COMPUTATIONAL METHODS FOR APPLIED
TECHNOLOGICAL PROBLEMS
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The scientific team had
to deal with the following two categories of equations: |
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NONLINEAR WAVE EQUATIONS |
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From this category it is
going to be examined the equations applied to shallow waters. More precisely
it is going to be examined the following: a. Classical equations otherwise the equations for which it is known the
theoretical solution. The research has to improve some of the existing
numerical methods and to propose new methods for their solution. The
equations which are going to be examined are i. Kortweg
& de Vries
and its extension in two
dimensions the Kadomtsev-Petviashvili equation given by
where |σ|=1 is a
parameter. ii. Davey-Stewartson
where
u=u(x,y,t), v=v(x,y,t) and η1, η2 ρ1, ρ2, μ, λ1, λ2 are parameters. iii. Boussinesq
and
its extensions in two dimensions. iv nonlinear cubic Schrödinger
equation
Where
i is the imaginary unity and q is a real parameter, as well as the coupled
Schrödinger equation given by the following system
where u1, u2 are
the wave amplitudes and e, δ are parameters. |
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b. Equations describing real physical phenomenaWe consider for study the following problem describing the wave propagation in swallow water under true natural conditions, which is an application of the Boussinesq equation under real natural circumstances. The problem is briefly described as follows: The system of wave equations for a wave approaching a coast with variable deepness is defined by the following equations ζ t+px+qy=0 (2.1) (continuity
equation) pt
+(p^2/h)x+(pq/h)y+gh ζx+(gp/c^2h^2)(p^2+q^2)^{1/2}-E(pxx+pyy)=Dh(pxxt+qxyt)/3 (2.2) (impulse
preservation equation in the x direction) qt
+(q^2/h)y+(pq/h)x+gh ζy+(gp/c^2h^2)(p^2+q^2)^{1/2}-E(qyy+qyy)=Dh(qxxt+pxyt)/3 (2.3) (impulse preservation equation in the y
direction) where ζ(x,y,t) is the elevation of the free surface of the
sea above a reference level, p(x,y,t) is the density of the flow volume (m^3/s/m)
in the x direction, q(x,y,t) is the density of the flow volume (m^3/s/m) in
the y direction, h(x,y,t) is the deepness from the bottom of the sea, D(x,y)
is the deepness from the bottom of the sea in still surface, E(x,y) is the
coefficient of noisy flow, C is a constant, H is the wave height, L is the
wave length etc. These equations include non-linear terms as well as
frequency distribution. They can be applied in the study of wave potential at
sea shelters, ports and small coast territories. The wave disturbance at
ports is considerable for the run aground of the boats as well as for the
cargo loading and unloading. The disturbance within the port basins is one of
the most important factors for the engineers, who are called to choose the
construction place for the port and to determine the highest efficiency of
the port. The wave movement from the open sea into a port or a sea shelter is
a process that basically involves the phenomena of decrease of the depth of
the sea, the diffraction, the reflection and the refraction. At this point we
must point out that according to the conclusions from extended studies the
solution of the above equations is accomplished by the National Center for Marine
Researches (NCMR) with the help of credible programs available in the free
market. The access to these programs is impossible since their availability
depends on the owner companies. As a result, the scientific personnel of the
NCMR has not the ability to apply and use productively their specific
knowledge on these programs. They are merely called to use them by entering
some data. |
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REACTION-DIFFUSION EQUATIONS
u t=k uxx+f(u)
(3.1) where u=u(x,t), k the contactivity and f(u) a nonlinear function. |
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BIBLIOGRAPHY |
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a. Classical equations i. Kortweg & de Vries Freeman, N.C.,
Soliton solutions of Non-linear Evolution Equations}, IMA
Journal of Applied Mathematics, Vol. 32 (1984), pp 125-145. Ablowitz, J. M.
and Villarroel, J., Solutions of the Time Dependent Schrödinger
and the Kadomtsev-Petviashvili Equations, Phys. Rev. Let., Vol. 78, No
4 (1997). Bratsos A. G.,
Twizell E.H., An Explicit Finite-difference scheme for the solution of the
Kadomtsev-Petviashvili equation, Intern. J. Computer Math., 68 (1998),
pp 157-187. ii. Davey-Stewartson J. Satsuma, M. J.
Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math.
Phys., 20, No 7 (1979), pp 1496-1503. D. Anker, N. C.
Freeman, On the soliton solutions of the Davey-Stewartson equation for long
waves, Proc. R. Soc. Lond., A. 360 (1978), pp 529-540. A. Davey, K.
Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc.
Lond., A. 338 (1974), pp 101-110. iii. Boussinesq J. Argyris and M.
Haase, An Engineer's Guide to Soliton Phenomena: Application of the Finite
Element Method, Comput. Methods Appl. Mech. Engrg. 61 (1987), pp
71-122. A.G. Bratsos, The
solution of the Boussinesq equation using the method of lines, Comput.
Methods Appl. Mech. Engrg. 157 (1998), pp 33-44. R. Hirota, Exact
N-soliton solutions of the wave of long waves in shallow-water and in nonlinear
lattices, J. Math. Phys. 14
(1973), pp 810-814. V.S. Manoranjan,
A.R. Mitchell and J.Li. Morris, Numerical solutions of the Good Boussinesq
equation, SIAM J. Sci. Stat. Comput. 5 (1984), pp 946-957. iv κυβική εξίσωση Schrödinger E.H. Twizell,
A.G. Bratsos and J.C. Newby, A finite-difference method for solving the cubic
Schrödinger equation,
Mathematics and Computers in Simulation, 43 (1997), pp 67-75. J.A.C Weideman
and B.M. Herbst, Split-step methods for the solution of the non-linear Schrödinger equation,
SIAM J. Numer. Anal. 23(3) (1986), pp
485-507. G.B. Whitham,
Linear and nonlinear Waves, New York: Wiley-Interscience (1974). v coupled Schrödinger M. S. Ismail,
Finite difference method with cubic spline for solving nonlinear Schrödinger equation,
Int. J. Computer Math., 62 (1996), pp 101-112. D. F. Griffiths ,
A. R. Mitchell, J. Li Morris, A numerical study of the nonlinear Schrödinger equation,
Comput. Meths. Appl. Mech. Engrg. 45 (1984), pp. 177-215. M. Wadati, T. Izuka, M. Hisakado, A
coupled Nonlinear Schrödinger equation and optical solitons, J. of Phys. Soc. of
Japan., 61, No 7 (1992). b. Equations describing real physical phenomena Bayram, A. & Larson, M. Wave transformation in
the nearshore zone: comparison between a Boussinesq model and field data,
Coastal Eng. Vol. 39 (2000), pp. 149-171 Beji, S. & Nadaoka, K. A formal derivation and
numerical modelling of the improved Boussinesq equations for varying depth,
Ocean Eng., Vol. 23 (1996), pp. 691-704. Larsen, J & Dancy, M., Open Boundaries in Short
Wave Simulations - A New Approach. Coastal Engineering, Vol 7 (1983),
pp 285-297. Madsen, P A, Murray, R & Sørensen, O R, A
New Form of the Boussinesq Equations with Improved Linear Dispersion
Characteristics (Part 1). Coastal Engineering, Vol 15, No 4 (1991), pp
371-388. Madsen, P A & Sørensen, O R., A New Form
of the Boussinesq Equations with Improved Linear Dispersion Characteristics,
Part 2: A Slowly-Varying Bathymetry. Coastal Engineering, Vol 18, No 1
(1992), pp 183-204. Nadaoka, K. & Raveenthiran, K., A phase-averaged
Boussinesq model with effective description of carrier wave group and
associated long wave evolution. Ocean Eng., Vol. 29 (2002), pp. 21-37. Oliveira, F.S.B.F., Improvement on open boundaries
on a time dependent numerical model of wave propagation in the nearshore
region. Ocean Eng., Vol. 28 (2000), pp. 95-115. Zou, Z.L., High order Boussinesq equations, Ocean
Eng., Vol. 26 (1999), pp. 767-792. |