1. Historical backgrounds: the Airy function in Physics 2 2. The area-Airy distributions: Brownian motion, linear probing hashing, additive parameters in grammars 4 2.1. Area under a Brownian excursion 4 2.2. On the analysis of linear probing hashing 4 2.3. Analytic variations on the Airy distribution 5 2.4. Hachage, arbres, chemins & graphes 6 3. Random matrices, Airy kernel and the Tracy–Wido...

in this paper rationalized haar (rh) functions method is applied to approximate the numerical solution of the fractional volterra integro-differential equations (fvides). the fractional derivatives are described in caputo sense. the properties of rh functions are presented, and the operational matrix of the fractional integration together with the product operational matrix are used to reduce t...

in this paper, a new identification of the lagrange multipliers by means of the sumudu transform, is employed to  btain a quick and accurate solution to the fractional black-scholes equation with the initial condition for a european option pricing problem. undoubtedly this model is the most well known model for pricing financial derivatives. the fractional derivatives is described in caputo sen...

although there are many methods for image denoising, but partial differential equation (pde) based denoising attracted much attention in the field of medical image processing such as magnetic resonance imaging (mri). the main advantage of pde-based denoising approach is laid in its ability to smooth image in a nonlinear way, which effectively removes the noise, as well as preserving edge throug...

in this paper, a new numerical method for solving time-fractional diffusion equations is introduced. for this purpose, finite difference scheme for discretization in time and chebyshev collocation method is applied. also, to simplify application of the method, the matrix form of the suggested method is obtained. illustrative examples show that the proposed method is very efficient and accurate.

An analytical method to find the flow generated by the basic singularities of Stokes flow in a wedge of arbitrary angle is presented. Specifically, we solve a biharmonic equation for the stream function of the flow generated by a point stresslet singularity and satisfying no-slip boundary conditions on the two walls of the wedge. The method, which is readily adapted to any other singularity typ...

The Cauchy problem for the modified KdV-equation ut + uxxx = (u 3)x, u(0) = u0 is shown to be locally wellposed for data u0 in the space Ĥr s (R) defined by the norm ‖u0‖ Ĥr s := ‖〈ξ〉sû0‖Lr′ ξ , provided 4 3 < r ≤ 2, s ≥ 1 2 − 1 2r . For r = 2 this coincides with the best possible result on the H-scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restrictio...

We introduce discrete fractional sum equations and inequalities. We obtain the equivalence of an initial value problem for a discrete fractional equation and a discrete fractional sum equation. Then we give an explicit solution to the linear discrete fractional sum equation. This allows us to state and prove an analogue of Gronwall's inequality on discrete fractional calculus. We employ a nabla...

Fractional Langevin equation to describe anomalous diffusion. Abstract A Langevin equation with a special type of additive random source is considered. This random force presents a fractional order derivative of white noise, and leads to a power-law time behavior of the mean square displacement of a particle, with the power exponent being noninteger. More general equation containing fractional ...

In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fra...