Numerical solution for the variable order linear cable equation with Bernstein polynomials

Chen, Yiming; Liu, Liqing; Li, Baofeng; Sun, Yannan

doi:10.1016/j.amc.2014.03.066

Cited by 104 publications

(90 citation statements)

References 44 publications

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“…Over the past 30 years, the spectral methods have been considered one of the most important main tools for various applications such as computational fluid dynamics, fractional differential equations, chemistry and other science . In fact, lately, the spectral methods are successfully used for widely diverse applications, such as wave propagation, biomechanics, solid and structural analysis, astrophysics, and even financial engineering (see, eg, previous works()). The spectral methods lead to accurate approximate solutions because it uses basis functions of orthogonal polynomials that are non‐zero over the whole domain in addition to it involve representing the solution to a problem as a truncated series of known functions of the independent variables.…”

Section: Introductionmentioning

confidence: 99%

“…Also, these methods based on orthogonal systems, such as Legendre polynomials, Chebyshev polynomials, Bernstein polynomials, and modified generalized Laguerre polynomials, are available for bounded and unbounded domains for the approximation of FDEs. ()…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, researchers are interested to obtain the numerical solutions of the models described by the variable‐order fractional differential equations. Some of these methods are Legendre wavelets, second‐kind Chebyshev polynomials, Bernstein polynomials, Legendre polynomials, and the operational matrix of fractional integration founded by the shifted second‐kind Chebyshev polynomials . In addition, some other numerical techniques for solving the variable‐order fractional differential equations are given in Chen et al, Shen et al, Tavares et al, and Nagy et al…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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“…Many researchers such as Liu et al [46], Hu et al [47] and Cavlak et al [48] have proposed various numerical methods to solve such problem using various discretizations. Indeed, numerical methods for the oneand two-dimensional variable-order fractional cable equations are quite limited and difficult to construct (see [49,50]). This motivates our interest to introduce Jacobi spectral collocation method as an efficient alternative approach to solving two-dimensional variableorder fractional nonlinear cable equation in the form:…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

208

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The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one-and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.Keywords One-dimensional cable equation · Two-dimensional variable-order nonlinear cable equation · Collocation method · Jacobi polynomials · Operational matrix of fractional derivative · Variable-order derivative

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“…Bernstein polynomials are used to numerically solve the VO fractional partial differential equations by Wang et al [35]. In [7], numerical solutions of the VO linear cable equation with adopted Bernstein polynomials basis on the interval [0, R] are presented. Although in most cases continuous orthogonal polynomials are used as basis functions for the approximate solution of equations, recently discrete orthogonal polynomials have been noticed for solving stochastic differential equations [36] and in numerical fluid dynamics problems [13] because of the behavior and property of these polynomials [15].…”

Section: Introductionmentioning

confidence: 99%

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

2018

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In this paper, we consider the discrete Hahn polynomials fH n g and investigate their application for numerical solutions of the time fractional variable order mobile-immobile advection-dispersion model which is advantageous for modeling dynamical systems. This paper presented the operational matrix of derivative of discrete Hahn polynomials. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy in the interval [0, N]. Furthermore, for computing the coefficients of the expansion uðxÞ ¼ P 1 n¼0 c n H n ðxÞ, we have to only compute a summation and the calculation of coefficients is exact. Also an upper bound for the error of the presented method, with equidistant nodes, is investigated. Illustrative examples are provided to show the accuracy and efficiency of the presented method. Using a small number of Hahn polynomials, significant results are achieved which are compared to other methods.

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Numerical solution for the variable order linear cable equation with Bernstein polynomials

Cited by 104 publications

References 44 publications

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

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