MATLAB: How to get MATLAB to solve a system of equations containing differential and algebraic equations

ode45

I have a system of about 40 equations, about half are ordinary differential equations (that must be solved numerically), the other half are simple algebraic equations, but all rely on variables solved for in the other type. I have been trying to use ode45 to solve the system, but the variables solved for in alegraic equations do not update with time steps, even though the variables used to calculate them are changing. For example, if i have dx1/dt = t, x2 = x1+5, and I use t=0:10, x1 changes with t, but x2 remains constant, even though it should change with x1. Has anybody else had this problem or know a way around it?

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Marc,

Thanks for your help. I've never used a mass matrix before and I have a couple of questions. First, how do I determine the index of my DAE (I thought it was only 1, but MATLAB is telling me it appears to be greater than 1)? Each equation contains only first derivatives, but they also contain other variables and first derivatives (e.g. dx3 = x3 + 4*x1 + 2*dx2). Since after looking at some information on indices of DAEs I'm pretty sure mine is greater than 1 and thus not solvable using MATLAB (or at least ODE15s), do you have any suggestions of programs that actually do this?

Related Solutions

MATLAB: What is the meaning of “This DAE appears to be of index greater than 1” using ODE solvers for solving Differential Algebraic Equations (DAE) in MATLAB 7.4 (R2007a)

The ODE15S and ODE23T solvers can solve DAEs of the form:

M(t,y)*y'=f(t,y)

where M(t,y) is singular. A singular M(t,y) means that you are trying to solve a Differential Algebraic Equation (DAE).

If so, the DAE must be of index 1.

Rewriting in semi-explicit form, the DAE is in semi-explicit form:

u' = f(t,u,v) (1a)

0 = g(t,u,v) (1b)

The system is of index 1 if the matrix of partial derivatives of the algebraic system g with respect to the algebraic variables (i.e. dg/dv) is non-singular.

Rewriting this in Mass Matrix form:

M(t,y)*y’=F(t,y)

the index 1 condition is satisfied when the matrix (M+lambda*dF/dy) is nonsingular, for all non-zero lambda.

An example of the use of ODE15s used in this type of Differential Algebraic Equation may be found

at the following link:

https://www.mathworks.com/help/releases/R2007a/techdoc/index.html?/help/releases/R2007a/techdoc/ref/ode23.html

MATLAB: Can an ODE be solved with an implicit form in MATLAB

The general form f(x',x,t) = 0 is a differential algebraic equation (DAE). Implicit ODEs are a special case of DAE's with no algebraic equations. The function ODE15I available in MATLAB version 7.0 (R14) and above can solve fully implicit ODEs and DAEs of index 1.

For previous versions of MATLAB, you may be able to use the function ODE15S as described in the attached document, 'bvp_paper.pdf'.

This paper has many examples of solving DAEs, in addition to the HB1DAE and AMP1DAE examples included in MATLAB 5.3 (R11). Additionally, the following paper may also be helpful:

 <http://audiophile.tam.cornell.edu/~als93/SR97.pdf>

For a more complete list of the types of integration available in MATLAB, see the Related Solution listed at the bottom of the page.

You may want to look at "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations" by K.E. Brenan, S.L. Campbell, and L.R. Petzold for more information about DAEs.

If you have a DAE with index greater than 1, MATLAB’s ODE solvers cannot solve it directly. However, you can apply index reduction on the DAE to reduce it to a DAE with an index of 1 and use ODE15S on that reduced DAE.

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Related Solutions

MATLAB: What is the meaning of “This DAE appears to be of index greater than 1” using ODE solvers for solving Differential Algebraic Equations (DAE) in MATLAB 7.4 (R2007a)

MATLAB: Can an ODE be solved with an implicit form in MATLAB

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