Search Results (26)

This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral equation to facilitate analysis; (ii) development of a novel L1 scheme for temporal discretization, which is rigorously proven to realize second-order accuracy in time; (iii) derivation of positive definiteness properties for discrete kernel coefficients; (iv) discretization of the spatial derivative using the classical second-order centered difference scheme, for which its second-order spatial convergence is rigorously verified through numerical experiments (this results in a fully discrete scheme, enabling second-order accuracy in both temporal and spatial dimensions); (v) a fast algorithm leveraging sum-of-exponential approximation, reducing the computational complexity from $O\left(N^2\right)$ to $O(NlogN)$ and memory requirements from $O\left(N\right)$ to $O(logN)$, where N is the number of grid points on a time scale. Our numerical experiments demonstrate the stability of the scheme across diverse parameter regimes and quantify significant gains in computational efficiency. Compared to the direct method, the fast algorithm substantially reduces both memory requirements and CPU time for large-scale simulations. Although a rigorous stability analysis is deferred to subsequent research, the proven properties of the coefficients and numerical validation confirm the scheme’s reliability. Full article

This study is concerned with the class of p-valent meromorphic functions, represented by the series $f\left(\zeta \right)={\zeta }^{-p}+{\sum }_{k=1-p}^{\infty }{d}_k{\zeta }^k$, with the domain characterized by $0<\left|\zeta \right|<1$. We apply an Erdelyi–Kober-type integral operator to derive two recurrence relations. From this, we draw specific conclusions on differential subordination and differential superordination. By looking into suitable classes of permitted functions, we obtain various outcomes, including results analogous to sandwich-type theorems. The operator used can provide generalizations of previous operators through specific parameter choices, thus providing more corollaries. Full article

In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of a fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If th formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, particularly when they involve operators of the Bessel type. Applying this operator simplifies the problem at hand to a comparable problem, even in the absence of the Bessel operator. An exact solution to the original problem is constructed and analyzed based on the solution to the supplementary problem. Full article

This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with $(j\in \mathbb{N})$ positively continuous and decaying functions in the finite interval $a\le t\le x$, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. Full article

In this paper, we consider and study in detail the generalized Fox–Wright function ${}_p\tilde{\Psi }_q$ introduced in our recent work as an extension of the Fox–Wright function ${}_p\Psi _q$. This special function can be seen as an important case of the so-called I-functions of Rathie and $\overline{H}$-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, ${}_p\tilde{\Psi }_q$ is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of ${}_p\tilde{\Psi }_q$ are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal. Full article

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem. Full article

The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them. Full article

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions ${}_p{\Psi }_q$ and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and ${}_p{F}_q$-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the $\overline{H}$-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and $\overline{H}$-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of $\Gamma$-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and $\overline{H}$-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

We provide and prove some new fundamental properties of the Erdélyi–Kober ($\mathcal{EK}$) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces ($L_p$) and Orlicz spaces ($L_{\phi }$). We employ these properties with the concept of the measure of noncompactness ($\mathcal{MNC}$) associated with the fixed-point hypothesis ($\mathcal{FPT}$) in solving a quadratic integral equation of fractional order in $L_p,p\ge 1$ and $L_{\phi }$. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems. Full article

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov’s new conservation theorem, the conservation laws are derived for the system. Full article

The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment of invariant solutions of initial value and boundary value problems, and for the deduction of laws of conservations. This article is aimed at applying Lie symmetry to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. This system is reduced to nonlinear fractional ordinary differential equations (FODEs) by using symmetries and explicit solutions. The reduced equations are exhibited in the form of an Erdelyi–Kober fractional (E-K) operator. The series solution of the fractional-order system and its convergence is investigated. Noether’s theorem is used to devise conservation laws. Full article

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. Full article

This paper is concerned with a class of ten time-fractional polynomial evolution equations. The one-parameter Lie point symmetries of these equations are found and the symmetry reductions are provided. These reduced equations are transformed into nonlinear ordinary differential equations, which are challenging to solve by conventional methods. We search for power series solutions and demonstrate the convergence properties of such a solution. Full article

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional ($\kappa ,\varphi$)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space $C\left(\right[1,\mathtt{T}\left]\right)$ arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations. Full article

In this paper, we first introduce a linear integral operator ${\Im }_p(a,c,\mu )(\mu >0;a,c\in \mathbb{R};c>a>-\mu p;p\in {\mathbb{N}}^{+}:=\{1,2,3,\dots \}),$ which is somewhat related to a rather specialized form of the Riemann–Liouville fractional integral operator and its varied form known as the Erdélyi–Kober fractional integral operator. We then derive some differential subordination and differential superordination results for analytic and multivalent functions in the open unit disk $\mathbb{U},$ which are associated with the above-mentioned linear integral operator ${\Im }_p(a,c,\mu )$. The results presented here are obtained by investigating appropriate classes of admissible functions. We also obtain some Sandwich-type results. Full article