Fractional differential equations with singular coefficients

Shishkina, Elina; Sitnik, Sergei M.

doi:10.1016/b978-0-12-819781-3.00017-3

Cited by 61 publications

(42 citation statements)

References 3 publications

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“…For every x ∈ [0, 𝜋], the series converge uniformly on any compact set of the complex plane of the variable 𝜌, and the remainders of their partial sums admit estimates independent of Re𝜌. This last feature of the series representations (the independence of Re𝜌 of the estimates for the remainders) is a direct consequence of the fact that the representations are obtained by expanding the integral kernels of the transmutation operators (for their theory we refer to previous works [24][25][26] ) into Fourier-Legendre series (see Kravchenko et al 1 and Kravchenko 6, Sect. 9.4 ).…”

Section: Preliminariesmentioning

confidence: 97%

“…This last feature of the series representations (the independence of

\operatorname{Re}\rho

of the estimates for the remainders) is a direct consequence of the fact that the representations are obtained by expanding the integral kernels of the transmutation operators (for their theory we refer to previous works 24–26 ) into Fourier–Legendre series (see Kravchenko et al 1 and Kravchenko 6 , Sect. 9.4 ).…”

Section: Preliminariesmentioning

confidence: 99%

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Spectrum completion and inverse Sturm–Liouville problems

Kravchenko

2022

Math Methods in App Sciences

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Given a finite set of eigenvalues of a regular Sturm–Liouville problem for the equation −y″+qfalse(xfalse)y=λy$$ -{y}^{{\prime\prime} }+q(x)y=\lambda y $$, the potential qfalse(xfalse)$$ q(x) $$ of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential qfalse(xfalse)$$ q(x) $$. Moreover, considering the Sturm–Liouville problem with the boundary conditions y′false(0false)−hyfalse(0false)=0$$ {y}^{\prime }(0)- hy(0)=0 $$ and y′false(πfalse)+Hyfalse(πfalse)=0$$ {y}^{\prime}\left(\pi \right)+ Hy\left(\pi \right)=0 $$, where h,0.1emH$$ h,H $$ are some constants, we complete its spectrum without additional information neither on the potential qfalse(xfalse)$$ q(x) $$ nor on the constants h$$ h $$ and H$$ H $$. The eigenvalues are computed with a uniform absolute accuracy. Based on this result, we propose a new method for numerical solution of the inverse Sturm–Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential qfalse(xfalse)$$ q(x) $$ is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm–Liouville problems.

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Section: Preliminariesmentioning

confidence: 97%

“…This last feature of the series representations (the independence of

\operatorname{Re}\rho

Section: Preliminariesmentioning

confidence: 99%

Spectrum completion and inverse Sturm–Liouville problems

Kravchenko

2022

Math Methods in App Sciences

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“…In transmutation theory, there are problems for next varied types of operators: integral, integro-differential, difference-differential (e.g., the Dunkl operator), pseudodifferential and abstract differential operators, cf. [1][2][3][4][5][6][7][8]. In quantum physics, in the study of the Shrödinger equation and inverse scattering theory, underlying transmutations are called wave operators.…”

Section: Introductionmentioning

confidence: 99%

“…For the explicit construction of transmutations, a special method was introduced and developed by the first author-the integral transforms composition method (ITCM) , thoroughly studied in [3,4,9] (and more references therein). The essence of this method is to construct the necessary transmutation operator and corresponding connection formulas among the solutions of perturbed and nonperturbed equations, as a composition of classical integral transforms with properly chosen weighted functions.…”

Section: Introductionmentioning

confidence: 99%

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On an Integral Equation with the Riemann Function Kernel

Sitnik

Arian

2022

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This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written.

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Comparison of simulation and analytical models for the distribution of a group of agents moving in random directions

Kuznetsov

Shishkina

Rataj

2022

Math Methods in App Sciences

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This study focused on simulation and analytical models of a multidimensional random walk of many agents. At each step, any agent can rotate by an arbitrary angle and continue moving in the direction selected. The analytical model used gives the probability of finding an agent in the circular area of the radius r in the given time moment. The model includes parameters that control the intensity of the walking process and the characteristics of the environment. Work on this topic mainly concerns the discrete case of a random walk. In this article, we consider the case of a random walk continuous in spatial coordinates. We obtain certain theoretical results for the analytical model with the help of the mathematical apparatus aimed at working with a generalized translation. With the help of a simulation, we reveal the meaning of the analytical model's parameters.Results can be used for modeling diffusion-like processes such as the spread of an epidemic, mechanical vibrations, forest fires, migration, and dissemination of information over a network.

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Fractional differential equations with singular coefficients

Cited by 61 publications

References 3 publications

Spectrum completion and inverse Sturm–Liouville problems

Spectrum completion and inverse Sturm–Liouville problems

On an Integral Equation with the Riemann Function Kernel

Comparison of simulation and analytical models for the distribution of a group of agents moving in random directions

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