In this paper, a method for solving singular differential algebraic equations combining Legendre wavelets with a collocation technique is presented. Numerical examples show that the method is easy to implement and yields very accurate results.Facultad de Ingenierí

Vampa, Victoria

English

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NUMERICAL SOLUTIONS FOR DAES USING LEGENDRE
WAVELETS
Victoria Vampa†
†Departamento de Ciencias Ba´sicas, Facultad de Ingenierı´a , Universidad Nacional de La Plata, Argentina,
victoriavampa@gmail.com
Abstract: In this paper, a method for solving singular differential algebraic equations combining Legendre wavelets
with a collocation technique is presented. Numerical examples show that the method is easy to implement and yields
very accurate results.
Keywords: Differential algebraic equations, Legendre wavelets, Collocation methods
2000 AMS Subject Classification: 21A54 - 55P54
1 INTRODUCTION
Differential algebraic equations (DAEs) play an important role in many branches of science and engi-
neering. DAEs arise in a wide variety of systems, for instances, circuit analysis, computer-aided design,
simulation of mechanical systems and optimal control problems.
DAEs are a type of differential equation where one or more derivatives of dependent variables are not
present in the equations. Variables that appear in the equations without their derivative are called algebraic,
and the presence of algebraic variables means that you cannot write down the equations in the explicit form
y0 = f(t, y). They are characterized by their differential index, which is a measure of their singularity. By
differentiating equations you can eliminate algebraic variables, and if this is done enough times, then the
equations take the form of a system of explicit ODEs. The differential index of DAEs is the smallest number
of derivatives you must take to express the system as an equivalent system of explicit ODEs. In particular,
ODEs have a differential index of 0.
In recent years, DAEs have attracted the attention of numerical analysists. One of the reasons is that
when the index is greater than 1 they can be difficult to solve numerically. Vanani and Aminataei (see
[1]) obtain numerical solutions using a multiquadric approximation scheme, using as basis functions a type
of spline approximation with good regularity properties. Other authors, as Hosseini ([3]), have proposed
reducing index methods for semi-explicit DAEs.
The aim of this work is to present the Legendre wavelets approximation method (LWM) for DAEs (see
[2]) and to make a comparison with multiquadric scheme and ODE solvers in case they can be used.
This paper is organized as follows. In Section 2 some notations and definitions of the DAE forms are
given. Section 3 describes basic concepts of the Legendre wavelet method and how they are applied on
DAEs. In Section 4 two illustrative numerical experiments are presented. Finally, Section 5 consists of
some obtained conclusions.
2 DIFFERENTIAL ALGEBRAIC EQUATIONS. DAES
Let us consider a system of differential equations of the form:
M(t, y)y0 = f(t, y) (1)
where the mass matrix M(t, y) is singular. That means that you have a Differential Algebraic Equation
(DAE) rather than an ODE system. And it is important to note that solving DAEs is more complicated
because, as it is demonstrated in ([4]), DAEs has a solution only if the initial conditions are consistent .
Rewriting Eq.(1) in semi-explicit form,
u0 = f1(t,u, v) (2)
0 = f2(t,u, v)
where y = (u, v), the system is of index 1 if the matrix of partial derivatives of the algebraic system f2
with respect to the algebraic variables (i.e. df2/dv) is non-singular. And in mass matrix form (Eq.(1)), the
index 1 condition is satisfied when the matrix (M+ λ ∗ df2/dv) is nonsingular, for all non-zero λ.
In Matlab, ode15s and ode23t solvers can be used to find numerical solutions for DAEs of index 1. In
that case it is demonstrated that there is any distinction between solving a DAE and an ODE system (see [4]
for details). If the index of the equations is 2 or higher, then it is necessary to rewrite the equations as an
equivalent system of index-1 DAEs.
3 PROPERTIES OF LEGENDRE WAVELETS. COLLOCATION METHOD
Wavelet theory is a relative new and emerging area in mathematical research ([5], [6], [7]). Different
wavelets have found their way into many fields of science and engineering, as they constitute a family of
functions generated from dilation and translation of a single function called the mother wavelet. When the
dilation parameter a and the translation parameter b vary continuosly, the following family of continuos
wavelets is constructed:
ψa,b(t) = |a|−1/2ψ( t− b
a
), a, b ∈ R, a 6= 0 (3)
If the parameters a and b are restricted to discret values as a = a−k0 and b = nb0a
−k
0 where a0 > 1,
b0 > 0, and n and k are positive integers, the family of discrete wavelets,
ψk,n(t) = |a0|−1/2ψ(ak0t− nb0), (4)
is a wavelet basis for L2(R). When a0 = 2 and b0 = 1 they form an orthonormal basis.
Legendre wavelets (see [2])ψnm(t) = ψ(k, nˆ,m, t) have four arguments nˆ = 2n−1, n = 1, 2, · · · , 2k−1,
k is any positive integer, m is the order of the Legendre polynomials and t is the normalized time (they are
defined on the interval [0, 1)). These polynomials are orthogonal with respect to the weight function 1 on
the interval [−1, 1] and they can be constructed easily by a recursive formula (see [2]).
Using the above family, a function f(t) defined over [0, 1) may be expanded as
f(t) =
∞X
n=1
∞X
m=0
cnmψnm(t), (5)
where cnm = hf(t), ψnm(t)i, and the following approximation can be obtained if the series is truncated:
f(t) ≈
2k−1X
n=1
M−1X
m=0
cnmψnm(t) (6)
In this way, the approximate solution for the function y in Eq.(1) can be expanded as follows :
y(t) = CTΨ(t) (7)
where C and Ψ(t) are 2k−1M × 1 matrices with the unknown coefficients and Legendre wavelets, respec-
tively (see [2] for details).
Thus for the solution of above DAEs Eq.(2) it is sufficient to suppose that
u(t) = CT1 Ψ(t), v(t) = C
T
2 Ψ(t) (8)
and therefore,
CT1 Ψ
0(t) = f1(t, CT1 Ψ, C
T
2 Ψ) (9)
0 = f2(t, C
T
1 Ψ, C
T
2 Ψ)
Collocation points ti are required in order to apply LWM . They are selected in [0, 1]. Thus, the following
algebraic system is obtained,
CT1 Ψ
0(ti) = f1(ti, CT1 Ψ, C
T
2 Ψ) (10)
0 = f2(ti, C
T
1 Ψ, C
T
2 Ψ)
and adding initial conditions, it can be solved and the unknown coefficients are found.
4 NUMERICAL EXAMPLES
In this section we present two numerical examples concerning a problem of the form
A(t)y0(t) + B(t)y = 0 0 ≤ t ≤ 1 (11)
where,
A =

