reduceDifferentialOrder

Reduce system of higher-order differential equations to equivalent system of first-order differential equations

Syntax

[newEqs,newVars] = reduceDifferentialOrder(eqs,vars)

[newEqs,newVars,R] = reduceDifferentialOrder(eqs,vars)

Description

[newEqs,newVars] = reduceDifferentialOrder(eqs,vars) rewrites a system of higher-order differential equations eqs as a system of first-order differential equations newEqs by substituting derivatives in eqs with new variables. Here, newVars consists of the original variables vars augmented with these new variables.

[newEqs,newVars,R] = reduceDifferentialOrder(eqs,vars) returns the matrix R that expresses the new variables in newVars as derivatives of the original variables vars.

Examples

Reduce Differential Order of DAE System

Reduce a system containing higher-order DAEs to a system containing only first-order DAEs.

Create the system of differential equations, which includes a second-order expression. Here, x(t) and y(t) are the state variables of the system, and c1 and c2 are parameters. 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 x(t) y(t) c1 c2
eqs = [diff(x(t),t,t) + sin(x(t)) + y(t) == c1*cos(t),...
       diff(y(t),t) == c2*x(t)];
vars = [x(t),y(t)];

Rewrite this system so that all equations become first-order differential equations. The reduceDifferentialOrder function replaces the higher-order DAE by first-order expressions by introducing the new variable Dxt(t). It also represents all equations as symbolic expressions.

[newEqs,newVars] = reduceDifferentialOrder(eqs,vars)

newEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }\mathrm{Dxt}\left(t\right)+\sin \left(x\left(t\right)\right)+y\left(t\right)-c_1\hspace{0.17em}\cos \left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }y\left(t\right)-c_2\hspace{0.17em}x\left(t\right)\\ \mathrm{Dxt}\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\end{array}\right)$$

newVars = 
$$\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\\ \mathrm{Dxt}\left(t\right)\end{array}\right)$$

Show Relations Between Generated and Original Variables

Reduce a system containing a second- and a third-order expression to a system containing only first-order DAEs. In addition, return a matrix that expresses the variables generated by reduceDifferentialOrder via the original variables of this system.

Create a system of differential equations, which includes a second- and a third-order expression. Here, x(t) and y(t) are the state variables of the system. 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 x(t) y(t) f(t)
eqs = [diff(x(t),t,t) == diff(f(t),t,t,t), diff(y(t),t,t,t) == diff(f(t),t,t)];
vars = [x(t),y(t)];

Call reduceDifferentialOrder with three output arguments. This syntax returns matrix R with two columns: the first column contains the new variables, and the second column expresses the new variables as derivatives of the original variables, x(t) and y(t).

[newEqs,newVars,R] = reduceDifferentialOrder(eqs,vars)

newEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }\mathrm{Dxt}\left(t\right)-\frac{{\partial }^3}{\partial t^3}\mathrm{ }f\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }\mathrm{Dytt}\left(t\right)-\frac{{\partial }^2}{\partial t^2}\mathrm{ }f\left(t\right)\\ \mathrm{Dxt}\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\\ \mathrm{Dyt}\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)\\ \mathrm{Dytt}\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }\mathrm{Dyt}\left(t\right)\end{array}\right)$$

newVars = 
$$\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\\ \mathrm{Dxt}\left(t\right)\\ \mathrm{Dyt}\left(t\right)\\ \mathrm{Dytt}\left(t\right)\end{array}\right)$$

R = 
$$\left(\begin{array}{cc}\mathrm{Dxt}\left(t\right)& \frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\\ \mathrm{Dyt}\left(t\right)& \frac{\partial }{\partial t}\mathrm{ }y\left(t\right)\\ \mathrm{Dytt}\left(t\right)& \frac{{\partial }^2}{\partial t^2}\mathrm{ }y\left(t\right)\end{array}\right)$$

Input Arguments

eqs — System containing higher-order differential equations

vector of symbolic equations | vector of symbolic expressions

System containing higher-order differential equations, specified as a vector of symbolic equations or expressions.

vars — Variables of original differential equations

vector of symbolic functions | vector of symbolic function calls

Variables of original differential equations, specified as a vector of symbolic functions, or function calls, such as x(t).

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

Output Arguments

newEqs — System of first-order differential equations

column vector of symbolic expressions

System of first-order differential equations, returned as a column vector of symbolic expressions.

newVars — Extended set of variables

column vector of symbolic function calls

Extended set of variables, returned as a column vector of symbolic function calls. This vector includes the original state variables vars followed by the generated variables that replace the higher-order derivatives in eqs.

R — Relations between new and original variables

symbolic matrix

Relations between new and original variables, returned as a symbolic matrix with two columns. The first column contains the new variables newVars. The second column contains their definition as derivatives of the original variables vars.

Version History

Introduced in R2014b