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The time-fractional Black-Scholes model governing European options in which the temporal derivative is focused on the Caputo fractional derivative with 0<β≤1is considered in this article. Approximating financial options with respect to their hereditary characteristics can be well understood and explained due to its outstanding memory effect current in fractional derivatives. Compelled by the stated cause. It is important to find reasonably accurate and successful numerical methods when approaching fractional differential equations. The presented numerical scheme is derived from the following manners: at first, the semi-discrete is constructed in the temporal sense based on a quadratic interpolation with accuracy order τ3-αand secondly the unconditional stability and convergence order are analyzed. For the constructed full-discrete scheme, the spatial derivative terms are approximated with the helping of the collocation method based on Legendre basis. Finally, the efficacy of the proposed method is confirmed with numerical experiments, which shows the results obtained are acceptable and in good agreement with earlier works in the literature.

The main purpose of this paper is to design a numerical method for solving the space–time fractional advection-diffusion equation (STFADE). First, a finite difference scheme is applied to obtain the semi-discrete in time variable with convergence order O(τ2-β). In the next, to discrete the spatial fractional derivative, the Chebyshev collocation method of the fourth kind has been applied. This discrete scheme is based on the closed formula for the spatial fractional derivative. Besides, the time-discrete scheme has studied in the L2space by the energy method and we have proved the unconditional stability and convergence order. Finally, we solve three examples by the proposed method and the obtained results are compared with other numerical problems. The numerical results show that our method is much more accurate than existing techniques in the literature.

This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the (2-α)-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order O(δτ2-α),0<α<1. Then, the Chebyshev spectral collocation of the third kind is implemented to approximate spatial variables and to obtain full discretization of the equation. Moreover, the unconditional stability and convergence of the proposed method are shown in the perspective H2-norm. Two numerical examples are presented to verify the effectiveness and the accuracy of the proposed method. The comparison between our obtained numerical results and the results of existing schemes in the literature shows that the proposed method is more reliable and precise.

Owing to rational deep understanding of intrinsic nature of groundwater as common pool and decayable resources, together with, lack of withdrawal control by local states, groundwater depletion has been incredibly under consistent withdrawal. Under these circumstances, attempts have been made by users to maximize withdrawal without considering any limitation on account of lack of reliable withdrawal pattern. In this research, sustainable management of groundwater in south western part of Tehran plain has been evaluated by coupling simulation and optimization models of cooperative and non-cooperative approaches. The MODFLOW model and the Artificial Neural Network model are used for simulation of different scenarios after calibration and validation. Then, a non-linear optimization model is used for optimal allocation of groundwater to different water users. Results indicate that users may gain more benefit in early years of withdrawal in non-cooperative approach. However, this may have an adverse effect which implies a decreasing trend in the benefit in such a way that it reaches null in long term period, as this is reflected by dramatic drawdown of groundwater level and its depletion. In cooperative approach, rate of withdrawal from aquifer is determined by considering renewable water volume, together with, groundwater table drawdown constraint. In this approach, water users may gain less benefit than non-cooperative in early years of withdrawal, however, that will offset in 50 years period by means of more benefit.

In this simulation study, the flow and thermal characteristics of a novel rotary-piston engine, which is a kind of internal combustion engines, were investigated by computational fluid dynamics and the finite volume method. The structure of this engine is different to others, mainly for having 24 cylinders during the motoring mode. As a novel engine, creation of numerical models based on Reynolds average Navier Stokes (RANS) simulation and analysis of various speed engines on the flow and thermal fields during intake and compression strokes are the focus of this work. The results were illustrated in term of the streamline patterns, in-cylinder temperature and pressure profile, swirl ratio (SR), wall heat flux, and turbulent velocity fluctuation. The present study indicates that, the mean pressure, temperature trace, and heat loss from the wall increase when switching to a higher engine speed. The temperature distribution reveals that the maximum temperature is restricted in the center of the combustion chamber near top dead center (TDC). Also, the maximum amount of turbulent velocity and swirl ratio are achieved at the beginning of the intake stroke and near TDC. It is observed that the obtained numerical results are in general agreement with the available experimental data.

