Haar wavelet technique for solving fractional differential equations with an application

Mechee, Mohammed S.; Al-Shaher, Oday I.; Al-Juaifri, Ghassan A.

doi:10.1063/1.5095110

Cited by 9 publications

(7 citation statements)

References 10 publications

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“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well-known algorithms can solve FDEs, such as operational matrix, [22][23][24][25][26] Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), [31][32][33] and finite difference. 34,35 In comparison to classic numerical approaches, the approximate calculation of ANN appears to be less sensitive to the spatial dimension.…”

Section: Introductionmentioning

confidence: 99%

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A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Shloof

Ahmadian

Senu

et al. 2023

Numerical Meth Engineering

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Artificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal‐fractional differential equations (FFDEs) of high‐order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs‐HOL, the suggested technique is simple, highly efficient, and resilient.

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Section: Introductionmentioning

confidence: 99%

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well‐known algorithms can solve FDEs, such as operational matrix, 22‐26 Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), 31‐33 and finite difference 34,35 …”

Section: Introductionmentioning

confidence: 99%

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Shloof

Ahmadian

Senu

et al. 2023

Numerical Meth Engineering

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“…Numerical integration and numerical solutions of fractional ordinary and fractional partial differential equations are some of the other applications of wavelet methods in applied mathematics. So, for now, wavelets such as the Haar-wavelet, B-spline, Daubechies, and Legendre wavelet are used [17][18][19][20][21]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation of Fractional-Order Differential Equations via $φ$ -Haar-Wavelet Method

Alharbi

Zidan

Naeem

et al. 2021

Journal of Function Spaces

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In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

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“…Wavelets are being used for analyzing signals, for representation of waveform and segmentation, optimal control, numerical analysis, fast algorithm for easy implementation, and time-frequency analysis [26]. There are many kinds of wavelets, for example, Haar [27][28][29][30], Daubechies [31], B-spline [32], Battle-Lemarie [33], Legender [34], as well as Green-CAS [30]. A naive form of orthonormal wavelets which employ compact support has been used by many researchers and is called the Haar wavelet.…”

Section: Introductionmentioning

confidence: 99%

Green–Haar wavelets method for generalized fractional differential equations

et al. 2020

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The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

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Haar wavelet technique for solving fractional differential equations with an application

Cited by 9 publications

References 10 publications

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Numerical Investigation of Fractional-Order Differential Equations via $φ$ -Haar-Wavelet Method

Green–Haar wavelets method for generalized fractional differential equations

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Haar wavelet technique for solving fractional differential equations with an application

Cited by 9 publications

References 10 publications

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

Green–Haar wavelets method for generalized fractional differential equations

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Product

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Numerical Investigation of Fractional-Order Differential Equations via $φ$ -Haar-Wavelet Method