13,874 research outputs found
Integrating factors for second order ODEs
A systematic algorithm for building integrating factors of the form mu(x,y),
mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can
determine the existence and explicit form of the integrating factors themselves
without solving any differential equations, except for a linear ODE in one
subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries
are shown to be solvable using this algorithm. The scheme was implemented in
Maple, in the framework of the "ODEtools" package and its ODE-solver. A
comparison between this implementation and other computer algebra ODE-solvers
in tackling non-linear examples from Kamke's book is shown.Comment: 21 pages - original version submitted Nov/1997. Related Maple
programs for finding integrating factors together with the ODEtools package
(versions for MapleV R4 and MapleV R5) are available at
http://lie.uwaterloo.ca/odetools.ht
Finding Liouvillian first integrals of rational ODEs of any order in finite terms
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher
and others, that if a given rational ODE has a Liouvillian first integral then
the corresponding integrating factor of the ODE must be of a very special form
of a product of powers and exponents of irreducible polynomials. These results
lead to a partial algorithm for finding Liouvillian first integrals. However,
there are two main complications on the way to obtaining polynomials in the
integrating factor form. First of all, one has to find an upper bound for the
degrees of the polynomials in the product above, an unsolved problem, and then
the set of coefficients for each of the polynomials by the
computationally-intensive method of undetermined parameters. As a result, this
approach was implemented in CAS only for first and relatively simple second
order ODEs. We propose an algebraic method for finding polynomials of the
integrating factors for rational ODEs of any order, based on examination of the
resultants of the polynomials in the numerator and the denominator of the
right-hand side of such equation. If both the numerator and the denominator of
the right-hand side of such ODE are not constants, the method can determine in
finite terms an explicit expression of an integrating factor if the ODE permits
integrating factors of the above mentioned form and then the Liouvillian first
integral. The tests of this procedure based on the proposed method, implemented
in Maple in the case of rational integrating factors, confirm the consistence
and efficiency of the method.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs
The chemical kinetics ODEs arising from operator-split reactive-flow
simulations were solved on GPUs using explicit integration algorithms. Nonstiff
chemical kinetics of a hydrogen oxidation mechanism (9 species and 38
irreversible reactions) were computed using the explicit fifth-order
Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster
than single- and six-core CPU versions by factors of 126 and 25, respectively,
for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for
hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane
(53 species and 634 irreversible reactions) oxidation, were computed using the
stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The
GPU-based RKC implementation demonstrated an increase in performance of nearly
59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than
the single- and six-core CPU-based RKC algorithms using the
hydrogen/carbon-monoxide mechanism. With the methane mechanism, RKC-GPU
performed more than 65 and 11 times faster, for problem sizes consisting of
131,072 ODEs and larger, than the single- and six-core RKC-CPU versions, and up
to 57 times faster than the six-core CPU-based implicit VODE algorithm on
65,536 ODEs. In the presence of more severe stiffness, such as ethylene
oxidation (111 species and 1566 irreversible reactions), RKC-GPU performed more
than 17 times faster than RKC-CPU on six cores for 32,768 ODEs and larger, and
at best 4.5 times faster than VODE on six CPU cores for 65,536 ODEs. With a
larger time step size, RKC-GPU performed at best 2.5 times slower than six-core
VODE for 8192 ODEs and larger. Therefore, the need for developing new
strategies for integrating stiff chemistry on GPUs was discussed.Comment: 27 pages, LaTeX; corrected typos in Appendix equations A.10 and A.1
Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating factor form. First of all, one has to find an upper bound for the degrees of the polynomials in the product above, an unsolved problem, and then the set of coefficients for each of the polynomials by the computationally-intensive method of undetermined parameters. As a result, this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors for rational ODEs of any order, based on examination of the resultants of the polynomials in the numerator and the denominator of the right-hand side of such equation. If both the numerator and the denominator of the right-hand side of such ODE are not constants, the method can determine in finite terms an explicit expression of an integrating factor if the ODE permits integrating factors of the above mentioned form and then the Liouvillian first integral. The tests of this procedure based on the proposed method, implemented in Maple in the case of rational integrating factors, confirm the consistence and efficiency of the method
Methods in Mathematica for Solving Ordinary Differential Equations
An overview of the solution methods for ordinary differential equations in
the Mathematica function DSolve is presented.Comment: 13 page
Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
Lie symmetry analysis is one of the powerful tools to analyze nonlinear
ordinary differential equations. We review the effectiveness of this method in
terms of various symmetries. We present the method of deriving Lie point
symmetries, contact symmetries, hidden symmetries, nonlocal symmetries,
-symmetries, adjoint symmetries and telescopic vector fields of a
second-order ordinary differential equation. We also illustrate the algorithm
involved in each method by considering a nonlinear oscillator equation as an
example. The connections between (i) symmetries and integrating factors and
(ii) symmetries and integrals are also discussed and illustrated through the
same example. The interconnections between some of the above symmetries, that
is (i) Lie point symmetries and -symmetries and (ii) exponential
nonlocal symmetries and -symmetries are also discussed. The order
reduction procedure is invoked to derive the general solution of the
second-order equation.Comment: 31 pages, To appear in the proceedings of NMI workshop on nonlinear
integrable systems and their applications which was held at Centre for
Nonlinear Dynamics, Tiruchirappalli, Indi
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