Comparing and Contrast Different Numerical Approaching to the IVP and BVP Problems

This paper focuses on the comparison of various numerical solutions for Initial Value Problem(IVP) and Boundary Value Problem(BVP) in differential equations. The practical problem is solved by comparing each numerical method solution. In the part of IVP problem, this paper established an Susceptible Infectious Recovered Model (SIR) to measure the disease situation in India during the new coronavirus period of 2020-2021, and calculated the corresponding parameters through machine learning genetic algorithm(GA). The differential equations are solved by using first and second-order Taylor expansions(TM), Euler(EM), improved Euler(ME), Runge-Kutta 2(RK2), optional Runge-Kutta(RK2*), and Runge-Kutta 4(RK4) methods. Then, based on the model, the situation of the novel coronavirus epidemic in the next year is predicted; In the part of BVP problem, this paper establishes a linear spring model(LSM) to measure the displacement of elastomer under the force distribution, that is, the force density, by using a simulation algorithm(SA) to expand a small amount of measurement data, and by using machine learning algorithm linear regression(LR) to calculate the spring stiffness coefficient. Numerical solutions are calculated by using forward, center and backward finite difference method(FDM), Galerkin method in variational method(VGLM), collocation method(CM) and Galerkin method in finite element method(FEM), and mean square error(MSE) is used to evaluate the model. Finally, the analytical solutions especially Laplace Transformation(LT) of the two models are given to compare the numerical solutions.