An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations

Behera, S.; Ray, S. Saha

doi:10.1016/j.cam.2021.113825

Cited by 20 publications

(6 citation statements)

References 29 publications

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“…Most applications of the wavelet concept are found in applied science and engineering. Wavelets have been found to be effective in a variety of applications, and they are particularly useful in signal processing [24][25][26][27][28]. The operational matrix of fractional integration, the fractional Taylor wavelet basis, function approximation, and the Lagrange multiplier approach have been used in this study to solve a specific FOCP.…”

Section: Introductionmentioning

confidence: 99%

A numerical technique for solving fractional optimal control problems using fractional wavelet method

Singh,

Ray

2024

Preprint

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This manuscript presents an efficient numerical method for solving fractional optimal control problems using an operational matrix for a fractional wavelet. Using well-known formulae such as Caputo and Riemann-Liouville operators to determine fractional derivatives and integral fractional wavelets, operational matrices were devised and utilized to solve fractional optimal control problems. The proposed method reduced the fractional optimal control problems into a system of algebraic equations and apply the Lagrange multiplier technique for determining the cost value. The convergence analysis and error bound of the proposed scheme have been established. To validate the effectiveness of the presented numerical approach, some illustrative examples were solved using fractional Taylor and Taylor wavelets, and the approximate cost function value derived by approximating state and control functions was compared. MSC Classification: 49J15 , 49N10 , 65T60 , 26A33.

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Section: Introductionmentioning

confidence: 99%

A numerical technique for solving fractional optimal control problems using fractional wavelet method

Singh,

Ray

2024

Preprint

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“…This wavelet basis originated from a single function called the mother wavelet ψ(x), which is a small beat. In literature, wavelet methods such as the Euler wavelet scheme for volterra delay integral DEs [14], Hermite wavelet scheme for nonlinear singular initial value problems (IVPs) [15], Legendre wavelet method for nonlinear DDEs [16], continuous wavelet series method for Lane-Emden equations [17], B-spline method for Burgers-Huxley equation [18], Haar wavelet method for the Chen-Lee-Liu equation [19], DDEs based on Euler wavelets [20], R-K method for the DDEs [21] and A novel approach for Pantograph equations [22], and so on [23,24] have been presented.…”

Section: Introductionmentioning

confidence: 99%

Wavelets approach for the solution of nonlinear variable delay differential equations

Kumbinarasaiah

Mundewadi

2023

International Journal of Mathematics and Computer in Engineering

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In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

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“…The authors in [16] proposed a hybrid method to solve the distributed-order fractional Black-Scholes European option. Also, the Euler wavelet method was presented to find an approximate solution of fractional pantograph Volterra delay integro-differential equations [17]. The nonlinear fractional differential equations were solved by the Adomian decomposition method [18].…”

Section: Introductionmentioning

confidence: 99%

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Rahimkhani¹,

Ordokhani

Sabermahani

2023

Math Methods in App Sciences

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In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

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An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations

Cited by 20 publications

References 29 publications

A numerical technique for solving fractional optimal control problems using fractional wavelet method

A numerical technique for solving fractional optimal control problems using fractional wavelet method

Wavelets approach for the solution of nonlinear variable delay differential equations

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

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