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Yuriy Povstenko - One of the best experts on this subject based on the ideXlab platform.
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fundamental solutions to the fractional heat conduction equation in a ball under robin boundary condition
Open Mathematics, 2014Co-Authors: Yuriy PovstenkoAbstract:The central symmetric time-fractional heat conduction equation with Caputo Derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal Derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.
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Fundamental Solutions to Robin Boundary-Value Problems for the Time-Fractional Heat-Conduction Equation in a Half Line
Journal of Mathematical Sciences, 2013Co-Authors: Yuriy PovstenkoAbstract:The time-fractional heat-conduction equation with the Caputo Derivative of the order 0 < α ≤ 2 is considered in a half line. Two types of Robin boundary condition are examined: the mathematical condition with prescribed linear combination of the values of temperature and the values of its normal Derivative and the physical condition with prescribed linear combination of the values of temperature and the values of heat flux on the boundary of the domain. These two types of Robin boundary condition coincide only for the classical heat-conduction equation.
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time fractional heat conduction in an infinite medium with a spherical hole under robin boundary condition
Fractional Calculus and Applied Analysis, 2013Co-Authors: Yuriy PovstenkoAbstract:The time-fractional heat conduction equation with the Caputo Derivative of the order 0 < α ≤ 2 is considered in an infinite medium with a spherical hole in the central symmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal Derivative at the boundary and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary. The integral transforms techniques are used. Several particular cases of the obtained solutions are analyzed. The numerical results are illustrated graphically.
Delfim F M Torres - One of the best experts on this subject based on the ideXlab platform.
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on a fractional oscillator equation with natural boundary conditions
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Assia Guezanelakoud, R. Khaldi, Delfim F M TorresAbstract:We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional Derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with such approach, we first prove a result on the monotonicity of the right Caputo Derivative.
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Optimality conditions for fractional variational problems with dependence on a combined Caputo Derivative of variable order
Optimization, 2015Co-Authors: Dina Tavares, Ricardo Almeida, Delfim F M TorresAbstract:We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo Derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.
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Multiobjective fractional variational calculus in terms of a combined Caputo Derivative
Applied Mathematics and Computation, 2012Co-Authors: Agnieszka B Malinowska, Delfim F M TorresAbstract:Abstract The study of fractional variational problems in terms of a combined fractional Caputo Derivative is introduced. Necessary optimality conditions of Euler–Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems.
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fractional calculus of variations for a combined Caputo Derivative
arXiv: Optimization and Control, 2011Co-Authors: Agnieszka B Malinowska, Delfim F M TorresAbstract:We generalize the fractional Caputo Derivative to the fractional Derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional Derivative of order $\alpha$ and the right Caputo fractional Derivative of order $\beta$. The fractional variational problems under our consideration are formulated in terms of ${{^CD}^{\alpha,\beta}_{\gamma}}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.
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Fractional calculus of variations for a combined Caputo Derivative
Fractional Calculus and Applied Analysis, 2011Co-Authors: Agnieszka B Malinowska, Delfim F M TorresAbstract:We generalize the fractional Caputo Derivative to the fractional Derivative C D γ α,β , which is a convex combination of the left Caputo fractional Derivative of order α and the right Caputo fractional Derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γ α,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.
Fenghui Huang - One of the best experts on this subject based on the ideXlab platform.
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the fundamental solution of the space time fractional advection dispersion equation
Journal of Applied Mathematics and Computing, 2005Co-Authors: Fenghui HuangAbstract:A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time Derivative is replaced with Caputo Derivative of order α ∈ (0, 1], and the second-order space Derivative is replaced with a Riesz-Feller Derivative of order β ∈ (0, 2]. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time Derivatives both replacing with Caputo Derivatives.
Margarita Miranda Hernandez - One of the best experts on this subject based on the ideXlab platform.
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space time fractional diffusion advection equation with Caputo Derivative
Abstract and Applied Analysis, 2014Co-Authors: Jose Francisco Gomez Aguilar, Margarita Miranda HernandezAbstract:An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the Derivative is considered as , for the space and time domain, respectively. The fractional Derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both Derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters and are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters and . The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.
Ricardo Almeida - One of the best experts on this subject based on the ideXlab platform.
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Optimality conditions for fractional variational problems with dependence on a combined Caputo Derivative of variable order
Optimization, 2015Co-Authors: Dina Tavares, Ricardo Almeida, Delfim F M TorresAbstract:We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo Derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.
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fractional variational problems with the riesz Caputo Derivative
Applied Mathematics Letters, 2012Co-Authors: Ricardo AlmeidaAbstract:Abstract In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz–Caputo Derivative. First we prove a generalized Euler–Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional Derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.
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Fractional variational problems with the Riesz–Caputo Derivative
Applied Mathematics Letters, 2012Co-Authors: Ricardo AlmeidaAbstract:Abstract In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz–Caputo Derivative. First we prove a generalized Euler–Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional Derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.
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Necessary and sufficient conditions for the fractional calculus of variations with Caputo Derivatives
Communications in Nonlinear Science and Numerical Simulation, 2011Co-Authors: Ricardo Almeida, Delfim F M TorresAbstract:Abstract We prove optimality conditions for different variational functionals containing left and right Caputo fractional Derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler–Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo Derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo Derivatives. Normal and abnormal extremals are considered.