Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Staněk, Svatoslav

doi:10.2478/s11533-012-0141-4

Cited by 22 publications

(23 citation statements)

References 13 publications

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“…In , Stanek investigated the existence and uniqueness of solutions of the fractional differential equation (which is called a generalized Bagley–Torvik fractional differential model )

u^{''} + A^{c} D_{0^{+}}^{α} u = f (t, u,^{c} D_{0^{+}}^{μ} u, u^{'}), t \in [0, T],

subject to the boundary conditions

u^{'} (0) = 0,

u (T) + a u^{'} (T) = 0

. Here

α \in (1, 2)

μ \in (0, 1)

, f is a Carathéodory function and

^{c} D_{0^{+}}

is the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

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Boundary value problems of singular multi-term fractional differential equations with impulse effects

Liu

2016

Math. Nachr.

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We study a class of boundary value problems for nonlinear impulsive fractional differential equations. A weighted function Banach space and a completely continuous nonlinear operator are constructed firstly. Some existence results for solutions of these problems are established continuously. Our analysis relies on the well known Schauder's fixed point theorem. Examples are given to illustrate the main results finally.

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“…In , Stanek investigated the existence and uniqueness of solutions of the fractional differential equation (which is called a generalized Bagley–Torvik fractional differential model )

u^{''} + A^{c} D_{0^{+}}^{α} u = f (t, u,^{c} D_{0^{+}}^{μ} u, u^{'}), t \in [0, T],

subject to the boundary conditions

u^{'} (0) = 0,

u (T) + a u^{'} (T) = 0

. Here

α \in (1, 2)

μ \in (0, 1)

, f is a Carathéodory function and

^{c} D_{0^{+}}

is the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…In [33], Stanek investigated the existence and uniqueness of solutions of the fractional differential equation (which is called a generalized Bagley-Torvik fractional differential model)…”

Section: Introductionmentioning

confidence: 99%

Boundary value problems of singular multi-term fractional differential equations with impulse effects

Liu

2016

Math. Nachr.

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“…It is noted that for the generalized nonlinear Bagley-Torvik equation, Stanek [16] has studied the two-point boundary value problem. Moreover, the coefficients A, B and C may also change with the changes of fluid density and viscosity.…”

Section: Ay (T) + B D α Y(t) + Cy(t) = F(t)mentioning

confidence: 99%

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Huang

Zhong

Guo

2015

Int. J. Appl. Comput. Math

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The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is 1 < α < 2, the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is 0 < α < 1, the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.

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“…Here, the solution of BTE expressed by (14) is modified, in order to be written in terms of Wiman's functions and their derivatives. The explicit forms of these functions are given by…”

Section: Bte Solution Via Wiman's Functions and Derivativesmentioning

confidence: 99%

“…Analytical exact solutions for BTE were obtained in [13] for the particular initial condition (0) = (0) = 0, considering the boundary condition given by (0) = (1) = 1. Besides, by using a modified generalized Laguerre spectral method for fractional differential equations, BTE was solved in [14] for some specific conditions.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of General Bagley-Torvik Equation

Labecca

Guimarães

Piqueira

2015

Mathematical Problems in Engineering

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Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.

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Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Cited by 22 publications

References 13 publications

Boundary value problems of singular multi-term fractional differential equations with impulse effects

Boundary value problems of singular multi-term fractional differential equations with impulse effects

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Analytical Solution of General Bagley-Torvik Equation

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