Kingdom of Saudi Arabia
Majmaah University
Deanship of Scientific Research
Engineering and Applied Sciences
Research Center
Research Project Grant
Project No. 15 (2014-2015)
Research Title: Evolution Equations on the Heinsenberg Group
Duration of the project grant : 10 months
Total Accorded Budget: 22000 SR
Principal Researcher: Dr Rabah Kellil
Associated Researcher: Pr Dr Mokhtar Kirane
Department: : Mathematics
College: Of Science Al-Zulfi
Mobile: +966564070512
Official E-mail: r.kellil@mu.edu.sa
Dr Rabah Kellil *
Abstract
The Heisenberg group plays an important role in many branches of physics,
mathematics, such as number theory, representation theory, several complex variables,
quantum mechanics and Partial Differential Equations of any type (elliptic, parabolic,
hyperbolic, ultra-parabolic or ultra-hyperbolic).
We will study some evolution equations with fractional derivatives in time, we intend to
present some results concerning nonexistence of global solution which indicate that
some solutions become infinite in a finite time revealing some singularities. Our
equations may contain one or may derivatives in time; they may interpolate between the
heat equation and the wave equation. Our equations may contain many times as in
physics some models contain two or may times.
Moreover, we will extend the study on one equation to a system of two
equations strongly coupled via the non-linear coupling terms. Our method of proof will
relay on the weak formulation of the sought solution with a judicious choice of the test
function and a scaling argument.
Pr Dr Mokhtar Kirane * *
Expected Results
Our objectives in studying non-global existence (blow-up) of solutions to
evolution equations is to capture their essential futures; a blowing-up solutions may tell
two things: either there is a physical singularity or the non-linearity taken in the model
is not suitable and hence the model has to be changed.
Our objectives are clear:
1-Obtain the conditions leading to non-existence solutions (which in presence of a
local existence theorem means blow-up) of some evolutions equations.
2- Extend the results to systems of at least two equations.
3- Extend the study to equations with at least two times.
4- Extend the study to equations with fractional time derivatives.
5- Extend the study to systems with fractional time derivatives.
Conclusions
Introduction
In particles physics, it was proved by Weyl that the Schrodinger picture and
the Heisenberg picture are physically equivalent. The Heisenberg picture is based on
the Heisenberg group. From here emerged a large number of problems in many fields:
physics, mathematics, number theory, representation theory, several complex
variables, quantum mechanics and Partial Differential Equations.
In partial differential equation, all articles deal with equations with usual
derivatives. In recent years, appears the necessity to study evolutions equations (and
even stationary equations) with derivatives of non-integer orders. Most of the articles
deal with linear equations whose studies are easy as they are performed using the
Laplace transform.
Our aim is to tackle non-linear equations with many (at least two) fractional
derivatives on one hand and non-linear evolutions equations with multi-times on
another hand. The main problem with non-linear equations is that solutions may
experience blow-up in a finite time. The eventual non-boundedness of the sought
solutions depends on the non- linearity, the dimension of the space and the underlying
functional space in which the equation is studied.
In our project, we will study some evolution equations with or without
fractional derivatives in time; we intend to present some results concerning
nonexistence of global solution which indicate that some solutions become infinite in a
finite time revealing some singularities in the physical problem. Our equations may
contain one or many derivatives in time; they may interpolate between the heat
equation and the wave equation. Also our equations may contain many times as in
physics some models contain two or many times.
Moreover, we will extend the study on one equation to a system of two
equations strongly coupled via the non-linear coupling terms. Our method of proof will
relay on the weak formulation of the sought solution with a judicious choice of the test
function and a scaling argument. Our results will shed light not only on the physics of
the problems but also on the methods of study.
Literature Review
Non-autonomous evolutions equations with at least two fractional
derivatives in time and even a fractional derivative in but linear have been studied in a
series of papers by R. K. Saxena, A. M. Mathai and H. J. Haubold using the Laplace
transform. They obtained explicit representation of solutions. These type of equations
appear naturally in engineering as they describe hereditary phenomena or memory
effects. They also appear
in several other complex systems such as atmospheric diffusion of pollution, traffic
network, anomalous diffusion ( in rocks for example: oil modeling), biology, electronics,
wave propagation (as in geophysics for example), mechanics, astrophysics, signal
and
image processing, optics, financial engineering, etc.
