A fractional measles model having monotonic real statistical data for constant transmission rate of the disease

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R0 < 1, which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.publishe

Adaptive step-size approach for Simpson’s-type block methods with time efficiency and order stars.

[EN]In the present scientific literature, block methods to solve stiff and nonlinear initial value problems are in great use due to their better stability features and smaller computational cost. Adaptive step-size versions of such methods, however, are not presented in many research articles, although they are more efficient than their fixed step-size counterparts. Keeping in view the computational efficiency and accuracy obtained with adaptive step-size approaches, two three-step Simpson’s-type block methods based on the second derivative having sixth and eighth order of convergence are considered here. To prove their better performance, the results obtained from different numerical simulations with stiff differential systems under the adaptive step-size approach are compared to the results obtained using fixed step-size. Those problems include the Kaps problem, a Gear’s problem and the Blasius model from fluid dynamics. When compared to an adaptive step-size version of the well-known Lobatto-IIIA methods (implicit in nature), the superiority of the considered block methods is revealed. To fill the gap in previous research works on the block methods, the theory of order stars is also included herein

A Convergent Scheme for Solving Initial Value Problems with Polynomial and Exponential Functions

This paper presents the development of a convergent numerical scheme for the solution of initial value problems of first order ordinary differential equations. The scheme has been derived via the combination of two functions namely, polynomial and exponential functions. The local truncation error , order of convergence, consistency and stability of the proposed scheme have been analyzed in the present study.  The Taylor’s series expansion has been used to derive the principal term of . The Dahlquist’s test equation is used to investigate the linear stability region. It is observed that the newly proposed scheme is fourth order convergent, consistent and conditionally stable with the region of linear stability. Three IVPs of different nature have been solved numerically to check the applicability of a new proposed scheme. The absolute error has been calculated at each mesh point of the integration interval. The numerical results show that the scheme is computationally effective, adequate and compares favorably with exact solutions. The aid of MATLAB version: 9.2.0.538062 (R2017a) has been used to carry out all numerical calculations. Keywords: Local truncation error, Absolute error, stability, consistency, convergence. MSC: 34A12, 45L05, 65L05, 65L20, 65L7

An Efficient Three Step Method For finding the Root Of Non-linear Equation with Accelerated convergence.

We have made an effort to design an accurate numerical strategy to be applied in the vast computing domain of numerical analysis. The purpose of this research is to develop a novel hybrid numerical method for solving a nonlinear equation, That is both quick and computationally cheap, given the demands of today's technological landscape. Sixth-order convergence is demonstrated by combining the classical Newton method, on which this method is largely based, with another two-step third-order iterative process. The effectiveness index for this novel approach is close to 1.4309, and it requires only five evaluations of the functions without a second derivative. The findings are compared to standard practice. The provided technique demonstrates higher performance in terms of computational efficiency, productivity, error estimation, and CPU times. Moreover, its accuracy and performance are tested using a variety of examples from the existing literature. Keywords: efficient scheme, nonlinear application, nonlinear functions, error estimation, computational cost

An Improved L-Stable Scheme for Initial Value Problems Under Variable Step-Size Approach

A non-linear explicit scheme has been studied for autonomous and non-autonomous initial value problems in ordinary differential equations (ODEs). This research proposed fifth order of convergence. The stability region of the scheme is also shown, as is the evolution of the scheme's associated local truncation error. A few numerical experiments showed that the scheme is fit for initial value problems with singular solutions, blowup the ODEs, singularly perturbed and stiff problems. MATLAB R2019a was used for the numerical computations and plotting of results produced by all methods. Keywords. Nonlinear method, local truncation error, L-Stability, Variable step-size, autonomous and non-autonomous

