Generalized Fibonacci Operational Collocation Approach for Fractional Initial Value Problems

Atta, A. G.; Moatimid, Galal M.; Youssri, Y. H.

doi:10.1007/s40819-018-0597-4

Cited by 39 publications

(16 citation statements)

References 18 publications

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“…Table 8 Note. The best absolute maximum errors obtained in [33,34] are E = 2.73 × 10 −15 and E = 1.00 × 10 −15 , respectively, while here we obtained a smaller error with an inferior number of retained modes.…”

Section: Example 1 Consider the Linear Singular Perturbed Boundary Value Problemmentioning

confidence: 43%

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

et al. 2021

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In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Moreover, the convergence of the expansion coefficients and an upper estimate for the truncation error are included. The obtained results for several case studies illustrate the accuracy and reliability of the proposed approach.

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Section: Example 1 Consider the Linear Singular Perturbed Boundary Value Problemmentioning

confidence: 43%

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

et al. 2021

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“…The collocation method is the most popular technique and can be used for all differential equations. For some articles that utilize collocation approach, see Mahdy et al (2022), Wu and Wang (2022), Taghipour and Aminikhah (2022a) and Atta et al (2019).…”

Section: Introductionmentioning

confidence: 99%

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

Atta

Youssri

2022

Comp. Appl. Math.

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This research apparatuses an approximate spectral method for the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel (TFPIDE). The main idea of this approach is to set up a new Hilbert space that satisfies the initial and boundary conditions. The new spectral collocation approach is applied to obtain precise numerical approximation using new basis functions based on shifted first-kind Chebyshev polynomials (SCP1K). Furthermore, we support our study by a careful error analysis of the suggested shifted first-kind Chebyshev expansion. The results show that the new approach is very accurate and effective.

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“…There are very promising efforts in developing spectral methods for solving various types of differential equations. In this regard, collocation method is applied in [21][22][23][24][25], tau method is used in [26][27][28][29][30][31][32], and Galerkin method was employed in [33][34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

et al. 2022

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A new numerical scheme based on the tau spectral method for solving the linear hyperbolic telegraph type equation is presented and implemented. The derivation of this scheme is based on utilizing certain modified shifted Chebyshev polynomials of the sixth-kind as basis functions. For this purpose, some new formulas concerned with the modified shifted Chebyshev polynomials of the sixth-kind have been stated and proved, and after that, they serve to study the hyperbolic telegraph type equation with our proposed scheme. One advantage of using this scheme is that it reduces the problem into a system of algebraic equations that can be simplified using the Kronecker algebra analysis. The convergence and error estimate of the proposed technique are analyzed in detail. In the end, some numerical tests are presented to demonstrate the efficiency and high accuracy of the proposed scheme.

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Generalized Fibonacci Operational Collocation Approach for Fractional Initial Value Problems

Cited by 39 publications

References 18 publications

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

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