The effect of upstream velocity perturbations on the response of a premixed flame was investigated in terms of the flame transfer function dependency on excitation frequency. In this study, the assumption of constant flame speed was extended and the effect of flame speed development was considered; i.e., the flame speed would grow with the time after ignition or with the distance from a flame-holder. In the present study, the kinematics of a conical flame was investigated by linearization of the front tracking equation of flame to uniform and convected fluctuations of the flow velocity and the response was compared with that of a V-shaped flame and the experimental data in the previous studies. The results show that the effect of flame speed development could influence a decreasing gain and increase the phase of the flame response to the uniform velocity oscillations in low and moderate frequencies. Comparing the variations in the gain of flame response upon normalized frequency, show that a conical flame has lower values than the V-flame. In other words, these flames might be less susceptible to combustion instabilities than the V-flames. Furthermore, the variations in phase of the V-flames responses, which show a quasi-linear behavior with normalized frequency, have higher values than the saturated behavior in phase of the conical flame responses. Also, considering that the flame speed development induces an increase in the gain and phase of the conical flame response to the convected velocity oscillations in certain frequencies; because the developed flame front has longer length in comparison to the flame front in constant flame speed model. Therefore, the flame length may be longer than convective wavelength and the heat release would be generated in different points of the flame; consequently the flow oscillations might exert a stronger impact on the unsteady heat release fluctuations.

In this paper, the approximate solution u(x, t) of the one-dimensional fractional Rayleigh-Stokes equation with initial and boundary conditions has been studied. This equation has the main part in defining the dynamic character of some non-Newtonian fluids. Time discretization is performed by new finite methods of the first order, and the RBF method is used for spatial discretization. Additionally, the convergence and stability analysis of the specified methods are considered and proven. Finally, The numerical solutions of some examples were obtained and compared with their analytical solutions that demonstrate the high accuracy and feasibility of the proposed approach.

Nowadays, the connection of the insurance industry to financial markets is very important because of unpredictable fortuitous events. A type of these insurances is the insurance of life which has been considered in developed and developing countries. In this paper, in order to value the life of the policy, we construct a portfolio that includes commitments for assuring life insurance and the zero-coupon bond which pays off one dollar in the maturity to the holder of the bond. Investing in that sum is the amount of a money market account whose return will be determined by the short interest rate. Therefore, we need a model for pricing coupon bonds and to do this we use a model based on the short interest rate and the market price of risk. Finally, we will lead to a model for pricing life insurance under the circumstance of stochastic mortality where the studied model is considered based on an instant sharp ratio. Thus, we will show that this pricing model follows a partial differential equation and we explain the way of solving this equation via collocation method based on the Chebyshev orthogonal basis. The price of life policy is the solution of this PDE.

Due to the importance of gas-phase chemical reaction kinetics in low-emission combustion, stirred tank reactors have been used for decades as an experimental tool to study high- and low-temperature oxidation. A Jet-Stirred Reactor (JSR) setup is valuable to determine the evolution of species mole fractions. For the accuracy of the experimental results, it is important that a JSR is designed such that the concentration field is as homogeneous as possible in order to avoid disturbance of the chemical kinetics. In this work, numerical simulations were performed to investigate the mixing in a JSR chamber. The turbulent structures inside the JSR and the nozzles are captured using Large Eddy Simulations. We conducted numerically a parametric study to evaluate the effects of thermodynamic conditions and geometrical parameters on the mixing characteristics. More specifically, the diameter of the spherical chamber is modified together with the diameter of the nozzles through which fresh gases are fed. The characterization of the gas flow inside a typical spherical JSR layout and results derived by the normalized standard deviation of a tracer mass fraction show that a reduction of the JSR diameter at high pressures improves the homogeneity. Further, we propose a new optimized configuration consisting of six nozzles pointing to the center of the reactor which provides a more uniform composition compared to the standard JSR design.

Many researchers have expanded a new form of fractional diffusion models named the temporal-space fractional Bloch–Torrey equation (TSF-BTE) to evaluate the diffusion structure of human brain tissues as well as prepare additional insight into other studies of cells and tissues and the microenvironment. The main objective of this paper is to present an effective computational method of solving such models in two dimensions. The temporal and spatial directions are based on the Caputo and the Riemann–Liouville fractional derivative, respectively. The presented numerical scheme is derived from the following manners: at first, the semi-discrete is constructed in the temporal based on a quadratic interpolation with accuracy order O(τ2-α)and secondly, the unconditional stability and convergence order are analyzed. For the constructed full-discrete scheme, the spatial derivative terms approximated with the helping of the collocation method based on the Legendre basis. Finally, to illustrate the high precision of the proposed design, we use some test problems. Furthermore, the obtained results are compared with some other techniques under which the suggested methodology is highly accurate and feasible.

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order O(δτ2)is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.