On Effects of a New Method for Fractional Initial Value Problems

Sevindir, Hülya Kodal; Çetınkaya, Süleyman; Demir, Ali

doi:10.1155/2021/7606442

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…Fundamental definitions and their properties are presented in this subsection [7,10,13,4,8]. Definition 1. α th order of the time-fractional integral of u(t), a real valued function, in Riemann-Liouville sense is defined as…”

Section: Preliminary Resultsmentioning

confidence: 99%

Effects of the ARA transform method for time fractional problems

Çetınkaya¹,

Demir²

2022

Mathematica Moravica

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The aim of this study is to establish the solutions of time fractional mathematical problems with the aid of new integral transforms called the ARA transform. The fractional derivative is taken in the sense of Liouville-Caputo derivative. The fractional partial differential equations are reduced into ordinary differential equations. Later solving this fractional equation and applying inverse the ARA transform, the solution is acquired. The implementation of this transform for fractional differential equations is very similar to the implementation of the Laplace transform. However, the ARA transform allows us to take the integral transform of some functions for which we can not take the Laplace transform. The illustrated examples justify that the implementation and efficiency of this method are better than any other integral transforms to tackle time fractional differential equations (TFDEs).

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Section: Preliminary Resultsmentioning

confidence: 99%

Effects of the ARA transform method for time fractional problems

Çetınkaya¹,

Demir²

2022

Mathematica Moravica

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“…The existence proof of the ARA transformation for Liouville-Caputo derivative and Riemann-Liouville integral is introduced in [17].…”

Section: Definitionmentioning

confidence: 99%

The analytic solution of the fractional Rosenau--Hyman model in Liouville-Caputo sense

Çetınkaya

Demir

2023

Preprint

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The purpose of this work is to achieve the analytic solution of non-linear Liouville-Caputo fractional initial value problem including time fractional Rosenau–Hyman equation by means of ARA transform and iterative method. First of all, ARA transform is employed to reduce the problem into a simpler one. Then the series solution of the non-linear fractional initial value problem is acquired by applying iterative method which leads to the analytic solution of the Rosenau–Hyman equation (RHE) with initial condition. The obtained results verify that the combination of ARA transform with iterative method produces an effective method to tackle with non-linear fractional problems.

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“…The adversity of fractional differential equations is that solving them analytically is hard or impossible. Therefore, numerous numerical methods such as reduced differential transform method , Adomian decomposition method , homotopy perturbation method , variational iteration method , homotopy analysis method , fractional difference method, Euler Method, Runge-Kutta Method and new iterative method have been developed to establish numerical solution in series form [20][21][22]. Different from the numerical solutions, numerous analytical solutions of fractional models can be seen in literature [23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

A new application of fractional glucose-insulin model and numerical solutions

ÖZTÜRK

2024

Sigma J Eng Nat Sci - Sigma Müh Fen Bil Derg

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Along with the developing technology, obesity and diabetes are increasing rapidly among people. The identification of diabetes diseases, modeling, predicting their behavior, conducting simulations, studying control and treatment methods using mathematical methods has become of great importance. In this paper, we have obtained numerical solutions by considering the glucose-insulin fractional model. This model consists of three compartments: the blood glucose concentration (𝐺), the blood insulin concentration (𝐼) and the ready-to-absorb glucose concentration (𝐷) in the small intestine. The fractional derivative is used in the sense of Caputo. By performing mathematical analyzes for the Glucose-Insulin fractional mathematical model, numerical results were obtained with the help of the Euler method and graphs were drawn.

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On Effects of a New Method for Fractional Initial Value Problems

Cited by 5 publications

References 11 publications

Effects of the ARA transform method for time fractional problems

Effects of the ARA transform method for time fractional problems

The analytic solution of the fractional Rosenau--Hyman model in Liouville-Caputo sense

A new application of fractional glucose-insulin model and numerical solutions

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