Orthonormal piecewise Vieta-Lucas functions for the numerical solution of the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations

Heydari, Mohammad Hossein; Razzaghi, Mohsen; Băleanu, Dumitru

doi:10.1016/j.jare.2022.10.002

Cited by 12 publications

(2 citation statements)

References 37 publications

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“…Many accomplishments have been attributed to the assumption of strategy presence and consistency in the fractional IVP framework [11]. For additional scientific articles on fractional ordinary and PDEs emerging in various fields of scientific research, see [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Chu,

Sultana,

Karim

et al. 2024

Computer Modeling in Engineering &Amp; Sciences

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The goal of this research is to develop a new, simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations (PDEs) with variable coefficient. ARA-transform is a robust and highly flexible generalization that unifies several existing transforms. The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor's expansion. The process of finding approximations for dynamical fractional-order PDEs is challenging, but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts. This approach is effective and useful for solving a massive class of fractional-order PDEs. Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients. Additionally, several visualizations are drawn for different fractional-order values. Besides that, the estimated findings by the proposed technique are in close agreement with the exact outcomes. Finally, statistical analyses further validate the efficacy, dependability and steady interconnectivity of the suggested ARA-residue power series approach.

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Section: Introductionmentioning

confidence: 99%

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Chu,

Sultana,

Karim

et al. 2024

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

show abstract

“…Moreover, the idea of using piecewise derivatives of fractional orders has recently been explored in several studies. The concept of piecewise fractional differential equations has been employed by researchers in recent studies, after Atangana and Araz presented in [21], to explore various dynamical systems [22][23][24]. Such operators have proven useful in solving problems that exhibit crossover behavior and are gaining popularity among researchers.…”

Section: Introductionmentioning

confidence: 99%

Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions

Rahman

Arfan

Băleanu

2023

BBM

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This study introduces several updated results for the piecewise plant-pathogen-herbivore interactions model with singular-type and nonsingular fractional-order derivatives. A piecewise fractional model has developed to describe the interactions between plants, disease, (insect) herbivores, and their natural enemies. We derive essential findings for the aforementioned problem, specifically regarding the existence and uniqueness of the solution, as well as various forms of Ulam Hyers (U-H) type stability. The necessary results were obtained by utilizing fixed-point theorems established by Schauder and Banach. Additionally, the U-H stabilities were determined based on fundamental principles of nonlinear analysis. To implement the model as an approximate piecewise solution, the Newton Polynomial approximate solution technique is employed. The applicability of the model was validated through numerical simulations both in fractional as well as piecewise fractional format. The motivation of our article is that we have converted the integer order problem to a global piecewise and fractional order model in the sense of Caputo and Atangana-Baleanu operators and investigate it for existence, uniqueness of solution, Stability of solution and approximate solution along with numerical simulation for the validity of our obtained scheme.

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Initial data identification in advection–diffusion processes via a reversed fixed‐point iteration method

Cosgun

Sarı

2022

Math Methods in App Sciences

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In the present study, a novel method called the reversed fixed‐point iteration method (RFPIM) is applied to obtain numerical responses of the advection–diffusion mechanisms. Heretofore, the RFPIM has been employed to find out unstable equilibria of nonlinear mappings defined on Banach spaces. The current paper implements the method to recover the initial data from the final data. The method has been tested under measurements including different levels of noise such as 5%, 10%, 30%, 50%, and 100% in the final time data, and the results indicate that the present technique could be regarded as a powerful tool in handling such inverse problems.

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Orthonormal piecewise Vieta-Lucas functions for the numerical solution of the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations

Cited by 12 publications

References 37 publications

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions

Initial data identification in advection–diffusion processes via a reversed fixed‐point iteration method

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