Logarithmic Jacobi collocation method for Caputo–Hadamard fractional differential equations

Zaky, Mahmoud A.; Hendy, Ahmed S.; Сураган, Дурвудхан

doi:10.1016/j.apnum.2022.06.013

Cited by 20 publications

(10 citation statements)

References 51 publications

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“…Substituting Equations ( 32), (38), and (39) in Equation (1) and using the collocation points as x q = q−0.5 2M 1…”

Section: Methodsmentioning

confidence: 99%

“…In this section, we establish the error bound of the proposed method. To establish the convergence of the present method, we take the analytic form of Equation (38) as…”

Section: Convergencementioning

confidence: 99%

“…For example, a singularity preserving spectral collocation method is discussed for tempered fractional and nonlinear systems of FDEs in previous studies 36,37 . Jacobi collocation method for Caputo–Hadamard FDEs and multidimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions are proposed in previous works 38,39 …”

Section: Introductionmentioning

confidence: 99%

“…36,37 Jacobi collocation method for Caputo-Hadamard FDEs and multidimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions are proposed in previous works. 38,39 On the other hand, wavelet collocation methods have proven to be a powerful tool to estimate the solution of differential equations due to compact support and simple operation. A good deal of research is impelled toward the numerical solution of differential and FDEs using wavelets.…”

Section: Introductionmentioning

confidence: 99%

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A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

Faheem

Khan

2023

Math Methods in App Sciences

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A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann-Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann-Liouville fractional integral operator for Haar wavelet (RLFIO-H). The application of RLFIO-H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

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“…Substituting Equations ( 32), (38), and (39) in Equation (1) and using the collocation points as x q = q−0.5 2M 1…”

Section: Methodsmentioning

confidence: 99%

“…In this section, we establish the error bound of the proposed method. To establish the convergence of the present method, we take the analytic form of Equation (38) as…”

Section: Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

Faheem

Khan

2023

Math Methods in App Sciences

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“…Subsequently, Wang et al [27] successfully developed a second-order accurate finite difference scheme in time for CHFDEs. For CHFDEs with nonlinear terms, Zaky et al [31] developed a spectral logarithmic Jacobian collocation method and proved its convergence. The well-posedness of solutions for Caputo-Hadamard fractional diffusion equations was studied by Yang et al [30].…”

mentioning

confidence: 99%

A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation

Wang,

Sun

2024

DCDS-S

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We propose an L1-type scheme on nonuniform meshes to approximate the Caputo-Hadamard derivative. While this scheme shares a similar structure with the logarithmic L1 formula, it differs in the selection of mesh points, making it more applicable. Next, we consider the numerical solution of a class of variable-coefficient diffusion equation involving the time Caputo-Hadamard derivative. To provide a theoretical foundation for the design of the numerical scheme, we first study the regularity of the solution to this equation. Then, we discretize the time-fractional derivative using the derived L1 scheme and approximate the spatial derivative by the local discontinuous Galerkin (LDG) finite element method, resulting in a fully discrete scheme. We prove the stability and convergence of this scheme and validate its performance through numerical experiments.

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A robust scheme for Caputo variable-order time-fractional diffusion-type equations

Sadri¹,

Hosseini²,

Băleanu

et al. 2023

J Therm Anal Calorim

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Logarithmic Jacobi collocation method for Caputo–Hadamard fractional differential equations

Cited by 20 publications

References 51 publications

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation

A robust scheme for Caputo variable-order time-fractional diffusion-type equations

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