighting function

.…”

Section: Discretized Ldg Formulationmentioning

confidence: 99%

“…Besides the analytical investigations, numerous computational approaches have been proposed for the nonlinear FLE. Let us mention the predictor-corrector approaches [ 9 , 15 ], the finite difference schemes [ 14 , 16 ], the spectral methods [ 17 , 18 ], the Bessel collocation method [ 19 ], the Chebyshev wavelet method [ 20 ], the Laguerre collocation method [ 21 ], and the fractional spline collocation method [ 22 ].…”

Section: Introductionmentioning

confidence: 99%

A Discretization Approach for the Nonlinear Fractional Logistic Equation

Izadi

Srivástava

2020

Entropy

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The present study aimed to develop and investigate the local discontinuous Galerkin method for the numerical solution of the fractional logistic differential equation, occurring in many biological and social science phenomena. The fractional derivative is described in the sense of Liouville-Caputo. Using the upwind numerical fluxes, the numerical stability of the method is proved in the L∞ norm. With the aid of the shifted Legendre polynomials, the weak form is reduced into a system of the algebraic equations to be solved in each subinterval. Furthermore, to handle the nonlinear term, the technique of product approximation is utilized. The utility of the present discretization technique and some well-known standard schemes is checked through numerical calculations on a range of linear and nonlinear problems with analytical solutions.

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“…Collocation-based approaches along with (orthogonal) basis functions are very common in solving differential and integral equations due to their simplicity and efficiency. Among many different bases used in the literature, let us consider the methods based on the Legendre, Chebyshev, Jacobi, Genocchi, and Hermit functions reported in [33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

2021

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Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.

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A Comparative Study of Two Legendre-Collocation Schemes Applied to Fractional Logistic Equation

Cited by 19 publications

References 26 publications

A novel matrix technique for multi-order pantograph differential equations of fractional order

A novel matrix technique for multi-order pantograph differential equations of fractional order

A Discretization Approach for the Nonlinear Fractional Logistic Equation

Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

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