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0

and,
B1 =

1 0 −t 1
−1 1 −t2 t
−t3 t2 −1 0
t −1 t −1
 , B2 =

1 −t2 1 t
0 t 0 −3
−(1− t2) −t3 0 t2
t t −1 −1

correspond to two different problems, of index 1 (see [2]).
They were solved using a computer program in MATLAB, with k = 1 and M = 2j + 1 collocation
points, j = 3, 4, 5.
Exact solutions in each case are: y1(t) = (e−t, te−t, et, tet)T , y2(t) = (e−t
2
, et
2
, te−t2 , tet2)T . Us-
ing both, the multiquadric approximation ([1]) and the Legendre wavelet method, numerical results were
obtained. They were similar in accuracy (relative errors ≈ 10−6 for j = 4 ) but with the advantage that
the matrices in LWM method are better conditioned. In addition, they were solved using ode15s and a
comparison of the behaviour was made.
Acknowledgments
I would like to thank Marila Etchechoury for her helpful comments.
5 CONCLUSIONS
The results reveal that the technique introduced here is effective and convenient in solving DAEs and it
can be extended for solving nonlinear problems. Areas of future work include performing further validations
of the method to solve high-index semi-explicit DAEs and understanding the behaviour of singular solutions,
in particular, those arising in modelling electric circuits.
REFERENCES
[1] S. KARIMI VANANI, A. AMINATAEI, Numerical solution of differential algebraic equations using a multiquadric approxima-
tion scheme, (2011) Mathematical and Computing Modelling, 53, pp. 659-666, .
[2] S. KARIMI VANANI, F. SOLEYMANI AND M. AVAJI, Legendre Wavelet Method for Solving Differential Algebraic Equations,
(2011), Australian Journal of Basic and Applied Sciences, 5 (9), pp. 2105-2110.
[3] M. M. HOSSEINI, A domian decomposition method for solution of differential-algebraic equations, (2006), Journal of Com-
putational and Applied Mathematics, 197, pp. 495-501.
[4] L. SHAMPINE, M. W. REICHELT, AND J. KIERZENKA, Solving index 1 DAEs in Matlab and Simulink, (2001), Society for
Industrial and Applied Mathematics SIAM Rev, 41,3, pp.538-552.
[5] C. CHUI, An introduction to wavelets, Academic Press, New York, 1992.
[6] S. G. MALLAT, A Wavelet Tour of Signal Processing - The Sparse Way, Academic Press- Elsevier-MA EEUU, 2009.
[7] D. WALNUT, An Introduction to Wavelet Analysis, Applied and Numerical Harmonic Analysis Series, Birkha¨user Eds., Boston,
2009.


Numerical solutions for DAEs using Legendre wavelets

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