A new class of the entire functions: a study of two cases from Euler

In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.

On some classes of the entire functions

This article proves the products, behaviors and simple zeros for the classes of the entire functions associated with the Weierstrass-Hadamard product and the Taylor series.

The scaling-law flows: An attempt at scaling-law vector calculus

In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.

On nontrivial zeros for the Ramanujan zeta function

The Hardy hypothesis, as an analogue to the Riemann hypothesis for the Riemann zeta function, is a conjecture proposed by Hardy in 1940, that all of the nontrivial zeros for the Ramanujan zeta function have a real part equal to 6. In this paper, we propose the power series expansion for the entire Ramanujan zeta function using the work of Mordell. Then, we suggest an alternative infinite product for the entire Ramanujan zeta function derived from the work of Conrey and Ghosh. We also establish the class of the entire Ramanujan zeta function related to the functional equation coming from Wilton. Motivated by the work of Lekkerkerker, we prove an conjecture due to Bruijn that all of the zeros of the Ramanujan Xi function are nonzero real numbers. From theory of the entire functions, we also prove that the Hardy hypothesis is true.

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.

Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems

In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from mathematical view of point. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. They are very efficient in description of the complex phenomenon arising in heat transfer.

Non-differentiable solutions for local fractional nonlinear Riccati differential equations

We investigate local fractional nonlinear Riccati differential equations (LFNRDE) by transforming them into local fractional linear ordinary differential equations. The case of LFNRDE with constant coefficients is considered and non-differentiable solutions for special cases obtained.

A new fractional operator of variable order: application in the description of anomalous diffusion

In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.

Some new applications for heat and fluid flows via fractional derivatives without singular kernel

This paper addresses the mathematical models for the heat-conduction equations and the Navier-Stokes equations via fractional derivatives without singular kernel.

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.