The document discusses quasi-Lie schemes and their application to differential equations. It introduces quasi-Lie schemes as generalizations of Lie schemes that allow certain differential equations to be transformed into Lie systems. The main theorem of quasi-Lie scheme theory states that the flows of a quasi-Lie scheme transform a time-dependent vector field describing a differential equation into another vector field. This allows certain differential equations, like the second-order Riccati equation, to be transformed into Lie systems and obtain time-dependent superposition rules for their solutions.
Presentation used to defend the PhD thesis: "Lie systems and applications to Quantum Mechanics", held in Zaragoza Spain on 23th October 2009.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
Presentation used to defend the PhD thesis: "Lie systems and applications to Quantum Mechanics", held in Zaragoza Spain on 23th October 2009.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
A type system for the vectorial aspects of the linear-algebraic lambda-calculusAlejandro Díaz-Caro
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.
Those are the slides for my Master course on Monte Carlo Statistical Methods given in conjunction with the Monte Carlo Statistical Methods book with George Casella.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
A type system for the vectorial aspects of the linear-algebraic lambda-calculusAlejandro Díaz-Caro
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.
Those are the slides for my Master course on Monte Carlo Statistical Methods given in conjunction with the Monte Carlo Statistical Methods book with George Casella.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A complete list of Uq(sl2)-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the Uq(sl2)-module algebra structures are discussed.
A new approach to constants of the motion and the helmholtz conditions
Quasi Lie systems and applications
1. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie schemes: theory and
applications
J.F. Cariñena1 J. Grabowski2 J. de Lucas12
1 University of Zaragoza, Spain
2 Institute of Mathematics of the Polish Academy of Science, Poland
International Young Researches Workshop on Geometry,
Mechanics and Control, 2008
2. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Outline
1 Introduction
Lie Systems and superposition rules.
Lie’s theorem, problems and solutions.
2 Quasi-Lie schemes and quasi-Lie systems
Quasi-Lie schemes.
The main theorem of the theory of quasi-Lie schemes.
Quasi-Lie systems and time-dependent superposition rules.
3 Examples and applications
The second order Riccati equation.
Other applications.
Outlook.
3. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Lie Systems and superposition rules.
Differential equations to study.
Geometrical description
Suppose a differential equation on Rn
dx i
= Y i (t, x), i = 1, . . . , n,
dt
From a differential eometric point of view, this is the equation of
the integral curves for the time-dependent vector field
n
∂
Y (t, x) = Y i (t, x) .
∂x i
i=1
4. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Lie Systems and superposition rules.
Superposition Rules.
Definition
The initial differential equation admits a superposition rule if
there exists a map Φ : Rn(m+1) → Rn such that its general
solution x(t) can be written as
x(t) = Φ(x(1) (t), . . . , x(m) (t), k1 , . . . , kn ).
with {x(1) (t), . . . , x(m) } a family of particular solutions and
{k1 , . . . , kn } a set of constants.
Examples
Linear inhomogeneous differential equations.
Riccati equations.
5. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Lie Systems and superposition rules.
Definition of Lie system.
Definition of Lie system
The initial differential equation is called a Lie system if there
exist vector fields X(1) , . . . , X(r ) on Rn such that
The time-dependent vector field Y (t, x) verifies
r
Y (t, x) = bα (t)Xα (x).
α=1
The vector fields {X(α) | α = 1, . . . , r } verify
γ
X(α) , Xβ = cαβ X(γ)
γ
for certain r 3 constants cαβ .
6. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Lie’s theorem, problems and solutions.
Lie’s theorem and practical problems.
Lie’s Theorem
A first-order differential equation admits a superposition rule if
and only if it is a Lie system.
PROBLEMS
1 Generally, it is difficoult checking out that a differential
equation is a Lie system and even more finding a
superposition rule.
2 We maybe do not know if a differential equation admits
any, one or more superposition rules.
3 There exist many important differential equations which are
Lie systems, but in general, these cases are rare.
7. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Lie’s theorem, problems and solutions.
Solutions for the problems of Lie’s theorem.
Provided solutions
Quasi-Lie schemes and quasi-Lie systems.
Generalized Lie theorem.
8. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie schemes.
Fundamentals.
Denifition of quasi-Lie scheme
A quasi-Lie scheme sc(W , V ) is given by:
A non-null Lie algebra of vector fields W .
A linear space V spanned by a finite set of vector fields.
These two elements verify:
W ⊂ V.
[W , V ] ⊂ V .
9. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie schemes.
A scheme for the Abel equation.
Example
Define on R the set of vector fields
∂ ∂ ∂ ∂
X0 = , X1 = x , X2 = x 2 , X3 = x 3 .
∂x ∂x ∂x ∂x
and
WAbel = X0 , X1 , VAbel = X0 , X1 , X2 , X3 .
