Comparison of Various Fractional Basis Functions for Solving Fractional-Order Logistic Population Model

Cattani

2020

Symmetry

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The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order.

Section: Illustrative Test Problemsmentioning

confidence: 99%

Generalized Bessel Polynomial for Multi-Order Fractional Differential Equations

Cattani

2020

Symmetry

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“…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%

“…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%

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

Abbasi

Math Methods in App Sciences

Hosseini

2022

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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.

“…Consequently, most of the resulting FDEs do not have analytical solutions, and therefore approximation or numerical techniques are developed and applied to solve them. Solution techniques for FDEs have been developed extensively by many researchers such as the fractional linear multistep method [9], the Adomian decomposition method [33], the variational iteration method [25], the Adams-type predictor-corrector method [5], the spectral methods [11][12][13][14][15][16], and the local discontinuous Galerkin methods [17][18][19][20][21], to name but a few.…”

Section: Introductionmentioning

confidence: 99%

A computational algorithm for simulating fractional order relaxation–oscillation equation

2021

SeMA

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In the present work, a collocation approach is developed to find an approximate solution of fractional relaxation-oscillation differential equation describing the processes of relaxation and oscillation in many physical systems. The method is relied on generalized Chebyshev polynomials as basis functions, collocation points, and the matrix operations. The proposed scheme converts the underlying fractional initial value problems into a matrix equation, which corresponds to a set of linear algebraic equations consist of polynomial coefficients. An error estimation based on residual function is performed to show the accuracy of the results. Hence, an improvement of the approximate solutions are obtained based upon this error estimation. To illustrate the utility and applicability of the technique, three numerical examples are given and the comparisons between the numerical results of the proposed method with existing results are carried out. The experiments show the accuracy and efficiency the present work.