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Differential-Algebraic Equations in Maple 9.5\251 Maplesoft, a division of Waterloo Maple Inc., 2004

Differential-Algebraic Equations (mixed systems of differential and algebraic equations) are crucial in the modeling of complex physical phenomena. The usual strategy for solving them is to differentiate the algebraic equations until one obtains a pure ODE system that can be solved by classical algorithms. However, this often results in stiff systems that are very hard to solve. Maple 9.5 provides three powerful new solvers for solving DAEs efficiently.

restart: interface(warnlevel=0): interface(displayprecision=4): with( VectorCalculus ): with( VariationalCalculus ): with( PDEtools ): with( plots ): with( plottools ): declare( x(t), y(t), h(t), prime=t, quiet ):interface(warnlevel=1):

Here is an 3-variable problem from physical chemistry on which Maple's default numeric ODE solver fails but the new DAE solvers succeed. In this case, the solution is constrained to lie on the plane NiMvLCgtSSJhRzYiNiNJInRHRiciIiItSSJiR0YnRihGKi1JImNHRidGKEYqRio=.sys := { diff( a(t), t ) = -a(t)/10 + 10000*b(t)*c(t), diff( b(t), t ) = a(t)/10 - 10000*b(t)*c(t) - 10000000*b(t)^2, a(t) + b(t) + c(t) = 1, a(0)=1, b(0) = 0 };Default numeric solver fails on the desired range. dsolve( sys, range=0..40000, numeric ):We now try one of the DAE solvers, the aptly named Modified Extended Backward-Differentiation Implicit method (mebdfi).DAESolution := dsolve( sys, numeric, method=mebdfi );odeplot( DAESolution, [t, a(t)], 0..40000 );

We model the behavior of a pendulum with a bob of mass NiNJIm1HNiI= and a string of length NiMlJHJob0c= that is fixed at the origin. The position of the bob is given by functions NiMtJSJ4RzYjJSJ0Rw== and y(t), constrained by the equationNiMvLCYqJC0lInhHNiNGJyIiIyIiIiokLSUieUc2IyUidEdGKUYqKiQlJHJob0dGKQ== We formulate the problem as an exercise in constrained Lagrangian dynamics that results in a DAE system. We then solve the system numerically using the DAE solver in Maple 9.5.Write the constraint equation asConstraint := x(t)^2 + y(t)^2 - rho^2 = 0;and the position vector for the bob asR := Vector([x(t), y(t)]);

The velocity vector for the bob isv := diff( R, t );so that the kinetic energy of the bob isT := m/2*DotProduct( v, v );The potential energy of the bob is taken asV := m*g*y(t);The Lagrangian is thenL := T - V;According to the theory of constrained minimization for functionals, we write the modified LagrangianML := L - h(t)*lhs( Constraint );where NiMtJSJoRzYjJSJ0Rw== is a Lagrange multiplier introduced because of the algebraic constraint. (The functional to be stationarized is the action integral, subject to the constraint NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoKiQlJHJob0dGJw==. Thus, a new functional is formed, and its integrand is the modified Lagrangian given above.)

The Euler-Lagrange equations are obtained with Maple's EulerLagrange command in the VariationalCalculus package. The Euler-Lagrange equations NiMsJiYlIkxHNiMlInFHIiIiKiYlImRHRiglI2R0RyEiIkYs NiMvJiUiTEc2IyklInFHJSIuRyIiIQ== becomeQ := remove(has, EulerLagrange( ML, t, [x(t), y(t), h(t)]), K ): (EQy, Q1) := selectremove( has, Q, g ): (EQx, EQh) := selectremove( has, Q1, h ): ELx := isolate( EQx[1], diff(x(t), t, t) ); ELy := isolate( EQy[1], diff(y(t), t, t) ); Constraint;

