Matlab solves differential algebraic equations (DAE)

A classmate asked a question at the weekend , He asked to solve a system of differential equations , But there is also a system of linear equations between the variables of the differential equation , This is a typical differential algebraic equation ,Matlab There is a special solution ,

What is differential algebraic equation ？

Differential algebraic equations are a class of differential equations , One or more of the dependent variable derivatives do not appear in the equation . Variables appearing in the equation that do not contain their derivatives are called algebraic variables , The existence of algebraic variables means that these equations cannot be written in explicit form y′=f(t,y).

ode15s and ode23t The solver can use singular mass matrix M(t,y)y′=f(t,y) To solve the differential index as 1 Linear implicit problem of , Include the following form of semi explicit DAE

y′0=f(t,y,z)

0 =g(t,y,z) 

In this form , Because there are one or more zero values in the main diagonal , Therefore, the existence of algebraic variables will produce a singular mass matrix .

By default , The solver will automatically check the singularity of the quality matrix , To detect DAE Equations . If you know the singularity in advance , Then you can put odeset Of MassSingular Option set to 'yes'. about DAE, You can also use odeset Of InitialSlope Property provides for the solver y′(0) Initial condition estimate of .

for instance

among x1(0)=0.8;x2(0)=x3(0)=0.1;

1） Equation writing DAE form

2） Programming to solve

%% 
clc
clear
close all
odefun = @(t,x)[-0.2*x(1)+x(2)*x(3)+0.3*x(1)*x(2);
                2*x(1)*x(2)-5*x(2)*x(3)-2*x(2)^2;
                x(1)+x(2)+x(3)-1]; % Differential equations 
M = [1 0 0;0 1 0;0 0 0]; %  Mass matrix 
options=odeset('mass',M); %  Definition mass attribute 
x0=[0.8;0.1;0.1];
[t,x]=ode15s(odefun,[0 10],x0,options);
figure
plot(t,x(:,1),t,x(:,2),t,x(:,3))
grid on
legend('x1','x2','x3')

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