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- PN-II-RU-TE-2012-3-0190
- Advances on the theory of generalized Bessel functions and applications to multiphoton processes
- Differential Subordinations Involving Generalized Bessel Functions
- Some Fractional Operators with the Generalized Bessel–Maitland Function

Fractional calculus and fractional differential equations FDE have many applications in different branches of sciences. But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on generalized Bessel function of the first kind GBF , spectral methods and Newton—Krylov subspace method to solve nonlinear FDEs. In this paper, we use the GBFs as the basis functions. Then, we introduce explicit formulas to calculate Riemann—Liouville fractional integral and derivative of GBFs that are very helpful in computation and saving time. In the presented method, a nonlinear FDE will be converted to a nonlinear system of algebraic equations using collocation method based on GBF, then the solution of this nonlinear algebraic system will be achieved by using Newton-generalized minimum residual Newton—Krylov method. To illustrate the reliability and efficiency of the proposed method, we apply it to solve some examples of nonlinear Abel FDE.

Ali, S. Mubeen, I. Nayab, Serkan Araci, G. Rahman, K. In this paper, we aim to determine some results of the generalized Bessel—Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel—Maitland functions and the Mittag-Leffler functions are considered.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. A derivation of the ionization rate for the hydrogen-like ion in the strong linearly polarized laser field is presented. This derivation utilizes the famous Keldysh probability amplitude in the length gauge in the dipole approximation and without Coulomb effects in the final state of the ionized electron. No further approximations are being made, because the amplitude has been expanded in the double Fourier series in a time domain with the help of the generalized Bessel functions. View PDF on arXiv.

Because of their remarkable properties, special functions have been used frequently by scientists. For example, a wide range of problems concerning the most important areas of mathematical physics and various engineering problems are linked into application of Bessel and hypergeometric functions. These functions are often used in the solution of problems of hydrodynamics, acoustics, radio physics, atomic and nuclear physics, information theory, wave mechanics and elasticity theory.

The linear second order ordinary differential equation of type. The Bessel function can be represented by a series, the terms of which are expressed using the so-called Gamma function :. The Gamma function is the generalization of the factorial function from integers to all real numbers. It has, in particular, the following properties:.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Deniz and M. In this paper our aim is to present some subordination and superordination results, using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions.

Bessel functions , first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are canonical solutions y x of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. For example:. Bessel functions also appear in other problems, such as signal processing e. Because this is a second-order linear differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient.

All authors must sign the Transfer of Copyright agreement before the article can be published. The copyright transfer covers the exclusive rights to reproduce and distribute the article, including reprints, photographic reproductions, microfilm or any other reproductions of similar nature and translations. Authors are responsible for obtaining from the copyright holder permission to reproduce any figures for which copyright exists. Skip to main content Skip to main navigation menu Skip to site footer. Furthermore, we consider an integral operator related to the generalized Bessel functions. References Baricz, A.

The generalized Bessel functions GBF are presented within the context of a more comprehensive formalism. Numerical results are given for the first-kind MGBF as well as for the GBF with the imaginary parameter, whose importance in multiphoton processes is extensively discussed. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve.

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