Surprisingly, because of the algebraic constraint, the five numbers x(0), y(0), x'(0), y'(0) and NiMtJSJoRzYjIiIh are not independent. Indeed, only two of these numbers are independent because they are linked by three equations, namely, the two Euler-Lagrange equations and the equation of constraint.However, the Euler-Lagrange equations contain the second derivatives NiMtJSR4JydHNiMlInRH and NiMtJSR5JydHNiMlInRH that can be eliminated by differentiating the constraint equation twice. Along with the constraint equation itself, this yields the equationsConstraint; Q1 := diff( Constraint, t); Q2 := diff( Constraint, t, t);Using the Euler-Lagrange equations to eliminate the second derivatives from this last equation, we obtainQ3 := eval( Q2, { ELx, ELy } );In conjunction with the equation of constraint and its first derivative, solve this equation for, say, h(t), y(t) and y'(t), obtaining the two solutions_EnvExplicit := true: Q4 := solve(convert( {Q1, Q3, Constraint}, D), {h(t), y(t), D(y)(t)} );Note that for general motion of the pendulum, NiMtJSJ5RzYjJSJ0Rw== is negative. Hence, choose the solution for which NiMtJSJ5RzYjIiIh is negative, obtainingQ5 := eval( Q4[2], t=0 );IntroduceQ6 := { x(0)=x[0], D(x)(0)=nu[x] };to simplify the notation toINIT := eval( Q5, Q6 );then list all five initial conditions asINITS := INIT union Q6;

Select specific values. Take the gravitational constant as NiMvJSJnRy0lJkZsb2F0RzYkIiMpKiEiIg==,PARAM := { x[0]=0, m=1, rho=1, nu[x]=2, g=9.8 };leading to the complete set of initial conditionsinits := eval( INITS, PARAM );For this specific system, the Euler-Lagrange equations themselves becomeeqns := eval({ ELx, ELy, Constraint}, PARAM );

A numeric solution of this DAE system is obtained with Maple's dsolve command, implemented as follows. In this case, Maple detects that the system is a DAE and calls the appropriate solver.eqns union inits;sol := dsolve( eqns union inits, numeric );The trajectory of the bob in the NiMlI3h5Rw==-plane is shown in Figure 1.pendulum := proc( z ) xt := eval( x(t), sol(z) ): yt := eval( y(t), sol(z) ): display( disk([xt(z),yt(z)], .05, color=gold ), line([0,0],[xt(z),yt(z)], color=blue, thickness=2 ) ): end proc: animate( pendulum, [.1*k], k=0..20, axes=frame, scaling=constrained, frames=50 );The motion in the plane is periodic, but not sinusoidal. (If it were sinusoidal, the plot below would be a circle.) The periodicity is best seen in the phase plane where orbits are closed paths. Figure 2 shows a closed orbit in the phase plane for NiMlInhH.odeplot( sol, [x(t), diff(x(t), t)], 0..10, numpoints=1000, scaling=constrained, tickmarks=[3, 3], title="Figure 2");Figure 3 shows a closed orbit in the phase plane for , consistent with periodicity in NiMlInlH.odeplot( sol, [y(t), diff(y(t), t)], 0..10, numpoints=1000, scaling=constrained, tickmarks=[2, 3], title="Figure 3" );A graph of the Lagrange multiplier NiMtJSJoRzYjJSJ0Rw== is seen in Figure 4.odeplot( sol, [t, h(t)], 0..10, numpoints=1000, view=[0..10, 0..8], scaling=constrained, title="Figure 4" );Figure 5 shows the graphs of NiMtJSJ4RzYjJSJ0Rw== and NiMtJSJ5RzYjJSJ0Rw== in the time domain, again indicating periodicity in both NiMtJSJ4RzYjJSJ0Rw== and NiMtJSJ5RzYjJSJ0Rw==.odeplot( sol, [[t, x(t)], [t, y(t)]], 0..10, color=black, style=point, numpoints=1000, title="Figure 5" );

The default DAE solver in Maple reduces the index and applies a projection back onto the manifold determined by the algebraic constraints. Reduction of index merely means that the algebraic constraints are differentiated to form a set of differential equations that accompany the other differential equations of the system. As a step is taken by the numeric ODE solver, the results are tested in the algebraic equations. The size of the residual determines if the solution at that step is close enough to the manifold to continue. The ordinary solver is rkf45_dae, combining the adaptive Runge-Kutta-Fehlberg algorithm with the projection step. Unfortunately, reduction of index often leads to stiff systems of ODEs, and therefore, Rosenbrock's stiff solver has also been adapted as the DAE solver rosenbrock_dae.A direct technique based on finite differences is implemented in mebdfi, the Modified Extended Backward-Differentiation Forula Implicit method that works for semi-implicit systems. Semi-implicit systems are those in which the differential equations have been solved for the derivative of highest order. A backward-difference approximation for the derivatives then leads to an implicit scheme where a Newton solver is applied to the resulting algebraic equations for the nodal values.

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