An Efficient Numerical Scheme for Biological Models in the Frame of Bernoulli Wavelets

This article considers three types of biological systems:the dengue fever disease model,the COVID-19 virus model,and the transmission of Tuberculosis model.The new technique of creating the integration matrix for the Bernoulli wavelets is applied.Also,the novel method proposed in this paper is called the Bernoulli wavelet collocation scheme(BWCM).All three models are in the form system of coupled ordinary differential equations without an exact solution.These systems are converted into a system of algebraic equations using the Bernoulli wavelet collocation scheme.The numerical wave distributions of these governing models are obtained by solving the algebraic equations via the Newton-Raphson method.The results obtained from the developed strategy are compared to several schemes such as the Runge Kutta method,and ND solver in mathematical software.The convergence analyses are discussed through theorems.The newly implemented Bernoulli wavelet method improves the accuracy and converges when it is compared with the existing methods in the literature.

Transmission and Reflection of Water-Wave on a Floating Ship in Vast Oceans

In this paper,we study the water-wave flow under a floating body of an incident wave in a fluid.This model simulates the phenomenon of waves abording a floating ship in a vast ocean.The same model,also simulates the phenomenon of fluid-structure interaction of a large ice sheet in waves.According to this method.We divide the region of the problem into three subregions.Solutions,satisfying the equation in the fluid mass and a part of the boundary conditions in each subregion,are given.We obtain such solutions as infinite series including unknown coefficients.We consider a limited number only of the coefficients by truncating the infinite series and satisfy the remaining boundary conditions approximately.Numerical experiments show that the results are acceptable.Tables are given along with the graph of the system of the resulting streamlines and the dynamical pressure acting on the obstacle.The drawn system of streamlines shows the correctness of the solution and the pressure is maximum on the side facing the upstream extremity of the channel.The same procedure can be adequately applied for problems with more complicated geometry and other phenomenon can thus be simulated.

A Compressed Sensing Approach for Partial Differential Equations with Random Input Data

In this paper,a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented.As with Monte-Carlo and stochastic collocation methods,only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest.The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory.The technique allows the retrieval of the modes that contribute most significantly to the approximation of the solution using a minimal amount of information.The generation of this information,via many solver calls,is almost always the bottle-neck of an uncertainty quantification procedure.If the stochastic model output has a reasonably compressible representation in the retained approximation basis,the proposedmethod makes the best use of the available information and retrieves the dominantmodes.Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of the new method,requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos(Smolyak scheme)to achieve comparable approximation accuracy.