We proposed a continuous blocks method for the solution of second order initial value problems with constant step size in this paper. The method was developed by interpolation and collocation of power series approximate solution to generate a continuous linear multistep method;this is evaluated for the independent solution to give a continuous block method which is evaluated at selected grid point to give discrete block method. The basic properties of the method were investigated and was found to be zero stable, consistent and convergent. The efficiency of the method was tested on some numerical examples and found to give better approximation than the existing methods.
There are several classifications of linear Integral Equations. Some of them include; Voltera Integral Equations, Fredholm Linear Integral Equations, Fredholm-Voltera Integrodifferential. In the past, solutions of higher-order Fredholm-Volterra Integrodifferential Equations [FVIE] have been presented. However, this work uses a computational techniques premised on the third kind Chebyshev polynomials method. The performance of the results for distinctive degrees of approximation (M) of the trial solution is cautiously studied and comparisons have been additionally made between the approximate/estimated and exact/definite solution at different intervals of the problems under consideration. Modelled Problems have been provided to illustrate the performance and relevance of the techniques. However, it turned out that as M increases, the outcomes received after every iteration get closer to the exact solution in all of the problems considered. The results of the experiments are therefore visible from the tables of errors and the graphical representation presented in this work.
In this work, a collocation technique is used to determine the computational solution to fractional order Fredholm-Volterra integro-differential equations with boundary conditions using Caputo sense. We obtained the linear integral form of the problem and transformed it into a system of linear algebraic equations using standard collocation points. The matrix inversion approach is adopted to solve the algebraic equation and substituted it into the approximate solution. We established the uniqueness and convergence of the method and some modelled numerical examples are provided to demonstrate the method’s correctness and efficiency. It is observed that the results obtained by our new method are accurate and performed better than the results obtained in the literature. The study will be useful to engineers and scientists. It is advantageous because it addresses the difficulty in tackling fractional order Fredholm-Volterra integro-differential problems using a simple collocation strategy. The approach has the advantage of being more accurate and reducing computer running time.
In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent and zero stable hence convergent. The new method was tested on real life problems namely: SIR model, Growth model and Mixture Model. The results were found to compete favorably with the existing methods in terms of accuracy and error bound.
Ebola virus (EBV) disease is a globally acknowledged public health emergency, endemic in the west and equatorial Africa. To understand the epidemiology especially the dynamic pattern of EBV disease, we analyse the EBV case notification data for confirmed cases and reported deaths of the ongoing outbreak in the Democratic Republic of Congo (DRC) between 2018 and 2019, and examined the impact of reported violence on the spread of the virus. Using fully Bayesian geo‐statistical analysis through stochastic partial differential equations (SPDE) allows us to quantify the spatial patterns at every point of the spatial domain. Parameter estimation was based on the integrated nested Laplace approximation (INLA). Our findings revealed a positive association between violent events in the affected areas and the reported EBV cases (posterior mean = 0.024, 95% CI: 0.005, 0.045) and deaths (posterior mean = 0.022, 95% CI: 0.005, 0.041). Translating to an increase of 2.4% and 2.2% in the relative risks of EBV cases and deaths associated with a unit increase in violent events (one additional Ebola case is associated with an average of 45 violent events). We also observed clusters of EBV cases and deaths spread to neighbouring locations in similar manners. Findings from the study are therefore useful for hot spot identification, location‐specific disease surveillance and intervention.
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