<i>ψ</i>-Haar wavelets method for numerically solving fractional differential equations

Ali, Amjid; Minamoto, Teruya; Saeed, Umer; Rehman, Mujeeb ur

doi:10.1108/ec-01-2020-0050

Cited by 3 publications

(2 citation statements)

References 44 publications

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“…The main focus of the paper (Almeida, 2019) is to discuss the fractional integrals and derivatives of a function with respect to the ψ function. Haar wavelet method (Ali et al ., 2020) is used for the solution of ψ -Caputo fractional differential equation. A numerical method is proposed in Sadiq and Rehman (2022a) for the solution of ψ -Caputo fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

Saeed

2023

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PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

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Section: Introductionmentioning

confidence: 99%

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

Saeed

2023

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“…In literature, we can find several numerical methods for the solutions of Riemann–Liouville fractional and Caputo fractional differential equations. Some of the methods are: Adomian decomposition method (Li and Pang, 2020), Fractional variational iteration method (Prakash et al , 2019), collocation method (Shi et al , 2020) and wavelet methods (Saeed et al , 2021; Ali et al , 2020; Saeed, 2019).…”

Section: Introductionmentioning

confidence: 99%

A wavelet method for solving Caputo–Hadamard fractional differential equation

Saeed

2021

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PurposeThe purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.Design/methodology/approachThe author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.FindingsThe author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.Originality/valueMany scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

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Fractional-order Boubaker wavelets method for solving fractional Riccati differential equations

Rabiei

Razzaghi

2021

Applied Numerical Mathematics

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ψ-Haar wavelets method for numerically solving fractional differential equations

Cited by 3 publications

References 44 publications

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

A wavelet method for solving Caputo–Hadamard fractional differential equation

Fractional-order Boubaker wavelets method for solving fractional Riccati differential equations

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