Related papers: Trace Formulas for a Conformable Fractional Diffus…
The study of fractional order differential operators is receiving renewed attention in many scientific fields. In order to accommodate researchers doing work in these areas, there is a need for highly scalable numerical methods for solving…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
In this paper we establish some (presumably new) interesting expressions for the composition of some well known fractional integral operators $ I^{\mu}_{a+}, D^{\mu}_{a+} $,$ I^{\gamma , \mu}_{a+}$ and also derive an integral operator…
This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $\alpha$-order. Solutions for these equations with constant coefficients are…
We consider Schr\"odinger operators with complex decaying potentials on the lattice. Using some classical results from Complex Analysis we obtain some trace formulae and using them estimate globally all zeros of the Fredholm determinant in…
In this paper we study generalized time-fractional diffusion equations on the Poincar\`e half plane $\mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function,…
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…
In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given…
In this work, we give the power series solutions around an ordinary point, in the case of variable coefficients, homogeneous sequential linear conformable fractional differential equations of order 2\alpha. Further, we introduce the…
In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$ \noindent with $m\in \mathbb{N}$,…
This note is devoted to exploring some analytic-geometric properties of the regularity and capacity associated to the so-called fractional dissipative operator $\partial_t+(-\Delta)^\alpha$, naturally establishing a diagonally sharp…
We show that there exists $0<\alpha_0<1$ (depending on the parameters) such that the fractal percolation is almost surely purely $\alpha$-unrectifiable for all $\alpha>\alpha_0$.
We compute the Selberg trace formula for Hecke operators (also called the trace formula for modular correspondences) in the context of cocompact Kleinian groups with finite-dimentional unitary representations. We give some applications to…
We develop proper correction formulas at the starting $k-1$ steps to restore the desired $k^{\rm th}$-order convergence rate of the $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order…
The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical…
We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional…
We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$…
Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists.…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…