This paper presents a comparative assessment of low Reynolds number k- models against standard k- model in an eulerian framework. Three different low-Re number k- models: Launder-Sharma (LS), YangShih (YS) and AbeKondoh-Nagano (AKN) have been used for the description of bubble plume behaviour in stratified water. The contribution of the gas phase movement into the liquid phase turbulence has been achieved by using the Dispersed with Bubble Induced Turbulence approach (DIS+BIT). The results reveal that the oscillation frequency of gas-liquid flow are correctly reproduced by standard k- and LS models. In fact, we found for standard K- and LS a clear dominant peak at a frequency equal to 0. 1 Hz. On the other hand, YS and AKN models have predicted chaotic oscillations. The oscillation amplitude of the bubble plume predicted from LS model seems to be in good agreement with the PIV measurements of Besbes et al. (2015). However, for the standard K- model the oscillation amplitude is low. The air-water interface shows that the bubble plume mixing with the stratified water is predicted to be stronger compared to standard k- model.

Wave-field extrapolation based on solving the wave equation is an important step in seismic modeling and needs a high level of accuracy. It has been implemented through a various numerical methods such as finite difference method as the most popular and conventional one. Moreover, the main drawbacks of the finite difference method are the low level of accuracy and the numerical dispersion for large time intervals (Δ t). On the other hand, the symplectic integrators due to their structure can cope with this problem and act more accurately in comparison to the finite difference method. They reduce the computation cost and do not face numerical dispersion when time interval is increased. Therefore, the aim of the current paper is to present a symplectic integrator for wave-field extrapolation using the euler method. Then, the extrapolation is implemented for rather large time intervals using a simple geological model. The extrapolation employed for both symplectic euler and finite difference methods showed a better quality image for the proposed method. Finally the accuracy was compared to the finite difference method.

In this paper we introduce euler method for solving one dimensional fuzzy differential inclusion. Fuzzy reachable set can be approximated by euler method with complete analysis. The method is illustrated by numerical example.

In this paper, we propose an exponential euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion Ba, b under some assumptions on a and b. We obtain also the convergence rate of the method to the true solution after proving an L2-maximal bound for the stochastic integrals in this case.

The fractional-order differential equations (FDEs) have the ability to model the real-life phenomena better in a variety of applied mathematics, engineering disciplines including diffusive transport, electrical networks, electromagnetic theory, probability and so forth. In most cases, there are no analytical solutions therefore a variety of numerical methods have been developed for the solution of the FDEs. In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the euler Wavelet method (EWM). The euler wavelet operational matrix method converts the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and efficiency of the technique.

In this paper, the extended euler deconvolution method is studied for interpreting magnetic and gravity anomalies. This method, overcoming some limitations of the conventional euler deconvolution method, is utilized for simultaneous and automatic estimate of the depth, structural index and horizontal location of potential field sources. The main limitation of the conventional euler deconvolution method is non-linear dependency of structural index and background field; hence a simultaneous estimation of these parameter is not possible. For overcoming this problem, a value of structural index is presumed and the obtained results are evaluated according to various criteria. A wrong structural index affects the final results. In the extended euler deconvolution, the euler differential equation is solved for Hilbert transform of the field and its derivatives. Since the Hilbert transform of a constant value is zero, the linear dependency of structural index and background filed will be removed, and therefore the automatic calculation of structural index will be possible and the presumption of structural index is not required anymore. Moreover, since Hilbert transform has two components of x and y, the number of equations to be solved at each point is increased, and consequently the solutions are more reliable. In this paper, firstly, a background theory of the extended euler deconvolution is discussed in detail. Then the method is applied to a magnetic anomaly produced over eighteen magnetic sphere (dipole) having different magnetic properties. Finally, the method is used for interpreting a Bouguer gravity anomaly of Noranda in Quebec province of Canada and also a magnetic anomaly of an area located near Anar city of Kerman province of Iran.

Efficiency of numerical methods is an important problem in dynamic nonlinear analyses. It is possible to use of numerical methods such as beta- Newmark in order to investigate the structural response behavior of the dynamic systems under random sea wave loads but because of necessity to analysis the offshore systems for extensive time to fatigue study it is important to use of simple stable methods for numerical integration. The modified euler method (MEM) is a simple numerical procedure which can be effectively used for the analysis of the dynamic response of structures in time domain. It is also very effective for response dependent systems in the field of offshore engineering. An important point is investigating the convergence and stability of the method for strongly nonlinear dynamic systems when high initial values for differential equation or large time steps are considered for numerical integrating especially when some frequencies of the system is very high. In this paper the stability of the method for solving differential equation of motion of a nonlinear offshore system (tension leg platform, TLP) under random wave excitation is presented. In this paper the stability criterion and the convergence of the numerical solution for critical time steps are presented.

In this paper, the functional Volterra integral equations of the Hammerstein type are studied. First, some conditions that ensure the existence and uniqueness of the solutions to these equations within the space of square-integrable functions are established and then the euler operational matrix of integration is constructed and applied within the collocation method for approximating the solutions. This approach transforms the integral equation into a set of nonlinear algebraic equations, which can be efficiently solved by employing standard numerical methods like Newton's method or Picard iteration. One significant advantage of this method lies in its ability to avoid the need for direct integration to discretize the integral operator. Error estimates are provided and two illustrative examples are included to demonstrate the method’s effectiveness and practical applicability.

Purpose: The purpose of this paper is to develop a set of identities for euler type sums of products of harmonic numbers and reciprocal binomial coefficients.method: We use analytical methods to obtain our results.Results: We obtain identities for variant euler sums of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.