Related papers: Self-similar Cauchy problems and generalized Mitta…
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary…
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also…
We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and…
In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative…
This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from…
The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…
A fractional Stefan problem with a boundary convective condition is solved, where the fractional derivative of order $ \alpha \in (0,1) $ is taken in the Caputo sense. Then an equivalence with other two fractional Stefan problems (the first…
Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here…
We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces…
We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of…
We study dynamic minimization problems of the calculus of variations with Lagrangian functionals containing Riemann-Liouville fractional integrals, classical and Caputo fractional derivatives. Under assumptions of regularity, coercivity and…
In this paper we give an explicit representation of the solutions of a characteristic Cauchy problem for a class of PDEs with singular coefficients. We give the explicit solutions in terms of the Gauss hypergeometric functions, which enable…
It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to…
In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the…
We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the…
The well-known Caputo fractional derivative and the corresponding Caputo fractional integral occur naturally in many equations that model physical phenomena under inhomogeneous media. The relationship between the two fractional terms can be…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…
The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation for a convolution type operator. In this equation we use a Caputo time derivative of order $\alpha$ with $\alpha\in(0,1)$,…
We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in…
The $k$-Cauchy-Fueter operators and complexes are quaternionic counterparts of the Cauchy-Riemann operator and the Dolbeault complex in the theory of several complex variables. To develop the function theory of several quaternionic…