The initial value problem model of differential equation is a common mathematical model in mathematical modeling competition. For some simple and typical differential equation models, such as linear equations and some special first-order nonlinear equations, we can try to find their analytical solutions, and the theoretical results can be used. However, the initial value problem model of ordinary differential equations encountered in mathematical modeling is usually very difficult, and even its analytical solution can not be found at all, only its approximate solution can be found.
Therefore, in order to get the numerical solution quickly, it is of great significance to study its numerical method. In view of this, this paper summarizes and studies the existing numerical solutions of the initial value problem model of ordinary differential equations. This paper mainly discusses some commonly used numerical solutions: Euler method, backward Euler method, 0- 1 method, improved Euler method, Runge-Kutta method, Addums extrapolation formula and interpolation formula.
By tracing the history of numerical solutions and using examples of numerical solutions, the advantages and disadvantages of various numerical solutions are summarized, which provides reference for seeking reasonable algorithms to meet various accuracy requirements in mathematical modeling and mathematical modeling competitions.
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Differential equation refers to the relationship with unknown function and its derivative. Solving differential equations means finding unknown functions.
Differential equations are developed by calculus. Newton and Leibniz, the founders of calculus, both dealt with problems related to differential equations in their works.
Differential equations are widely used and can solve many problems related to derivatives. Many kinematics and dynamics problems involving variable forces in physics, such as falling bodies with air resistance as speed function, can be solved by differential equations. In addition, differential equations have applications in chemistry, engineering, economics and demography.
The research on differential equations in the field of mathematics focuses on several different aspects, but most of them are related to the solutions of differential equations. Only a few simple differential equations can be solved analytically.
However, even if the analytical solution is not found, some properties of the solution can still be confirmed. When the analytical solution cannot be obtained, the numerical solution can be found by means of numerical analysis and computer. ? Dynamic system theory emphasizes the quantitative analysis of differential equation system, and many numerical methods can calculate the numerical solution of differential equation with certain accuracy.