• Korean Journal of Mathematics
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• 제29권4호
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• pp.749-763
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• 2021

This research article deals with the Mittag-Leffler-Hyers-Ulam stability of linear and impulsive fractional order differential equation which involves the Caputo derivative. The application of the generalized fractional Fourier transform method and fixed point theorem, evaluates the existence, uniqueness and stability of solution that are acquired for the proposed non-linear problems on Lizorkin space. Finally, examples are introduced to validate the outcomes of main result.

• Nonlinear Functional Analysis and Applications
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• 제26권1호
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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.

• Nonlinear Functional Analysis and Applications
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• 제26권4호
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• pp.793-809
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• 2021

This paper gives sufficient conditions to ensure the existence and stability of solutions for generalized nonlinear fractional integrodifferential equations of order α (1 < α < 2). The main theorem asserts the stability results in a weighted Banach space, employing the Krasnoselskii's fixed point technique and the existence of at least one mild solution satisfying the asymptotic stability condition. Two examples are provided to illustrate the theory.

• Computers and Concrete
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• 제34권1호
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• pp.63-77
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• 2024

In this research, we considered fractional order optimal control models for cancer, HIV treatment and glucose.These models are based on fractional order differential equations that describe the dynamics underlying the disease.It is formulated in term of left and right Caputo fractional derivative. Pontryagin's Maximum Principle is used as a necessary condition to find the optimal curve for the respective controls over fixed time period. The formulated problems are numerically solved using forward backward sweep method with generalized Euler scheme.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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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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• 제34권3_4호
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• pp.193-206
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• 2016

In this paper, we study boundary value problems for fractional integrodifferential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제25권3호
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• pp.117-131
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• 2021

The problem of generalized thermoelasticity of two-temperature for finite piezoelectric rod will be modified by applying three different types of heating applications namely, thermal shock, ramp-type heating and harmonically vary heating. The solutions will be derived with direct approach by the application of Laplace transform and the Caputo-Fabrizio fractional order derivative. The inverse Laplace transforms are numerically evaluated with the help of a method formulated on Fourier series expansion. The results obtained for the conductive temperature, the dynamical temperature, the displacement, the stress and the strain distributions have represented graphically using MATLAB.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.221-228
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• 2013

The method of upper and lower solutions and the generalized quasilinearization technique is developed for the existence and approximation of solutions to boundary value problems for higher order fractional differential equations of the type $^c\mathcal{D}^qu(t)+f(t,u(t))=0$, $t{\in}(0,1),q{\in}(n-1,n],n{\geq}2$ $u^{\prime}(0)=0,u^{\prime\prime}(0)=0,{\ldots},u^{n-1}(0)=0,u(1)={\xi}u({\eta})$, where ${\xi},{\eta}{\in}(0,1)$, the nonlinear function f is assumed to be continuous and $^c\mathcal{D}^q$ is the fractional derivative in the sense of Caputo. Existence of solution is established via the upper and lower solutions method and approximation of solutions uses the generalized quasilinearization technique.

• Steel and Composite Structures
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• 제24권3호
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• pp.297-307
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• 2017

A unified mathematical model of phase-lag Green-Naghdi magneto-thermoelasticty theories based on fractional derivative heat transfer for perfectly conducting media in the presence of a constant magnetic field is given. The GN theories as well as the theories of coupled and of generalized magneto-thermoelasticity with thermal relaxation follow as limit cases. The resulting nondimensional coupled equations together with the Laplace transforms techniques are applied to a half space, which is assumed to be traction free and subjected to a thermal shock that is a function of time. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of magneto-thermoelasticity with one relaxation time. The effects of Alfven velocity and the fractional order parameter on copper-like material are discussed in different types of GN theories.

• 대한수학회보
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• 제51권2호
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• pp.329-338
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• 2014

In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.