The linear space WAbel ⊂ VAbel is a Lie algebra because
[X0 , X1 ] = X1 . Moreover, as
[X0 , X2 ] = 2X1 , [X0 , X3 ] = 3X2 ,
[X1 , X2 ] = X2 , [X1 , X4 ] = 2X3 ,
then [WAbel , VAbel ] ⊂ VAbel .
10. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie schemes.
Differential equations described by a quasi-Lie scheme.
Definition
A differential equation described by means of a time-dependent
vector field Y (t, x) can be described by a scheme sc(W , V ) if
s
Y (t, x) = bα (t)Xα (x)
α=1
with {Xα | α = 1, . . . , s} a basis which spans V .
Theorem
Every Lie system can be described through a quasi-Lie
scheme.
11. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie schemes.
Example
The Abel equation
x = f3 (t)x 3 + f2 (t)x 2 + f1 (t)x + f0 (t),
˙
describes the integrals for the time-dependent vector field
X (t) = f3 (t)X3 + f2 (t)X2 + f1 (t)X1 + f0 (t)X0 .
Hence, it can be described by the quasi-Lie scheme
sc(WAbel , VAbel ).
12. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The main theorem of the theory of quasi-Lie schemes.
Flows for a quasi-Lie scheme.
Generalized flows of a quasi-Lie scheme
Given any time-dependent vector field X given by
s
X (t) = bα (t)Xα
α=1
with {Xα | α = 1, . . . , s} a basis for W . This vector field can be
considered as a curve b(t) = (b1 (t), . . . , bs (t)). Then, there
exists a generalized flow for X (t) which we denote
gt (b1 (t), . . . , bs (t)).
Let us denote this family of flows as fl(W ).
13. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The main theorem of the theory of quasi-Lie schemes.
Actions of flows.
Time-dependent vector fields and flows
Any time-dependent vector field admits a genealized flow gtX .
The knownledge of a time-dependent vector field is equivalent
to the knowledge of its flow.
Action of flows
Given a time-dependent vector field Y (t, x) with generalized
flow gtY and any other generalized flow gt , we define the action
of gt on Y , denoted by (gt )⋆ Y = Y ′ , with Y ′ the time-dependent
vector field with generalized flow gt ◦ gtY .
14. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The main theorem of the theory of quasi-Lie schemes.
Main theorem of the theory of quasi-Lie schemes
Suppose a quasi-Lie scheme sc(W , V ). Given a generalized
flow gt ∈ fl(W ) and a time-dependent vector field
s
Y (t, x) = bα (t)Xα ,
α=1
with {Xα | α = 1, . . . , s} a basis for V , then
s
′
(gt )⋆ Y = bα (t)Xα .
α=1
15. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Quasi-Lie systems and time-dependent superposition rules.
Time-dependent superposition rules and quasi-Lie systems.
Definition of quasi-Lie systems
We say that a time-dependent vector field Y (t, x) is a quasi-Lie
system respecto to a quasi-Lie scheme sc(W , V ) if
1 Y can be considered as a curve in V .
2 There exists an element gt ∈ fl(W ) such that (gt )⋆ Y is a
Lie system.
Main property of quasi-Lie systems
If Y is a quasi-Lie system respect a quasi-Lie scheme it can be
shown that its general solution can be written as
x(t) = Ψ(t, x(1) (t), . . . , x(m) (t); k1 , . . . , kn ),
with {x(1) (t), . . . , x(m) (t)} a family of particular solutions and
{k1 , . . . , kn } a set of constants.
16. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
A second order Riccati equation.
Problem
Let us study the second-order Riccati equation
x + (b0 (t) + b1 (t)x)x + a0 (t) + a1 (t)x + a2 (t)x 2 + a3 (t)x 3 = 0,
¨ ˙
with
a2 (t) ˙
a (t)
b1 (t) = 3 a3 (t), b0 (t) = − 3 .
a3 (t) 2a3 (t)
˙
By means of the change of variables x = v, we transform the
latter second-order differential equation into the first-order one
˙
x = v,
v = −(b0 (t) + b1 (t)x)v − a0 (t) − a1 (t)x − a2 (t)x 2 − a3 (t)x 3 .
˙
17. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
A quasi-Lie scheme for the second-order Riccati equation.
Example
Define on TR the set of vector fields
∂ ∂ ∂ ∂
Y1 =v , Y2 = v , Y3 = xv , Y4 = ,
∂x ∂v ∂v ∂v
∂ ∂ ∂ ∂
Y5 =x , Y6 = x 2 , Y7 = x 3 , Y8 = x .
