On the decomposition of solutions: From fractional diffusion to fractional Laplacian

This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virus

We investigate a model of spatio-temporal spreading of human immunodeficiency virus HIV-1. The mathematical model considers the presence of various components in a human tissue, including the uninfected CD4+T cells density, the density of infected CD4+T cells, and the density of free HIV infection particles in the blood. These three components are nonnegative and bounded variables. By expressing the original model in an equivalent exponential form, we propose a positive and bounded discrete model to estimate the solutions of the continuous system. We establish conditions under which the nonnegative and bounded features of the initial-boundary data are preserved under the scheme. Moreover, we show rigorously that the method is a consistent scheme for the differential model under study, with first and second orders of consistency in time and space, respectively. The scheme is an unconditionally stable and convergent technique which has first and second orders of convergence in time and space, respectively. An application to the spatio-temporal dynamics of HIV-1 is presented in this manuscript. For the sake of reproducibility, we provide a computer implementation of our method at the end of this work.

A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution u and a numerically computed approximate solution $${\hat{u}}$$ u ^ , we evaluate the number of sign-changes of u (the number of nodal domains) and determine the location of zero level-sets of u (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen–Cahn equation. The nodal line of solutions of this equation represents the interface between two coexisting phases.

Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

Space-time fractional diffusion: transient flow to a line source

Nonlocal diffusion to a line source well is addressed by space-time fractional diffusion to model transients governed by both long-range connectivity and distorted flow paths that result in interruptions in the geological medium as a consequence of intercalations, dead ends, etc. The former, superdiffusion, results in long-distance runs and the latter, subdiffusion, in pauses. Both phenomena are quantified through fractional constitutive laws, and two exponents α and β are used to model subdiffusion and superdiffusion, respectively. Consequently, we employ both time and space fractional derivatives. The spatiotemporal evolution of transients in 2D is evaluated numerically and insights on the structure of solutions described through asymptotic solutions are confirmed numerically. Pressure distributions may be classified through two situations (i) wherein 2α = β + 1 in which case solutions may be grouped on the basis of the classical Theis solution, and (ii) wherein 2α ≠ β + 1 in which case conventional expectations do not hold; regardless, at long enough times for the combined case, power-law responses are similar to those for pure subdiffusive flows. Pure superdiffusion on the other hand, although we consider a system that is infinite in its areal extent, interestingly, results in behaviors similar to steady-state flow. To our knowledge, documented behaviors are yet to be reported.