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.

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.

A Stochastic Numerical Analysis for Computer Virus Model with Vertical Transmission Over the Internet

We are presenting the numerical analysis for stochastic SLBR model of computer virus over the internet in this manuscript.We are going to present the results of stochastic and deterministic computer virus model.Outcomes of the threshold number C∗hold in stochastic computer virus model.If C∗<1 then in such a condition virus controlled in the computer population while C∗>1 shows virus spread in the computer population.Unfortunately,stochastic numerical techniques fail to cope with large step sizes of time.The suggested structure of the stochastic non-standard finite difference scheme(SNSFD)maintains all diverse characteristics such as dynamical consistency,bounded-ness and positivity as well-defined by Mickens.On this basis,we can suggest a collection of plans for eradicating viruses spreading across the internet effectively.

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 point𝑒𝑒∈𝑋𝑋that satisfying T e=e.One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems.Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Berinde(2004b);Agarwal,O’Regan and Sahu(2007)].The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can.We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al.[Sintunavarat and Pitea(2016);Agarwal,O’Regan and Sahu(2007);Mann(1953);Ishikawa(1974)].We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Abbas and Nazir(2014);Agarwal,O’Regan and Sahu(2007)]schemes.Moreover,we introduce a problem raised from Newton’s law of cooling as an application of our new iteration scheme.Also,we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.