∂v ∂v ∂v ∂x
Then, we construct WRicc and VRicc as WRicc = Y2 , Y8 ,
VRicc = Y1 , . . . , Y8 . Evidently WRicc ⊂ VRicc is an abelian two
dimensional Lie algebra and as
[Y2 , Y1 ] = Y1 , [Y2 , Y3 ] = 0, [Y2 , Y4 ] = −Y2 ,
[Y2 , Y5 ] = −Y5 , [Y2 , Y6 ] = −Y6 , [Y2 , Y7 ] = −Y7 ,
[Y8 , Y1 ] = −Y1 , [Y8 , Y3 ] = Y3 , [Y8 , Y4 ] = 0,
[Y8 , Y5 ] = Y5 , [Y8 , Y6 ] = 2Y6 , [Y8 , Y7 ] = 3Y7 .
then [WRicc , VRicc ] ⊂ VRicc .
18. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
Flows of the quasi-Lie scheme for the second-order Riccati
equation.
The second-Riccati equation can be understood as the
differential equation of the integral curves for the
time-dependent vector field given by
X (t) = Y1 − b0 (t)Y2 − b1 (t)Y3 − a0 (t)Y4 − a1 (t)Y5 − a2 (t)Y6 − a3 (t)Y7 .
So, the quasi-Lie scheme sc(WRicc , VRicc ) allows us to deal
with such an equation.
Flow of the quasi-Lie scheme
The set fl(WRicc ) is given by
x(t) = α(t)x ′ (t),
v(t) = β(t)v ′ (t).
19. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
Applications of the main theorem of the theory of quasi-Lie
schemes.
Consecuence of our main theorem.
The flows of our scheme transforms the initial second-order
Riccati equation into
dx ′ ˙
β(t) ′ α(t) ′
= v − x,
dt α(t) α(t)
dv ′ a0 (t) α(t)a1 (t) a2 (t)α2 (t) ′2
=− + −b1 (t)α(t)v ′ − x′ − x
dt β(t) β(t) β(t)
a3 (t)α3 (t) ′3 b0 (t)β(t) + β ′ (t) ′
− x − v.
β(t) β(t)
20. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
Fix b3 (t) = a3 (t) and α(t) = 1. We obtain that
′
dx = a (t)v ′ ,
3
dt
′
dv a (t) a (t) ′
=− 0 − a3 (t)(3v ′ x ′ + x ′3 ) − 1
x
dt
a3 (t) a3 (t)
a2 (t)
(v ′ + x ′2 ).
−
a3 (t)
But this is the system of differential equations related to the
time-dependent vector field
a0 (t) a1 (t) a2 (t)
X (t) = a3 (t)X1 − X2 − (X3 + X7 ) − X0
a3 (t) 2a3 (t) a3 (t)
And this is a Lie system related to a set of vector fields which
span a sl(3, R) Lie algebra.
21. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
The second order Riccati equation.
Time-dependent superposition rules.
The time-dependent superposition rule for the second-order
Riccati equation reads
−x4 F123 + G1234 Λ1 + G1243 Λ3 − x3 F124 Λ1 Λ3
x5 = ,
−F123 + (F124 − F324 )Λ3 + (F123 − F423 )Λ1 − Λ1 Λ3 F124
where
Gabcd = xa ((vd − vc )xb + (vb − vd )xc + (xb − xc )xb xc + (xc − xb )xa xd )+
xd ((vc − va )xb + (va − vb )xc + (xc − xb )xb xc + (xb − xc )xa xd )
and
Fabc = va (xc −xb )+vb (xa −xc )+vc (xb −xa )+(xa −xb )(xb −xc )(xc −xa ).
22. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Other applications.
1 Abel equations can be described by a quasi-Lie scheme.
Those Abel equations which are quasi-Lie systems seem
to be related to the so-called integrable cases.
2 Dissipative Ermakov systems can be studied through a
quasi-Lie schemes. Time-dependent superposition rules
have been obtained.
3 Emden-Fowler equations can be described by a quasi-Lie
schemes. This method can be used to obtain exact
solutions, i.e. the Lane-Emden equation can be solved
analytically.
4 Non-linear oscillators have been studied by means of this
method. Integrals of motion and other properties have
been obtained.
5 This method is related to a generalized Lie theorem which
allows to describe when a differential equations admits a
time-dependent superposition rule.
23. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Outlook.
Outlook
1 There are many differential equations that can be studied
by means of quasi-Lie schemes. This method allows to
obtain properties for them but it has not been applied yet to
many cases.
2 Still it has to be proved when there is a way to know if a
differential equation is not a quasi-Lie system.
3 Extensions of quasi-Lie schemes to deal with PDE’s and
Quantum Mechanics have partially been done.
4 Applications on many physical problems in Quantum
Mechanics.
24. Introduction Quasi-Lie schemes and quasi-Lie systems Examples and applications
Outlook.
THE END
THANK FOR YOUR ATTENTION