exponential function

In this paper, we establish algebraic bounds of the ratio type in nature for the natural exponential function $e^x$ involving two parameters, a and n, which become optimal as a tends to 0 or n tends to infinity. The proof is mainly based on Chebyshev's integral inequality and properties of the incomplete gamma function. Subsequently, we focus on the simple case obtained with n = 1, with comparisons to existing literature results. For the applications, we provide alternative proofs of inequalities involving ratio functions of trigonometric and hyperbolic functions. Graphics are given to illustrate the theory.

By using L'Hospital's rule for monotonicity, we provide an alternative proof of a monotonicity property of a certain function involving the exponential function. This new approach is very concise.

Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.

Through this treatise, a study has been submitted about modified of Szasz-Mirakyan-Schurer-Kantrovich operators which that preserving (e^{- bx },b > 0) function. We interpret and study the uniform convergence of the modern operators to (text{ f}). Also, by analyzing the asymptotic conduct of our operator.

The so called integrating factor method, used to find solutions of ordinary differential equations of a certain type, is well known. In this article, we extend it to equations with values in a Banach space. Besides being of interest in itself, this extension will give us the opportunity to touch on a few topics that are not usually found in the relevant literature. Our presentation includes various illustrations of our results.

Water quality prediction is a valuable tool for pollutant control in aquatic environments. A river system is selected and one-dimensional flow is assumed for the simulation of pollutant transport in this paper. Advective and diffusive terms of transport equation are considered separately. Different functions of advective transport are proposed and applied for a specified velocity on the discrete cells along the river and compared with each other. Abilities and flexibilities of these proposed functions are investigated and the compatible one is determined for the point source pollutant transport in the channel. Fluctuations (Maximum content of negative concentrations), tolerances (Maximum interval of negative concentrations) and attenuation (Decreasing peak of the pollutant distribution) are selected for this purpose. Quick and the proposed functions including trigonometric, simple exponential, symmetric exponential and quadratic exponential of advective transport are selected for the comparison. Results show that during the simulation period (times up to 100 seconds and times more than 100 seconds), for the estimation of attenuation, the peak of the pollutant distribution and the elimination of fluctuations and the tolerances, the proposed symmetric exponential and quadratic exponential functions of advective transport perform better than other numerical methods and presented functions. Also for all these scales both of quadratic exponential and symmetric exponential functions act similarly, therefore they can be used frequently for the simulation of pollutant transport in rivers without creating negative concentration.
Moreover, simple exponential function and quick method identically predict the peak of pollutant chemograph at each downstream point.