We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by so...

Exact and approximate solutions of the fractional diffusion equation for an assembly of fixed-axis dipoles are derived for anomalous noninertial rotational diffusion in a double-well potential. It is shown that knowledge of three time constants characterizing the normal diffusion, viz., the integral relaxation time, the effective relaxation time, and the inverse of the smallest eigenvalue of th...

Retina imaging technology is an effective control method for early diagnosis and early treatment of the diabetic retinopathy. In this paper, a fast robust inverse diffusion equation combining a blockwise filtering is presented to detect and evaluate diabetic retinopathy using retinal image enhancement. A flux corrected transport technique is used to control diffusion flux adaptively, which elim...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended fractional evolution equations characterized by anomalous dispersion using completeness suitable eigenfunctions the associated linear problem. In diffusion, mean squared displacement is proportional $t^{\alpha}$, $\alpha>0$, whi...

Abstract We derive a fundamental solution $${{\mathscr {E}}}$$ E to space-fractional diffusion problem on the half-line. The equation involves Caputo derivative. establish properties of as well formulas for solutions Dirichlet and fixed slope problems in terms convolution with data. also study integrability d...

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on boundary. We prove unique recovery together with their weights, which does not require full knowledge domain or medium properties, e.g. and potential coefficients, initial condition source model. The proof is based Laplace transform asy...

In this paper, we propose an algorithm for numerical solving an inverse nonlinear diffusion problem. The algorithm is based on the linearized nonlinear terms by Taylor ́s series expansion, removed the time-dependent terms by Laplace transform, and so, the results at a specific time can be calculated without step-by-step computations in the time domain. Finite difference technique used for discre...