A comparative study of analytical solutions of space-time fractional hyperbolic-like equations with two reliable methods

Singh, Brajesh Kumar; Gupta, Mukesh

doi:10.1080/25765299.2018.1538067

Cited by 10 publications

(1 citation statement)

References 40 publications

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“…and consequently the FDTM to Equation () for yields the following recurrence relation 53

({, \begin{matrix} leftarray \\ array 𝒰_{β} (0) = u (0) and 𝒰_{β} (k) = 0, for any k \in {1,2, \dots N - 1}, \\ array 𝒰_{β} (k + N) = \frac{Γ (k β + 1)}{Γ ((k + N) β + 1)} (δ (k) - \sum_{j = 0}^{k} (\frac{1}{2} δ (j - \frac{1}{β}) - 2 δ (j)) 𝒰_{β} (k - j) + \frac{2}{k β} \sum_{r = 0}^{k - \frac{1}{β}} \frac{1}{2^{k β - 1}} 𝒰_{β} (r) 𝒰_{β} (k - r - \frac{1}{β})), \end{matrix})

where

\frac{1}{\beta }

is the order of fraction of

\alpha

, that is,

\alpha &amp;amp;amp;amp;#x0003D; N\beta

and

<...

…”

Section: Test Examplesmentioning

confidence: 99%

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

Singh

Agrawal

2022

Math Methods in App Sciences

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This paper is devoted to the implementation of fractional differential transform method (FDTM) for the evaluation of new approximations for the fractional models of the system of multi-pantograph differential equations. Some new properties of FDTM have also been proposed for the fractional integro-differential equations. It is noteworthy that the integro-differential equations are studied for integers in the literature, and we have proposed the solutions of proportional delayed integro-differential equations for fractional order. The evaluated results are verified numerically by considering some test examples of fractional integro-differential equations and fractional models of the system of multi-pantograph differential equations. The outcomes conclude that evaluated series solutions via FDTM promise more appropriate approximations possess very negligible error and represent the convergent approximations. Thus, evaluated series solutions are comparatively more accurate and require less computational time. KEYWORDSfractional derivative of Caputo type, fractional differential transform method, fractional-order proportional delayed integro-differential equations, multi-pantograph system of differential equations of fractional order MSC CLASSIFICATION 34Axx, 34Kxx, 45Jxx

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“…and consequently the FDTM to Equation () for yields the following recurrence relation 53

({, \begin{matrix} leftarray \\ array 𝒰_{β} (0) = u (0) and 𝒰_{β} (k) = 0, for any k \in {1,2, \dots N - 1}, \\ array 𝒰_{β} (k + N) = \frac{Γ (k β + 1)}{Γ ((k + N) β + 1)} (δ (k) - \sum_{j = 0}^{k} (\frac{1}{2} δ (j - \frac{1}{β}) - 2 δ (j)) 𝒰_{β} (k - j) + \frac{2}{k β} \sum_{r = 0}^{k - \frac{1}{β}} \frac{1}{2^{k β - 1}} 𝒰_{β} (r) 𝒰_{β} (k - r - \frac{1}{β})), \end{matrix})

where

\frac{1}{\beta }

is the order of fraction of

\alpha

, that is,

\alpha &amp;amp;amp;amp;#x0003D; N\beta

and

<...

…”

Section: Test Examplesmentioning

confidence: 99%

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

Singh

Agrawal

2022

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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Revised residual power series method in Laplacian space for space‐time fractional PDEs with proportional delayed arguments

Singh

Agrawal

Singh³

2023

Math Methods in App Sciences

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This paper developed a new hybrid residual power series (RPS) method in the space of Laplace transform (in brief, ) for the study of the nonlinear space‐time fractional model of partial differential equations with proportional delayed arguments (STF‐PDEDAs). The main objective of this technique is to convert the target nonlinear STF‐PDEDAs into the algebraic equation in the space. The newly transformed equation is solved by utilizing the revised RPS method to achieve series approximations, and refer to this technique as RPSM. Different kinds of attractive applications of STF‐PDEDAs are discussed utilizing proposed RPSM to reveal its efficiency, effectiveness, and adaptability for constructing series solutions to the referred STF‐PDEDAs. The numerical experiments and graphical findings have been computed for different fractional parameter values to analyze the behavior of RPSM solutions and to test the authenticity and applicability of the present method in comparison to other existing results obtained by some rigorous techniques. The outcomes conclude that proposed RPSM is easy to understand and straightforward to provide results converging to its exact solution.

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Chebyshev spectral methods for multi-order fractional neutral pantograph equations

et al. 2020

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A comparative study of analytical solutions of space-time fractional hyperbolic-like equations with two reliable methods

Cited by 10 publications

References 40 publications

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

Revised residual power series method in Laplacian space for space‐time fractional PDEs with proportional delayed arguments

Chebyshev spectral methods for multi-order fractional neutral pantograph equations

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