23,888 research outputs found
A new fractional derivative involving the normalized sinc function without singular kernel
In this paper, a new fractional derivative involving the normalized sinc
function without singular kernel is proposed. The Laplace transform is used to
find the analytical solution of the anomalous heat-diffusion problems. The
comparative results between classical and fractional-order operators are
presented. The results are significant in the analysis of one-dimensional
anomalous heat-transfer problems.Comment: Keywords: Fractional derivative, anomalous heat diffusion, integral
transform, analytical solutio
A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow
In this article we propose a new fractional derivative without singular
kernel. We consider the potential application for modeling the steady
heat-conduction problem. The analytical solution of the fractional-order heat
flow is also obtained by means of the Laplace transform.Comment: 1 figur
Kernel Density Estimation with Linked Boundary Conditions
Kernel density estimation on a finite interval poses an outstanding challenge
because of the well-recognized bias at the boundaries of the interval.
Motivated by an application in cancer research, we consider a boundary
constraint linking the values of the unknown target density function at the
boundaries. We provide a kernel density estimator (KDE) that successfully
incorporates this linked boundary condition, leading to a non-self-adjoint
diffusion process and expansions in non-separable generalized eigenfunctions.
The solution is rigorously analyzed through an integral representation given by
the unified transform (or Fokas method). The new KDE possesses many desirable
properties, such as consistency, asymptotically negligible bias at the
boundaries, and an increased rate of approximation, as measured by the AMISE.
We apply our method to the motivating example in biology and provide numerical
experiments with synthetic data, including comparisons with state-of-the-art
KDEs (which currently cannot handle linked boundary constraints). Results
suggest that the new method is fast and accurate. Furthermore, we demonstrate
how to build statistical estimators of the boundary conditions satisfied by the
target function without apriori knowledge. Our analysis can also be extended to
more general boundary conditions that may be encountered in applications
Exactly solvable variable parametric Burgers type models
Exactly solvable variable parametric Burgers type equations in one-dimension
are introduced, and two different approaches for solving the corresponding
initial value problems are given. The first one is using the relationship
between the variable parametric models and their standard counterparts. The
second approach is a direct linearization of the variable parametric Burgers
model to a variable parametric parabolic model via a generalized Cole-Hopf
transform. Eventually, the problem of finding analytic and exact solutions of
the variable parametric models reduces to that of solving a corresponding
second order linear ODE with time dependent coefficients. This makes our
results applicable to a wide class of exactly solvable Burgers type equations
related with the classical Sturm-Liouville problems for the orthogonal
polynomials
Boundary integral formulation for interfacial cracks in thermodiffusive bimaterials
An original boundary integral formulation is proposed for the problem of a
semi-infinite crack at the interface between two dissimilar elastic materials
in the presence of heat flows and mass diffusion. Symmetric and skew-symmetric
weight function matrices are used together with a generalized Betti's
reciprocity theorem in order to derive a system of integral equations that
relate the applied loading, the temperature and mass concentration fields, the
heat and mass fluxes on the fracture surfaces and the resulting crack opening.
The obtained integral identities can have many relevant applications, such as
for the modelling of crack and damage processes at the interface between
different components in electrochemical energy devices characterized by
multi-layered structures (solid oxide fuel cells and lithium ions batteries).Comment: 43 pages, 9 figure
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