• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.305-316
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• 2022

We study the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach's contraction principle and the Schauder's fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.583-593
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• 2022

Lienard's equations are important nonlinear differential equations with application in many areas of applied mathematics. In the present article, a new approach known as the modified fractional Taylor series method (MFTSM) is proposed to solve the nonlinear fractional Lienard equations with Caputo fractional derivatives, and the convergence of this method is established. Numerical examples are given to verify our theoretical results and to illustrate the accuracy and effectiveness of the method. The results obtained show the reliability and efficiency of the MFTSM, suggesting that it can be used to solve other types of nonlinear fractional differential equations that arise in modeling different physical problems.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.545-558
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• 2024

In this study, a class of nonlinear boundary fractional Caputo Volterra-Fredholm integro-differential equations (CV-FIDEs) is taken into account. Under specific assumptions about the available data, we firstly demonstrate the existence and uniqueness features of the solution. The Gronwall's inequality, a adequate singular Hölder's inequality, and the fixed point theorem using an a priori estimate procedure. Finally, a case study is provided to highlight the findings.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.15-27
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• 2008

We present and discuss an algorithm for the numerical solution of initial value problems of the form $D_*^\alpha$y(t) = f(t, y(t)), y(0) = y0, where $D_*^\alpha$y is the derivative of y of order $\alpha$ in the sense of Caputo and 0<${\alpha}{\leq}1$. The algorithm is based on the fractional Euler's method which can be seen as a generalization of the classical Euler's method. Numerical examples are given and the results show that the present algorithm is very effective and convenient.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.197-223
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• 2021

In this paper, we investigate existence, uniqueness and four different types of Ulam's stability, that is, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability of the solution for a class of nonlinear fractional Pantograph differential equation in term of a proportional Caputo fractional derivative with mixed nonlocal conditions. We construct sufficient conditions for the existence and uniqueness of solutions by utilizing well-known classical fixed point theorems such as Banach contraction principle, Leray-Schauder nonlinear alternative and $Krasnosel^{\prime}ski{\breve{i}}{^{\prime}}s$ fixed point theorem. Finally, two examples are also given to point out the applicability of our main results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.1
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• pp.63-74
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• 2018

In this paper, we study the initial value problem for neutral functional differential equations involving Caputo fractional derivative of order ${\alpha}{\in}(0,1)$ with infinite delay. Some sufficient conditions for the uniqueness and continuous dependence of solutions are established by virtue of fractional calculus and Banach fixed point theorem. Some results obtained showed that the solution was closely related to the conditions of delays and minor changes in the problem. An example is provided to illustrate the main results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.163-177
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• 2018

This paper successfully applies the modified Adomian decomposition method to find the approximate solutions of the Caputo fractional integro-differential equations. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximation. Moreover, we proved the existence and uniqueness results and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.869-878
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• 2016

This paper presents an ecient fractional shifted Legendre polynomial method to solve the fractional Volterra's model for population growth model. The fractional derivatives are described based on the Caputo sense by using Riemann-Liouville fractional integral operator. The theoretical analysis, such as convergence analysis and error bound for the proposed technique has been demonstrated. In applications, the reliability of the technique is demonstrated by the error function based on the accuracy of the approximate solution. The numerical applications have provided the eciency of the method with dierent coecients of the population growth model. Finally, the obtained results reveal that the proposed technique is very convenient and quite accurate to such considered problems.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1639-1657
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• 2018

In this paper, we initiate the study of boundary value problems involving Hilfer fractional derivatives. Several new existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating our results are also presented.