Lucas Wavelet Scheme for Fractional Bagley–Torvik Equations: Gauss–Jacobi Approach

Koundal, Reena; Kumar, Rakesh; Srivastava, K.; Băleanu, Dumitru

doi:10.1007/s40819-021-01206-z

Cited by 9 publications

(8 citation statements)

References 45 publications

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“…Moreover, in the Examples 5.4) and 5.5 two-term linear and nonlinear fractional order differential equations are considered and the approximate results are obtained with the precision surpassing 10 −77 and with almost zero error, respectively. Further, the numerical results obtained in Example 5.6 are more accurate as compare to the existing results for the Caputo fractional derivative [35,42,43]. The salient feature of the suggested technique lies in its straightforward implementation,…”

Section: Discussionmentioning

confidence: 70%

“…Example 5.6. Let us consider the initial value Bagley-Trovik equation [35,42,43] derived by AB-fractional derivative…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In figure 7, we illustrate both the exact and computational outcomes, accompanied by the absolute error for better visualisation. The computational data tabulated in the table 5 is compared with existing findings with Caputo fractional derivative [35,42,43].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Further, section 4 introduces error bounds of function approximation. In the section 5, the discussed theory is applied to validate the methodology through comparisons of numerical outcomes with analytical solutions and the results available in literature [35,37,40,42,43]. Furthermore, graphical representation of numerical outcomes and absolute error are included for the proposed models.…”

Section: Introductionmentioning

confidence: 99%

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Solvability of fractional differential equations with applications of Morgan Voyce polynomials

Kumar,

Singh,

Kumar

2024

Phys. Scr.

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In this research, a new computational approach is presented to address multi-order fractional differential equations, including the renowned Bagley–Torvik and Painlevé equations. These equations are pivotal in scientific and engineering realms, like modelling the movement of a submerged plate restricted in a Newtonian fluid and gas in a fluid, along with simulating the coupled oscillations. We utilise the collocation approach employing a novel operational matrix derived for Morgan-Voyce polynomials via the Atangana-Baleanu fractional derivative. Initially, we introduce the fractional differential matrix to convert the problem and its constraints into a system of algebraic equations with unknown coefficients. These coefficients aid in finding numerical solutions for the given equations. To assess the efficiency of proposed method, various examples are simulated utilising the proposed approach and the outcomes are compared with existing results for different derivatives.

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Section: Discussionmentioning

confidence: 70%

“…Example 5.6. Let us consider the initial value Bagley-Trovik equation [35,42,43] derived by AB-fractional derivative…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

Kumar,

Singh,

Kumar

2024

Phys. Scr.

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“…Example Consider the following FFDE:

\begin{array}{ll}& {D}^{\alpha, \nu, \beta}\kern0.3em y(t)+y(t)=0,\\ {}& \kern1.20em y(0)=1,\kern0.3em {y}^{\prime }(0)=0.\end{array}

The exact solution when

\alpha =2

y(t)=\frac{\exp (it)+\exp \left(- it\right)}{2}

64 …”

Section: Numerical Simulation Of Fractal‐fractional Problemsmentioning

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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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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A Study of Fractional Order Financial Crime Model Using the Gegenbauer Wavelet Collocation Method

2024

Advcd Theory and Sims

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The manuscript investigates the numerical approximation of the fractional mathematical model of the financial crime population dynamics by the Gegenbauer wavelet collocation method. The study aims to enhance the accuracy and efficiency of solving the underlying differential equations that describe these phenomena by utilizing the proposed technique. The financial crime model is a nonlinear coupled system of ordinary differential equations. Using the Gegenbauer wavelets, the novel operational matrices of integration are created. A nonlinear system of ordinary differential equations are transformed into a system of algebraic equations using the characteristics of the Gegenbauer wavelet expansions and the operational matrix of integration, which speeds up processing. Then, this system of algebraic equations is solved using the Newton‐iterative technique to find the unknown Gegenbauer coefficients that help to obtain the approximate solution for the system. A numerical illustration is presented to show the efficacy and precision of the approach. The numerical results obtained from the projected approach are compared with the existing methods, such as NDSolve and Runge Kutta methods. These results show that the projected scheme is simple, reliable, and resilient. The findings suggest that this approach can be a powerful tool for researchers and practitioners in the financial sector, aiding in developing crime prevention and intervention strategies. The study concludes with suggestions for future research directions.

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Lucas Wavelet Scheme for Fractional Bagley–Torvik Equations: Gauss–Jacobi Approach

Cited by 9 publications

References 45 publications

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

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

A Study of Fractional Order Financial Crime Model Using the Gegenbauer Wavelet Collocation Method

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