Related papers: Fractional equations via convergence of forms
We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear…
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An…
In this article, we propose new proportional fractional operators generated from local proportional derivatives of a function with respect to another function. We present some properties of these fractional operators which can be also…
It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…
In this paper, we consider the backward problem for fractional in time evolution equations $\partial_t^\alpha u(t)= A u(t)$ with the Caputo derivative of order $0<\alpha \le 1$, where $A$ is a self-adjoint and bounded above operator on a…
We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $\alpha$ ($0 < \alpha < 1$) is in the Riemann-Liouville sense. The technique we…
The evolution of a quantity, described by a function of space and time, relates the first derivative in time of this function to a spatial operator applied to the function. The initial value of the function at time $t=0$ is given. The…
A form of the Laplace transform is reviewed as a paradigm for an entire class of fractional functional transforms. Various of its properties are discussed. Such transformations should be useful in application to differential/integral…
In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation $u(x_0,t,\omega).$ We build the representation of the…
We study invariant solutions of a certain class of time-fractional diffusion-wave equations with variable coefficients via Lie symmetry analysis. In physics, the fractional diffusion equation describes transport dynamics that are governed…
Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J.…
This article is devoted to presenting an abstract theory on time-fractional gradient flows for nonconvex energy functionals in Hilbert spaces. Main results consist of local and global in time existence of (continuous) strong solutions to…
We consider a class of fractional time stochastic equation defined on a bounded domain and show that the presence of the time derivative induces a significant change in the qualitative behaviour of the solutions. This is in sharp contrast…
This paper deals with the investigation of a closed form solution of a generalized fractional reaction-diffusion equation. The solution of the proposed problem is developed in a compact form in terms of the H-function by the application of…
The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment…
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…
This contribution considers the time-fractional subdiffusion with a time-dependent variable-order fractional operator of order $\beta(t)$. It is assumed that $\beta(t)$ is a piecewise constant function with a finite number of jumps. A proof…
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the…
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson…
This paper is in continuation of the authors' recently published paper (Journal of Mathematical Physics 55(2014)083519) in which computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo…