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A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the…
This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and…
In this paper, the fractional projective Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Burgers equation, the space-time…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
Time fractional advection-dispersion equations arise as generalizations of classical integer order advection-dispersion equations and are increasingly used to model fluid flow problems through porous media. In this paper we develop an…
The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional…
Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases…
This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them.…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
The numerical solution of a nonlinear and space-fractional anti-diffusive equation used to model dune morphodynamics is considered. Spatial discretization is effected using a finite element method whereas the Crank-Nicolson scheme is used…
We present and analyze a space-time Petrov-Galerkin finite element method for a time-fractional diffusion equation involving a Riemann-Liouville fractional derivative of order $\alpha\in(0,1)$ in time and zero initial data. We derive a…
Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the Advection Diffusion Equation supplemented by a time derivative of non-integer order. We are describing an inverse method…
We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[ \frac{\partial u}{\partial t}+(-\Delta)^{\sigma/2} (u^m)=0 \] posed for $x\in \mathbb{R}^N$, $t>0$, with $m\geq 1$, $\sigma \in (0,2)$,…
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations…
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of…
The time-fractional diffusion equation is considered, where the time derivative is either of Caputo or Riemann-Liouville type. The solution of a general initial-boundary value problem with time-dependent boundary conditions over bounded and…
Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short…
Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients.
This article examines a new approach to solving ordinary differential equations based on Fractional-Calculus theory. Poisson and Sturm-Liouville-type problems are studied, together with different boundary conditions. Each case is analyzed…