1.Description
The concept of numerical methods, in brief, is that when analytical solutions cannot be obtained or cannot be computed within a limited time, numerical methods are used to iteratively compute a series of data that satisfy the equation. Such results are called numerical solutions, and the methods used to obtain them are referred to as numerical methods. Currently, many algorithms have been developed for numerical methods, which have wide applications.
To use MATLAB for numerical solutions of differential equations, specific syntax must be provided. The formulation of the differential equation can be directly seen in the program example or refer to video 14.6 (recommended).
2.Understanding the Function
ode45 is used to solve the numerical solutions of differential equations. This function employs the Runge-Kutta algorithm principle for computation.
3.Programming Example
To solve the differential equation y”-(1-y^2)y’+2y=0, with initial conditions y(0) = 2, y'(0) = 0;
Program:
% Copy the following program into a script file and run it (very important)
tspan=[0 5];
y0=[2 0];
opts=odeset(‘reltol’,1e-2,’abstol’,1e-4);
[x,y]=ode45(@ode,tspan,y0,opts);
x1=1:5;
y1=interp1(x,y,x1);
function dy=ode(x,y)
dy=zeros(2,1);
dy(1)=y(2);
dy(2)=(1-y(1)^2)*y(2)-2*y(1);
end
Results:
x1 =
1 2 3 4 5
y1 =
0.9347 -1.8987
-1.7896 -1.7226
-1.4039 1.4465
1.0947 3.2556
1.7459 -1.0463
Explanation of Results: The variable x1 corresponds to the coordinate points, and the variable y1 contains two columns that correspond to the y values and y’ values at the x1 points.
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