massMatrixForm

Extract mass matrix and right side of semilinear system of differential algebraic equations

Syntax

[M,F] = massMatrixForm(eqs,vars)

Description

[M,F] = massMatrixForm(eqs,vars) returns the mass matrix M and the right side of equations F of a semilinear system of first-order differential algebraic equations (DAEs). Algebraic equations in eqs that do not contain any derivatives of the variables in vars correspond to empty rows of the mass matrix M.

The mass matrix M and the right side of equations F refer to this form.

$$M\left(t,x\left(t\right)\right)\dot{x}\left(t\right)=F\left(t,x\left(t\right)\right).$$

Examples

Convert DAE System to Mass Matrix Form

Convert a semilinear system of differential algebraic equations to mass matrix form.

Create the following system of differential algebraic equations. Here, the functions x1(t) and x2(t) represent state variables of the system. The system also contains symbolic parameters r and m, and the function f(t,x1,x2). Specify the equations and variables as two symbolic vectors: equations as a vector of symbolic equations, and variables as a vector of symbolic function calls.

syms x1(t) x2(t) f(t,x1,x2) r m
eqs = [m*x2(t)*diff(x1(t), t) + m*t*diff(x2(t), t) == f(t, x1(t), x2(t)),...
       x1(t)^2 + x2(t)^2 == r^2];
vars = [x1(t),x2(t)];

Find the mass matrix form of this system.

[M,F] = massMatrixForm(eqs, vars)

M = 
$$\left(\begin{array}{cc}m\hspace{0.17em}{x}_2\left(t\right)& m\hspace{0.17em}t\\ 0& 0\end{array}\right)$$

F = 
$$\left(\begin{array}{c}f\left(t,x_1\left(t\right),x_2\left(t\right)\right)\\ r^2-{x_1\left(t\right)}^2-{x_2\left(t\right)}^2\end{array}\right)$$

Solve this system using the numerical solver ode15s. Before you use ode15s, assign the following values to symbolic parameters of the system: m = 100, r = 1, f(t,x1,x2) = t + x1*x2. Also, replace the state variables x1(t), x2(t) by variables Y1, Y2 acceptable by matlabFunction.

syms Y1 Y2
M = subs(M, [vars, m, r, f], [Y1, Y2, 100, 1, @(t,x1,x2) t + x1*x2]);
F = subs(F, [vars, m, r, f], [Y1, Y2, 100, 1, @(t,x1,x2) t + x1*x2]);

Create the following function handles MM and FF. You can use these function handles as input arguments for odeset and ode15s. These functions require state variables to be specified as column vectors.

MM = matlabFunction(M, 'vars', {t, [Y1; Y2]});
FF = matlabFunction(F, 'vars', {t, [Y1; Y2]});

Solve the system using ode15s.

opt = odeset('Mass', MM, 'InitialSlope', [0.005;0]);
ode15s(FF, [0,1], [0.5; 0.5*sqrt(3)], opt)

Copyright 2014 The MathWorks, Inc

Input Arguments

eqs — System of semilinear first-order DAEs

vector of symbolic equations | vector of symbolic expressions

System of semilinear first-order DAEs, specified as a vector of symbolic equations or expressions.

vars — State variables

vector of symbolic functions | vector of symbolic function calls

State variables, specified as a vector of symbolic functions or function calls, such as x(t).

Example: [x(t),y(t)] or [x(t);y(t)]

Output Arguments

M — Mass matrix

symbolic matrix

Mass matrix of the system, returned as a symbolic matrix. The number of rows is the number of equations in eqs, and the number of columns is the number of variables in vars.

F — Right sides of equations

symbolic column vector of symbolic expressions

Right sides of equations, returned as a column vector of symbolic expressions. The number of elements in this vector is equal to the number of equations in eqs.