Multi-time models appear in theoretical physics and Multi-time models
with fractional derivatives appear in physics as time is fractal in essence.
Methods and Materials
We will use modern functional analysis: fixed point theorems when dealing with local
existence theorems, a priori estimates when we look for global solutions, and the
weak formulation of the problems with a judicious choice of the test function and a
scaling argument when we want to show non-existence results.
Planned processes of major tasks as follow:
1- Non-existence results for one equation with two fractional derivatives
2 Non-existence results for one system with two equations and two fractional
derivatives
3-Non-existence results for one equation posed on the Heisenberg group with two
fractional derivatives
4-Non-existence results for one system with two equations posed on the Heisenberg
group with two fractional derivatives
References
1- R. K. Saxena, A. M. Mathai and H. J. Haubold, Reaction-Diffusion Systems and
Nonlinear waves, Astrophysics Space Sciences (2006) 305:297-303.
2-R. K. Saxena, A. M. Mathai and H. J. Haubold, On generalized fractional kinetic
equations, Physica A (2006) 344: 657-664.
3-R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional
calculus to viscoelasticity, J. Rheology (1983) 27: 201-210.
4-R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the
behavior of real materials, J. Appl. Mechanics (1983) 51: 294-298.
4-R. Hilfer, Fractional time evolution, applications of fractional calculus in Physics,
World Scientific, N.J., London, H.K, 2000.
5-A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional
differential equations, North-Holland Mathematics Studies 204, 2006.
6-I. Podlubny, Fractional Differential Equations, Mathematics in Sciences and
Engineering, 198, Academic Press, 1999.
7-M. Kirane, Qualitative properties of solutions to a time-space fractional evolution
equation. Quarterly in Applied Mathematics (2012) 70: 133-157.
8-M. Kirane, The profile of blowing-up solutions to a nonlinear system of fractional
differential equations. Nonlinear Analysis, T. M. A, (2010): 3723-3736.
9-M. Kirane, Semilinear Volterra integro-differential problems with fractional
derivatives in the nonlinearities. Abstract and Applied Mathematics 2011, Art. ID
510314, 11p.
10-M. Kirane, Critical exponents of Fujita type for certain evolution equations and
systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl. (2005) 312:
488-501.
11-M. Kirane, Image structure preserving denoising using generalized fractional time
integrals. Signal Processing (2012) 92(2): 553-563.
12-M. Kirane, Determination of an unknown source term and the temperature
distribution for the linear heat equation involving fractional derivative in time. Applied
Mathematics and Computation (2011) 218(1): 163-170.
13-M. Kirane, An inverse source problem for a two dimensional time fractional
diffusion equation with non-local boundary condition. Math. Methods in the Applied
Sciences, DOI; 10. 1002/MMAS: 2161.
14-Bars I., Survey of two-time physics, Class. Quant. Grav., 18 (2001), pp. 3113-3130.
15-Barashenkov V. S., Mechanics in Six-Dimensional Space-time, Foundations of
Physics, Vol. 28, No 3, 1988, pp. 471-484.
16- Barashenkov V. S., Propagation of signals in space with multi-dimensional time,
JINR, E2-96-112, Dubna, 1996.
17- Brenner A., The mixed problems in several time variables and problems with
parameters. International Conference on Differential Equations, River Edge, NJ, Vol.
80, no. 9, pp. 901-918, 2001.
18-Courant R. and D. Hilbert, Methods of Mathematical Physics, Vol II: Partial
Differential Equations. Inter-science, New York, 1962.
19- Craig, W. and S. Weinstein, On determinism and well-posedness in multiple time
dimensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2110,
3023–3046.
20-Dorling J., The dimensionality of time, Am. J. Phys., 38 (1970), pp. 539-540.
* Dr Rabah Kellil , Department of Mathematics, College of Science Al-Zulfi Majmaah University, KSA.
** Pr Dr Mokhtar Kirane, Laboratoire de Mathématiques, Image et Applications, Pôle Scientifique et Technologique, Université de La Rochelle, Av:.M. Crépeau, 17000 La Rochelle, France.