Four-Step Iteration Scheme to Approximate Fixed Point for Weak Contractions

Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems. It is known that many problems in applied sciences and engineering can be formulated as functional equations. Such equations can be transferred to fixed point theorems in an easy manner. Moreover, we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations. Let X be a non-empty set. A fixed point for a self-mapping T on X is a point     

An Effective Numerical Method for the Solution of a Stochastic Coronavirus (2019-nCovid) Pandemic Model

Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model. The comparison of the results of deterministic and stochastic models is also presented. Our proposed efficient computational method well preserves the essential properties of the model. Comparison and convergence analyses of the method are presented.     

Chebyshev approximation problem of multidimensional IIR digital filters in the frequency domain

This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. Chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the Chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.

     

Application of Decision Support System for Sustainable Management of Water Resources in the Azraq Basin—Jordan

Abstract

Water scarcity in Jordan is a significant constraint to development, with limited available water and financial resources. As population and economic activity increase, it will be necessary to implement national strategies that seek to balance the present needs and those of future generations. Multiple variables associated with agricultural crops, industries, and the impact of climate change, were incorporated into a Decision Support System (DSS). The DSS utilized Analytical Hierarchy Process (AHP), which resulted in the prioritization of sustainable water policies for management in the Azraq Basin. The inputs to the DSS were generated through application of Modflow (groundwater), stochastic, and Penman Montieth models and through calculations of water productivity for agricultural and industrial sectors. The results of the DSS make recommendations as to how to enhance long-term sustainability of water resources in Azraq, while allowing for water utilization and economic growth. It is recommended for future planning that further research of the impacts to water resources must be conducted at local and national levels and linked to regional and global climate change prediction. It can be concluded that the DSS tool and AHP are potentially positive contributions to the process of decision- making for selection and ranking of alternatives and policies and for help in solving problems that include conflicting criteria.     

Effect of Khirbet As-Samra treated effluent on the quality of irrigation water in the Central Jordan Valley

Khirbet As-Samra (KA) treated wastewater is being used in irrigation in the Central Jordan Valley. The treated water is collected in King Talal Dam (KTD) and then mixed with King Abdullah Canal (KAC) water, which is diverted from the Yarmouk River for further use in agriculture. The treated water has adversely affected the water quality of Yarmouk River. Comparison of the results of water quality tests with guidelines for water to be used for irrigation, salt tolerance of agricultural crops and the influence of water quality on the potential clogging of drip irrigation systems reveals that treated effluent from KA can be used for irrigation with restrictions. Moreover, concentrations of all trace elements were found to be low and within guidelines for irrigation water. Clogging of drip emitters is expected due to high calcium and magnesium contents, besides the high bacterial counts and nutrients that promote algal growth.     

Effects of furfural activated crumb rubber on the properties of rubberized asphalt

The use of furfural (C₅H₄O₂) as an activation agent has been suggested as a method to improve the rheological properties of asphalt binders due to its compatibility with crumb rubber. This study uses five different crude sources and both ambient and cryogenic produced crumb rubber modifiers (CRM). The rheological properties for furfural activated and conventional CRM binders were evaluated using the dynamic shear rheometer (DSR) and the gel permeation chromatography (GPC). The results indicated that furfural activation has variable effects on the properties of the CRM binder. However, the most pronounced effect is shown in the storage stability improvement which will have an effect on the storage of CRM binder. Also, the activation caused a reduction in the ratio of the small molecular size distribution which is considered an improvement in the binder properties.     

N-Dimensional Phase Approximation in the L∞-Norm

Thephase functions of N-dimensional (N-D) digital all-pass filtersare investigated to approximate a prescribed phase response ina frequency region. The set of phase functions of the all-passfilters have common properties with some nonlinear approximatingfunctions. This similarity answers the question of characterizationof minimal approximation in the set of phase functions. The optimalapproximation is characterized by known theorems of TschebycheffApproximation Theory. Among the main tools of the theory, theGlobal and Local Kolmogoroff Criteria, are shown to give necessaryand sufficient conditions for best approximations in the phasefunctions of N-D all-pass filters. Moreover, this best approximationin the phase functions is shown to be a global minimum. The approximationon discrete point sets (H-sets) in a compact multidimensionaldomain is studied. Optimal N-dimensional approximation is notunique, an inherent property of functions of several variables.     

Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way. In it, we introduce a method called residual correction procedure, to correct some previous approximate solutions for such systems. We study the error analysis of our given method. We first introduce a new result to approximate the absolute solution by using the residual correction procedure. Second, we introduce a new result to get appropriate bound for the absolute error. The collocation method is used and the collocation points can be found by applying Chebyshev roots. Both techniques are explained briefly with illustrative examples to demonstrate the applicability, efficiency and accuracy of the techniques. By using a small number of Bernstein polynomials and correction procedure we achieve some significant results. We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method, continuous genetic algorithm, rational homotopy perturbation method and adomian decomposition method.