scholarly journals Some Results in the Theory of Fractional Order Integro-Differential Equation with Boundary Conditions

2010 ◽  
Vol 7 (2) ◽  
pp. 101-114
Author(s):  
Azzam Younes
2008 ◽  
Vol 15 (3) ◽  
pp. 531-539
Author(s):  
Temur Jangveladze ◽  
Zurab Kiguradze

Abstract Large time behavior of solutions to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. The rate of convergence is given, too. Dirichlet boundary conditions with homogeneous data are considered.


Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 121
Author(s):  
Tursun K. Yuldashev ◽  
Erkinjon T. Karimov

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.


2011 ◽  
Vol 90-93 ◽  
pp. 951-957
Author(s):  
Yuan Ping Li ◽  
Wei Zhang

The fractional dynamics equation of a viscoelastic two-member truss system, in which fractional derivative model introduced to simulate the materials’ characteristics, is proposed. The simplified single DOF differential equation is developed combined with boundary conditions and symmetry. Dynamic behaviors of the fractional single DOF system with harmonic loads are discussed by numerical calculations. The results show that: the system may lead to chaotic motion via period-doubling bifurcations or intermittent routes; the dynamical character is greatly inflected by the varying of excitation amplitude or damping coefficient or fractional order.


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