A numerical investigation of time-fractional modified Fornberg–Whitham equation for analyzing the behavior of water waves

Ray, S. Saha; Gupta, A. K.

doi:10.1016/j.amc.2015.05.045

Cited by 18 publications

(14 citation statements)

References 10 publications

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“…is absolutely convergent. Hence by means of relation in [34,35], the trial solution converges uniformly. Also deducted that,…”

Section: Convergence and Error Bound Analysismentioning

confidence: 99%

“…Therefore

{false\sum}_{i_{1} = 1}^{2^{k_{1} - 1}} {false\sum}_{j_{1} = 0}^{normal∞} {false\sum}_{i_{2} = 1}^{2^{k_{2} - 1}} {false\sum}_{j_{2} = 0}^{normal∞} σ_{i_{1}, j_{1}, i_{2}, j_{2}}

is absolutely convergent. Hence by means of relation in [34, 35], the trial solution converges uniformly. Also deducted that,

\begin{array}{l} boldE = \int_{0}^{1} \int_{0}^{1} (||, u ((), x, t) - \sum_{i_{1} = 1}^{2^{k_{1} - 1}} \sum_{j_{1} = 0}^{M_{1} - 1} \sum_{i_{2} = 1}^{2^{k_{2} - 1}} \sum_{j_{2} = 0}^{M_{2} - 1} σ_{i_{1}, j_{1}, i_{2}, j_{2}} ψ_{i_{1}, j_{1}, i_{2}, j_{2}} ((), x, t)) ϑ_{i_{1}, i_{2}}^{μ} ((), x, t) italicdxdt, \\ = \int_{0}^{1} \int_{0}^{1} (||, \sum_{i_{1} = 1}^{2^{k_{1} - 1}} \sum_{j_{1} = M_{1}}^{\infty} \sum_{i_{2} = 1}^{2^{k_{2} - 1}} \sum_{j_{2} = M_{2}}^{\infty} σ_{}) \end{array}

…”

Section: Convergence and Error Bound Analysismentioning

confidence: 99%

See 1 more Smart Citation

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

Usman

Hamid

Khalid

et al. 2020

Numerical Methods Partial

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In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative (D x and D t), fractional-order derivative (P x and P t), fractional-order integration (J x and J t) and delay terms (D a b How to cite this article: Usman M, Hamid M, Khalid MSU, Haq RU, Liu M. A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics.

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“…is absolutely convergent. Hence by means of relation in [34,35], the trial solution converges uniformly. Also deducted that,…”

Section: Convergence and Error Bound Analysismentioning

confidence: 99%

“…Therefore

{false\sum}_{i_{1} = 1}^{2^{k_{1} - 1}} {false\sum}_{j_{1} = 0}^{normal∞} {false\sum}_{i_{2} = 1}^{2^{k_{2} - 1}} {false\sum}_{j_{2} = 0}^{normal∞} σ_{i_{1}, j_{1}, i_{2}, j_{2}}

is absolutely convergent. Hence by means of relation in [34, 35], the trial solution converges uniformly. Also deducted that,

\begin{array}{l} boldE = \int_{0}^{1} \int_{0}^{1} (||, u ((), x, t) - \sum_{i_{1} = 1}^{2^{k_{1} - 1}} \sum_{j_{1} = 0}^{M_{1} - 1} \sum_{i_{2} = 1}^{2^{k_{2} - 1}} \sum_{j_{2} = 0}^{M_{2} - 1} σ_{i_{1}, j_{1}, i_{2}, j_{2}} ψ_{i_{1}, j_{1}, i_{2}, j_{2}} ((), x, t)) ϑ_{i_{1}, i_{2}}^{μ} ((), x, t) italicdxdt, \\ = \int_{0}^{1} \int_{0}^{1} (||, \sum_{i_{1} = 1}^{2^{k_{1} - 1}} \sum_{j_{1} = M_{1}}^{\infty} \sum_{i_{2} = 1}^{2^{k_{2} - 1}} \sum_{j_{2} = M_{2}}^{\infty} σ_{}) \end{array}

…”

Section: Convergence and Error Bound Analysismentioning

confidence: 99%

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

Usman

Hamid

Khalid

et al. 2020

Numerical Methods Partial

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show abstract

“…The Hermite wavelet has a restriction-free input range, which makes it more appropriate for solving highly nonlinear problems with a wide search space [ 41 ], [ 42 ]. Moreover, the series expansion of sufficient Hermite polynomials is used to represent any signal with a high degree of accuracy.…”

Section: Proposed Adaptive Control Paradigmmentioning

confidence: 99%

Adaptive control paradigm for photovoltaic and solid oxide fuel cell in a grid-integrated hybrid renewable energy system

Mumtaz

Khan

2017

PLoS ONE

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The hybrid power system (HPS) is an emerging power generation scheme due to the plentiful availability of renewable energy sources. Renewable energy sources are characterized as highly intermittent in nature due to meteorological conditions, while the domestic load also behaves in a quite uncertain manner. In this scenario, to maintain the balance between generation and load, the development of an intelligent and adaptive control algorithm has preoccupied power engineers and researchers. This paper proposes a Hermite wavelet embedded NeuroFuzzy indirect adaptive MPPT (maximum power point tracking) control of photovoltaic (PV) systems to extract maximum power and a Hermite wavelet incorporated NeuroFuzzy indirect adaptive control of Solid Oxide Fuel Cells (SOFC) to obtain a swift response in a grid-connected hybrid power system. A comprehensive simulation testbed for a grid-connected hybrid power system (wind turbine, PV cells, SOFC, electrolyzer, battery storage system, supercapacitor (SC), micro-turbine (MT) and domestic load) is developed in Matlab/Simulink. The robustness and superiority of the proposed indirect adaptive control paradigm are evaluated through simulation results in a grid-connected hybrid power system testbed by comparison with a conventional PI (proportional and integral) control system. The simulation results verify the effectiveness of the proposed control paradigm.

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“…Because these wavelet family have implicit expression, analytical differentiation or integration of Daubechies wavelets is not possible. And thus, more simple wavelets which are based on orthogonal polynomials such as Haar, Hermite, Legendre, Laguerre and Chebyshev are used in wavelet based numerical methods by many researchers [2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

Seçer

Özdemir

2019

Therm sci

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The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

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A numerical investigation of time-fractional modified Fornberg–Whitham equation for analyzing the behavior of water waves

Cited by 18 publications

References 10 publications

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

Adaptive control paradigm for photovoltaic and solid oxide fuel cell in a grid-integrated hybrid renewable energy system

Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

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