93,527 research outputs found
A Minimization Approach to Conservation Laws With Random Initial Conditions and Non-smooth, Non-strictly Convex Flux
We obtain solutions to conservation laws under any random initial conditions
that are described by Gaussian stochastic processes (in some cases
discretized). We analyze the generalization of Burgers' equation for a smooth
flux function for
under random initial data. We then consider a piecewise linear, non-smooth and
non-convex flux function paired with general discretized Gaussian stochastic
process initial data. By partitioning the real line into a finite number of
points, we obtain an exact expression for the solution of this problem. From
this we can also find exact and approximate formulae for the density of shocks
in the solution profile at a given time and spatial coordinate . We
discuss the simplification of these results in specific cases, including
Brownian motion and Brownian bridge, for which the inverse covariance matrix
and corresponding eigenvalue spectrum have some special properties. We
calculate the transition probabilities between various cases and examine the
variance of the solution in both and . We also
describe how results may be obtained for a non-discretized version of a
Gaussian stochastic process by taking the continuum limit as the partition
becomes more fine.Comment: 36 pages, 5 figures, small update from published versio
Numerical Computation of Exponential Functions of Nabla Fractional Calculus
In this article, we illustrate the asymptotic behaviour of exponential
functions of nabla fractional calculus. For this purpose, we propose a novel
matrix technique to compute these functions numerically
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Towards the Formalization of Fractional Calculus in Higher-Order Logic
Fractional calculus is a generalization of classical theories of integration
and differentiation to arbitrary order (i.e., real or complex numbers). In the
last two decades, this new mathematical modeling approach has been widely used
to analyze a wide class of physical systems in various fields of science and
engineering. In this paper, we describe an ongoing project which aims at
formalizing the basic theories of fractional calculus in the HOL Light theorem
prover. Mainly, we present the motivation and application of such formalization
efforts, a roadmap to achieve our goals, current status of the project and
future milestones.Comment: 9 page
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