Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices

Elgindy, Kareem T.; Smith‐Miles, Kate

doi:10.1016/j.cam.2012.05.024

Cited by 49 publications

(30 citation statements)

References 37 publications

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“…It is well known that PDMs are severely ill-conditioned when the number of collocation points is large, while PIMs are wellconditioned even for large number of collocation points [4,6]. This is basically because numerical differentiation is inherently sensitive, as small perturbations in input can cause large changes in output, while numerical integration is inherently stable [6].…”

Section: Introductionmentioning

confidence: 98%

“…In differential pseudospectral methods, the unknown solution is approximated using Lagrange interpolating polynomials and the differential equation is directly discretized using collocation at a specified set of points via pseudospectral differentiation matrices (PDMs) [1][2][3]. In integral pseudospectral methods, the differential equation is first recast as an equivalent integral equation and the latter is then discretized using collocation at the specified points via pseudospectral integration matrices (PIMs) [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

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Efficient and stable generation of higher-order pseudospectral integration matrices

Tang

2015

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Efficient and stable generation of higher-order pseudospectral integration matrices

Tang

2015

Applied Mathematics and Computation

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“…Even in constructing high-order numerical quadratures, line search optimization emerges in minimizing their truncation errors; thus boosting their accuracy and allowing to obtain very accurate solutions to intricate boundary-value problems, integral equations, integro-differential equations, and optimal control problems in short times via stable and efficient numerical schemes; cf. [Elgindy et al (2012), Elgindy and Smith-Miles (2013), Elgindy and Smith-Miles (2013a), Elgindy and Smith-Miles (2013b), , ].…”

Section: Introductionmentioning

confidence: 99%

Optimization via Chebyshev polynomials

Elgindy

2017

J. Appl. Math. Comput.

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SUMMARYThis paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one-and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.

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“…The Gegenbauer polynomials [12], or ultra spherical harmonics polynomials, C λ m (x), of order m are defined, for λ > − 1 2 , m ∈ Z + , on the interval [−1, 1] and given by the following recurrence formulae,…”

Section: Gegenbauer Polynomials and Gegenbauer Waveletsmentioning

confidence: 99%

“…Gegenbauer polynomials [12] or ultraspherical polynomials are orthogonal polynomials on the interval [−1, 1] with respect to the weight function. They generalize the Legendre and Chebyshev polynomials.…”

Section: Introductionmentioning

confidence: 99%

Gegenbauer Wavelets Operational Matrix Method for Fractional Differential Equations

Rehman¹,

Saeed²

2015

Journal of the Korean Mathematical Society

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Abstract. In this article we introduce a numerical method, named Gegenbauer wavelets method, which is derived from conventional Gegenbauer polynomials, for solving fractional initial and boundary value problems. The operational matrices are derived and utilized to reduce the linear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method.We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.

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Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices

Cited by 49 publications

References 37 publications

Efficient and stable generation of higher-order pseudospectral integration matrices

Efficient and stable generation of higher-order pseudospectral integration matrices

Optimization via Chebyshev polynomials

Gegenbauer Wavelets Operational Matrix Method for Fractional Differential Equations

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