Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

Yang, Yin; Huang, Yunqing

doi:10.1155/2013/821327

Cited by 47 publications

(27 citation statements)

References 24 publications

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“…Because of the computational complexity of fractional derivatives in the delay case, the exact analytical solution of the fractional delay differential equations is hardly available. Therefore, in the last decades many researchers have been attracted to deal with the numerical solution of this class of problems (see for example [23,24,25,26,27,28]).…”

Section: Introductionmentioning

confidence: 99%

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Nemati

Lima

Sedaghat

2020

Applied Numerical Mathematics

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In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss-Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss-Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.

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Section: Introductionmentioning

confidence: 99%

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Nemati

Lima

Sedaghat

2020

Applied Numerical Mathematics

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“…Yusufoglu [7] proposed an efficient algorithm for solving generalized pantograph equations with linear functional argument. In [8], Yang and Huang presented a spectral-collocation method for fractional pantograph delay integrodifferential equations and in [9] Yüzbasi and Sezer presented an exponential approximation for solutions of generalized pantograph delay differential equations. Chebyshev and Bessel polynomials are, respectively, used in [10,11] to obtain the solutions of generalized pantograph equations.…”

Section: Introductionmentioning

confidence: 99%

Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

Isah

Phang

2017

International Journal of Differential Equations

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An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

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“…Finite element methods have been introduced in [11][12][13] to obtain numerical solutions of FDEs; also the numerical treatments based on finite difference methods have been developed in [14][15][16]. Moreover, several spectral techniques were designed for such equations (see for instance, [17][18][19][20][21][22][23][24][25][26][27]). …”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

2015

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A spectral shifted Legendre Gauss-Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x), x ∈ [0, L] is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre-Gauss-Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge-Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.Mathematics Subject Classification. 34A08, 65M70, 33C45.

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Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

Cited by 47 publications

References 24 publications

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

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