Modified Fourier transform for solving fractional partial differential equations

Hasanah, Dahliatul; Sisworo,; Supeno, Imam

doi:10.1063/5.0004017

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…In this work, we develop the theory of the generalized Fourier transform for the generalized fractional derivative

{G}_T&#x0005E;{\alpha }

and state its main properties. It is important to note that the results obtained on the generalized Fourier transform contain as one of its particular cases the classical Fourier transform, when

T\left(t,\alpha \right)&#x0003D;1

, and the conformable fractional Fourier transform [37–39, 41], when

T\left(t,\alpha \right)&#x0003D;{t}&#x0005E;{1-\alpha }

in (i). Furthermore, the generalized Fourier transform allows to deal with the fractional derivatives in (ii) and (iii).…”

Section: Introductionmentioning

confidence: 99%

“…This theory was used in the study of fractional differential equations; see, for example, earlier studies [31][32][33][34][35][36]. In the same direction, taking the conformable fractional derivative as a basis, the conformable fractional Fourier transform was worked in previous work [37][38][39][40][41]. In these investigations, the authors present several properties of this fractional transform and its application to the solution of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

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On the generalized Fourier transform

Blaya

García

Sigarreta

2023

Math Methods in App Sciences

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In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the new corresponding convolution, inverse and Plancherel formulas, and Hausdorff–Young type inequality. We show that this generalized Fourier transform is useful in the study of several fractional differential equations (both ordinary and partial differential equations).

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“…In this work, we develop the theory of the generalized Fourier transform for the generalized fractional derivative

{G}_T&#x0005E;{\alpha }

and state its main properties. It is important to note that the results obtained on the generalized Fourier transform contain as one of its particular cases the classical Fourier transform, when

T\left(t,\alpha \right)&#x0003D;1

, and the conformable fractional Fourier transform [37–39, 41], when

T\left(t,\alpha \right)&#x0003D;{t}&#x0005E;{1-\alpha }

in (i). Furthermore, the generalized Fourier transform allows to deal with the fractional derivatives in (ii) and (iii).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the generalized Fourier transform

Blaya

García

Sigarreta

2023

Math Methods in App Sciences

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“…[10][11][12] In literature, there are many numerical methods for finding numerical solution of the fractional differential equations. Some of these methods include transform methods such as Fourier transform, 13 Laplace transform, 14 and differential transform 15 ; Adomian decomposition method 16 ; homotopy analysis method 17 ; power series method 18 ; variational iteration method 19 ; finite difference method 20 ; and wavelet methods. [21][22][23] Wavelet analysis is a new scheme and emerging area in the applied mathematics and other fields.…”

Section: Introductionmentioning

confidence: 99%

Generalized sine–cosine wavelet method for Caputo–Hadamard fractional differential equations

Idrees

Saeed

2022

Math Methods in App Sciences

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In this article, a wavelet method is introduced for solving Caputo-Hadamard fractional differential equations on an arbitrary interval. The proposed method is the fractional-order generalization of sine-cosine wavelets (FGSCWs). The operational matrices of fractional-order integration are constructed for solving initial value problem as well as boundary value problem. Furthermore, numerical solution of nonlinear Caputo-Hadamard fractional differential equation is obtained with the conjunction of proposed method with quasilinearization technique. We have constructed the FGSCW operational matrix, FGSCW operational matrix of Hadamard fractional integration of arbitrary order, and FGSCW operational matrix of Hadamard fractional integration for Caputo-Hadamard fractional boundary value problems. Convergence analysis of the proposed method is investigated. Numerical procedure is given for both Caputo-Hadamard initial and boundary value problems. Illustrative examples show the reliability and efficiency of the proposed method and give solution with less error.

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“…It also has corresponding Rolle's theorem and mean value theorem. Hasanah et al in [15] modified Fourier transform to handle fractional partial differential equations using conformal derivative. However, conformable fractional derivatives have disadvantages compared to Riemann-Liouville and Caputo fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

On continuity properties of the improved conformable fractional derivatives

Hasanah

2022

JFOURIER

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The conformable fractional derivative has been introduced to extend the familiar limit definition of the classical derivative. Despite having many advantages compared to other fractional derivatives such as satisfying nice properties as classical derivative and easy to solve numerically, it also has disadvantages as it gives large error compared to Riemann-Liouville and Caputo fractional derivatives. Modified types of conformable derivatives have been proposed to overcome the shortcoming. The improved conformal fractional derivatives are declared to be better approximations of Riemann-Liouville and Caputo derivatives in terms of physical behavior. In this paper, properties concerning continuity of the improved conformable fractional derivative are investigated. We prove the relation between -differentiable and continuity of a function and corresponding interior extremum theorem. We also prove the properties close to Rolle’s Theorem and Mean Value Theorem for the improved conformable fractional derivatives.

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Modified Fourier transform for solving fractional partial differential equations

Cited by 5 publications

References 8 publications

On the generalized Fourier transform

On the generalized Fourier transform

Generalized sine–cosine wavelet method for Caputo–Hadamard fractional differential equations

On continuity properties of the improved conformable fractional derivatives

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