• 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.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1123-1139
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• 2014

In this work, we investigate the existence of the extremal solutions for a class of fractional partial differential equations with order 1 < ${\alpha}$ < 2 by upper and lower solution method. Using the theory of Hausdorff measure of noncompactness, a series of results about the solutions to such differential equations is obtained.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.651-675
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• 2018

We present ${\alpha}$-type rational F-contractions in metric-like spaces, and respective fixed and common fixed point results for weakly ${\alpha}$-admissible mappings. Useful examples illustrate the effectiveness of the presented results. As applications, we obtain sufficient conditions for the existence of solutions of a certain type of integral equations followed by examples of nonlinear fractional differential equations that are verified numerically.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.1011-1020
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• 2021

In this paper, first order nonlinear Liouville-Caputo fractional differential equations is studied. The existence and uniqueness of a solution are investigated by using Krasnoselskii and Banach fixed point theorems and the method of lower and upper solutions. Finally, an example is given to illustrate our results.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.877-894
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• 2013

In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green's function, a nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem in a cone. Some examples are included to show the applicability of our results.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.479-489
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• 2019

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian sheet with Hurst parameters H, H' > 1/2, and a drift coefficient satisfying the linear growth condition. The result is obtained using a suitable Girsanov theorem for the fractional Brownian sheet.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.229-241
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• 2007

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractional-order dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.149-157
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• 2012

By means of fractional calculus techniques, we find explicit solutions of confluent hypergeometric equations. We use the N-fractional calculus operator $N^{\mu}$ method to derive the solutions of these equations.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1019-1036
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• 2016

In this paper, we study the existence of solutions and $L^2$-regularity for fractional order retarded neutral functional differential equations in Hilbert spaces. We no longer require the compactness of structural operators to prove the existence of continuous solutions of the non-linear differential system, but instead we investigate the relation between the regularity of solutions of fractional order retarded neutral functional differential systems with unbounded principal operators and that of its corresponding linear system excluded by the nonlinear term. Finally, we give a simple example to which our main result can be applied.

• 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.