TO DEVELOP EFFICIENT SCHEME FOR SOLVING INITIAL VALUE PROBLEM IN ORDINARY DIFFERENTIAL EQUATION

In this paper, a new scheme of Runge-Kutta (RK) type has been developed while evaluating two slope functions per step and maintaining the third order accuracy of the scheme. Local truncation error is obtained with the help of principal term which is obtained via multi variable Taylor series. It has been shown that the convergence order of the scheme is three and its stability polynomial is also derived. Some numerical examples are taken in order to compare the developed scheme with other existing schemes. It is observed that the developed scheme is better than other selected existing schemes and this comparison has been performed on the basis of slope evaluations per integration step, error analysis and computer time consumed by the scheme under consideration. KEY WORDS: Initial value problems, Runge-Kutta Scheme, Autonomous and non-autonomous differential equations, Zero stability. DOI: 10.7176/MTM/9-8-06 Publication date: August 31st 201

AN IMPROVED ROOT LOCATION METHOD FOR FAST CONVERGENCE OF NON-LINEAR EQUATIONS

In this paper an improved root location method has been suggested for nonlinear equations f(x)=0. The proposed improved root location method is very much effective for solving nonlinear equations and several numerical examples associated with algebraic and transcendental functions are present in this paper to investigate the new method. Throughout the study we have proved that proposed method is cubically convergent.  All the results are executed on MATLAB 16 which has a machine precision of around . Key words: Newton’s method, Iterative method, third order convergent, Root finding methods

Analysis of Accuracy, Stability, Consistency and Convergence of an Explicit Iterative Algorithm

In this work, an analysis is carried out vis-à-vis an explicit iterative algorithm proposed by Qureshi et al (2013) for initial value problems in ordinary differential equations. The algorithm was constructed using the well – known Forward Euler’s method and its variants. Discussion carries with it an investigation for stability, consistency and convergence of the proposed algorithm-properties essential for an iterative algorithm to be of any use. The proposed algorithm is found to be second order accurate, consistent, stable and convergent. The regions and intervals of absolute stability for Forward Euler method and its variants have also been compared with that of the proposed algorithm. Numerical implementations have been carried out using MATLAB version 8.1 (R2013a) in double precision arithmetic. Further, the computation of approximate solutions, absolute and maximum global errors provided in accompanying figures and tables reveal equivalency of the algorithm to other second order algorithms taken from the literature. Keywords: Iterative Algorithm, Ordinary Differential Equations, Accuracy, Consistency, Convergence

Frequency of Capillary leak syndrome in Dengue fever Patients

Abstract Background: Dengue fever is the world’s fastest spreading mosquito borne viral infection. It is prevalent throughout both subtropical and tropical region if the world. The severe form of dengue fever with bleeding manifestations called as dengue hemorrhagic fever. Some of the Dengue fever patients developed capillary leak during critical period of illness. This study aims at determining the frequency of capillary leak syndrome in hospitalized dengue fever patients of tertiary care hospital. Patients and Methods: The study was conducted over a period of one month from 1st October to 30th October 2019 at department of Medicine Federal Government Polyclinic Post Graduate Medical Institute, Islamabad. This cross sectional study comprising 200 consecutive hospitalized (≥14 years of both gender) dengue fever patients. Results: Capillary leak syndrome found in 75 patients of Dengue fever. All of them were Primary Dengue Patients. Both ascites and effusion was present in 31 patients. Ascites only found in 25 patients, Pleural effusion bilateral in 7, Right sided pleural effusion in 11 and Left sided in 1 patient. None of the patient had pericardial effusion. Conclusion: It is concluded that capillary leak syndrome is common in primary dengue fever patients and its early diagnosis helps us in better management during critical phase of illness with better outcome. Key Words: Dengue Fever, Capillary Leak Syndrome, Dengue Shoc

A Modified ODE Solver for Autonomous Initial Value Problems

In this work, modified version of a well-known variant of Euler method, known as the Improved Euler method,is proposed with a view to attain greater accuracy and efficiency. The attention is focused upon performance ofthe proposed method in autonomous initial value problems of ordinary differential equations. Order of accuracyof the proposed modified method is proved to be two using Taylor’s expansion. Numerical experiments areperformed using MS Excel 2010.Keywords: ODE solver, numerical solution, initial value problem