China
Africa
Explore chapters and articles related to this topic
The Dual Porosity Model
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Siphokazi Simnikiwe Manundu, Abdon Atangana
Mathematical models are largely used to depict real-world problems; however, to depict this problem, one needs to obtain a solution. The idea is to have exact solutions by using analytical methods, though the use of these methods is limited as some equations are so complex that they cannot be solved analytically. In these cases, researchers relied on numerical methods. The literature contains several numerical schemes, one of the most used being multi-step methods, for example, the Adams–Bashforth numerical scheme, the predictor-corrector method, or the Adams–Moulton method. These methods have been developed to solve mostly differential equations with classical derivatives. An extension has been made to solving fractional differential equations. For example, there is a version of Adams–Bashforth that has been developed in the case of the Caputo fractional derivative and its associate predictor-corrector. More versions have also been developed. In the case of non-singular kernels many new numerical schemes have been developed, for example by Batogna and Atangana (2019) using the Laplace transform for fractional differential operators. Toufik and Atangana (2017) suggested an alternative scheme for solving nonlinear fractional differential equations with Atangana–Baleanu fractional derivatives. Using the Newton polynomial interpolation, Atangana and Seda (2020) developed a numerical scheme that is able to handle fractional and fractal-fractional differential equations. This section will therefore present the numerical solution using the Newton polynomial scheme.
Deformable Models and Image Segmentation
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
Ahmed ElTanboly, Ali Mahmoud, Ahmed Shalaby, Magdi El-Azab, Mohammed Ghazal, Robert Keynton, Ayman El-Baz, Jasjit S. Suri
We summarize the idea of (HJ ENO) method as follows. We use the smoothest possible polynomial interpolation (with Newton polynomial interpolation [29]) to find φ and then differentiate to get φx. Define Di0φ=φi at each grid node located at xi. The first divided differences of φ are defined midway between grid nodes as Di+1/21φ=Di+10φ−Di0φΔx
Deterministic and fractional analysis of a newly developed dengue epidemic model
Published in Waves in Random and Complex Media, 2023
Rahat Zarin, Mohabat Khan, Amir Khan, Abdullahi Yusuf
Newton polynomial interpolation is a method of constructing a polynomial that passes through a set of given data points. It is named after Sir Isaac Newton, who developed the method in the 17th century. The basic idea behind this method is to create a polynomial that is as close as possible to the given data points. The polynomial is constructed by using the difference between the function and the polynomial at the given data points. The Newton polynomial is a unique polynomial that passes through all the given data points and can be computed efficiently using the divided difference method.