Date of Award
8-2024
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
School
Mathematics and Natural Sciences
Committee Chair
Haiyan Tian
Committee Chair School
Mathematics and Natural Sciences
Committee Member 2
Jiu Ding
Committee Member 2 School
Mathematics and Natural Sciences
Committee Member 3
James Lambers
Committee Member 3 School
Mathematics and Natural Sciences
Committee Member 4
Huiqing Zhu
Committee Member 4 School
Mathematics and Natural Sciences
Abstract
This research aims to solve nonlinear Poisson-type partial differential equations (PDEs) by the approach of the homotopy analysis method (HAM) incorporated with approximate particular solutions (APS) using Delta-shaped basis (DSB) approximations.
With the inclusion of the h auxiliary parameters, we tackle nonlinear problems by studying the mathematical characteristics of the h curve. This is to ensure the numerical convergence of the HAM.
In the solution process, we use the homotopy analysis method to convert a nonlinear PDE into linear inhomogeneous PDEs, which are solved using the method of approximate particular solutions with DSB.
A proper value of the h is used in each iteration. We solve the h-parameter dependent equations using the method of approximate particular solutions with DSB approximation. We use the analytical particular solutions of the DSB functions and the fundamental solutions associated with the differential operator to solve h-parameter dependent equations.
This is an accurate, efficient, and meshless method that uses the domain information and boundary conditions. The effectiveness of the delta-shaped basis approximation is demonstrated by allowing ghost points in the collocation approach for solving a Poisson equation.
Numerical examples for nonlinear Poisson-type PDEs are solved on various irregular shaped domains. The maximum error and mean square root error with an increasing number of iterations illustrate the accuracy and validity of the proposed method.
Copyright
2024, Cyril Ocloo
Recommended Citation
Ocloo, Cyril, "Delta-shaped Approximation Based Homotopy Analysis Method for Nonlinear Poisson-Type Partial Differential Equations" (2024). Dissertations. 2277.
https://aquila.usm.edu/dissertations/2277
Included in
Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons