Narrow Results

In this communication, a familiar physical phenomenon along with a time-dependent concentration source in a one-dimensional fractional differential advection–diffusion has been worked out. The problem is supported with the boundary with initial and boundary conditions. First of all, the results for the nondimensional classical advection–diffusion process are deliberated utilizing the Laplace coupled with finite sine-Fourier transforms analytically. Later on, the analysis is expanded for different fractional operators. The inspection of memory factors is presented through Mathcad. The impacts of the fractional (memory) parameter upon the solute concentration are discussed by making use of Mathcad15. A detailed physical significance of the fractional problem in view of the parameters is studied. It is noted that the decreasing change in concentration is associated with the larger values of noninteger parameter.

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain ${𝒟}\subset {ℝ}^n$ or, more generally, by a compact metric space ${𝒳}$ such as a compact $n$-manifold ${ℳ}$. A colored GRF ${𝒵}$, admissible for our algorithms, solves the stochastic fractional-order equation ${{𝒜}}^{\beta }{𝒵}={𝒲}$ for some $\beta >n/4$, where ${𝒜}$ is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and ${𝒲}$ is white noise on ${𝒳}$. We thus consider GRFs on ${𝒳}$ with covariance operators of the form ${𝒞}={{𝒜}}^{-2\beta }$. The proposed algorithms numerically approximate samples of ${𝒵}$ on nested sequences ${\{{{𝒯}}_{\ell }\}}_{\ell \ge 0}$ of regular, simplicial partitions ${{𝒯}}_{\ell }$ of ${𝒟}$ and ${ℳ}$, respectively. Work and memory to compute one approximate realization of the GRF ${𝒵}$ on the triangulation ${{𝒯}}_{\ell }$ of ${𝒳}$ with consistency ${𝒪}\left(\right.N_{\ell }^{-\rho }\left)\right.$, for some consistency order $\rho >0$, scale essentially linearly in $N_{\ell }=\#({{𝒯}}_{\ell })$, independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator ${{𝒜}}^{-\beta }$ to white noise ${𝒲}$. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

In this paper, the model for bovine babesiosis epidemic transmission is analyzed using a fractional operator with a Mittag-Leffler kernel. The existence and uniqueness of the solution of the considered model is studied using real analysis. The Hyers–Ulam (HU) stability is investigated with the help of nonlinear functional analysis. The numerical results of the proposed model are deduced through the Adams–Bashforth technique, which is based on the two-step Lagrangian interpolation method. All results are simulated for a few fractional orders to observe the dynamics of the proposed model.

Fractional-order techniques have many applications in real-life problems nowadays. The utilization of fragmentary math in many parts of science and engineering is wide and somewhat recent. There are various types of subsidiaries that can be valuable in various issues. In this paper, we focus on the effect of this kind of fractional derivative in the search for roots of nonlinear equations and its dependence on the initial estimations. We will check the convergence and stability analyses of the fractional Newton–Raphson (FNR) method for the proposed definitions. We will apply fractional Riemann–Liouville and Caputo-derivatives in the standard Newton root-finding method for different examples and also verify convergence and stability for these problems. Finding a root is very important for nonlinear equations which are used to solve chemical, biological, and engineering problems.

We consider the following fractional $p$-Laplacian logarithmic Schrödinger equation:

We derive a fractional relaxation equation in a setting which refers to the universal conductivity response of homogeneously disordered solid materials. Initial-time behavior of the decay function is a stretched exponential the so-called Kohlrausch-Williams-Watts (KWW) function, ρ(t) ∝ exp [–(t/τ)^β].