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Exact solutions of fractional oscillator eigenfunction problem with fixed memory length

Klimek Malgorzata, Blaszczyk Tomasz

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

45-58

EN

In this paper, the eigenproblem for a fractional oscillator under homogeneous Dirichlet and Neumann boundary conditions is considered. Key properties of fractional operators with fixed memory length are established, such as the connection between left and right operators, the product rule for fractional integrals, and the fractional integration by the parts rule for periodic/antiperiodic functions. Explicit solutions in the form of discrete sets of sine/cosine eigenfunctions are derived. The impact of fractional order and memory length on eigenvalues is presented on graphs. Finally, a comparison of eigenvalues of oscillator with a fixed memory length and infinite memory length is shown.

2

A computational method for time fractional partial integro-differential equations

Mojahedfar M., Tari Abolfazl, Shahmorad S.

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

315--323

EN

In this paper, a class of time fractional partial integro-differential equations (FPIDEs) with initial conditions is studied. Some operational matrices are used to reduce a FPIDE problem to a system of algebraic equations with special properties. The resulted system is solved to give an approximate solution to the problem. Error estimation is also discussed for the approximate solution. Finally, some numerical examples are given to show the accuracy of the proposed method.

3

Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems

Colli Pierluigi, Gilardi Gianni, Sprekels Jurgen

Control and Cybernetics

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2019

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Vol. 48, No. 1

153--197

EN

In this paper, we give an overview of results for Cahn–Hilliard systems involving fractional operators that have recently been established by the authors of this note. We address problems concerning existence, uniqueness, and regularity of the solutions to the system equations, and we study optimal control problems for the systems. The well-posedness results are valid for a wide class of fractional operators of spectral type and for the typical double-well nonlinearities appearing in the Cahn–Hilliard system equations, namely the classical diﬀerentiable, the logarithmic, and the nondiﬀerentiable double obstacle potentials. While this also applies to the existence of optimal controls in the related optimal control problems, the establishment of ﬁrst-order necessary optimality conditions requires imposing much stronger assumptions on the admissible class of fractional operators. One main reason for this is the necessity of deriving suitable diﬀerentiability properties for the associated control-to-state mapping. Nevertheless, it turns out that also in the singular case of logarithmic potentials, the ﬁrst-order necessary optimality conditions can be established under suitable assumptions, and a “deep quench” approximation, based on the results derived for logarithmic nonlinearities makes even the case of double obstacle potentials accessible.