]>
2008
1
1
ISSN 2008-1898
54
Variational principle for nonlinear Schrödinger equation with high nonlinearity
Variational principle for nonlinear Schrödinger equation with high nonlinearity
en
en
It is well-known that the Schrödinger equation plays an important
role in physics and applied mathematics as well. Variational formulations
have been one of the hottest topics. This paper suggests a simple but effective
method called the semi-inverse method proposed by Ji-Huan He to construct
a variational principle for the nonlinear Schrödinger equation with high nonlinearity.
1
4
Li
Yao
Department of Mathematics, Kunming College, Kunming,Yunnan, 650031, P.R. China
Jin-Rong
Chang
Department of Mathematics, Kunming College, Kunming,Yunnan, 650031, P.R. China
Variational principle
Semi-inverse method
nonlinear Schrödinger equation
Article.1.pdf
[
[1]
A. Bekir, A. Boz, Exact solutions for a class of nonlinear partial differential equations using exp-function method, Int. J. Nonlinear Sci., 8 (2007), 505-512
##[2]
J. Biazar, H. Ghazvini, Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method, Phys. Lett. A, 366 (2008), 79-84
##[3]
N. Bildik, A. Konuralp , The use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. , 7 (2006), 65-70
##[4]
M. Gorji, D. D. Ganji, S. Soleimani, New application of He’s homotopy perturbation method, Int. J. Nonlinear Sci. , 8 (2007), 319-328
##[5]
J. H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys., 2006 (B 20), 2561-2568
##[6]
J. H. He, Variational iteration method - Some recent results and new interpretations, J. Comput. and Appl. Math., 207 (2007), 3-17
##[7]
J. H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons, Fractals, 19 (2004), 847-851
##[8]
J. H. He, Variational approach to (2+1)-dimensional dispersive long water equations, Phys. Lett. A , 335 (2005), 182-184
##[9]
J. H. He, Some Asymptotic Methods for Strongly Nonlinear Equations, Int. J. Modern Phys. , B 20 (2006), 1141-1199
##[10]
J. H. He, Non-perturbative methods for strongly nonlinear problems, dissertation.de-Verlag im Internet GmbH, Berlin (2006)
##[11]
J. H. He , New interpretation of homotopy perturbation method, Int. J. Modern Phys. , B 20 (2006), 2561-2568
##[12]
J. H. He, X. H. Wu, Variational iteration method: New development and applications, Comput. & Math. with Appl. , 54 (2007), 881-894
##[13]
J. H. He, X. H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos Solitons, Fractals. , 29 (2006), 108-113
##[14]
J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons, Fractals , 30 (2006), 700-708
##[15]
Z. M. Odibat, S. Momani , Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. , 7 (2006), 27-34
##[16]
T. Ozis, A. Yidirim , Application of He’s semi-inverse method to the nonlinear Schrodinger equation, Comput. & Math. with Appl., 54 (2007), 1039-1042
##[17]
A. Sadighi, D. D. Ganji , Analytic treatment of linear and nonlinear Schrodinger equations: A study with homotopy-perturbation and Adomian decomposition methods, Phys. Lett. A , 372 (2008), 465-469
##[18]
N. H. Sweilam, Variational iteration method for solving cubic nonlinear Schrödinger equation, J. of Comput. and Appl. Math., 207 (2007), 155-163
##[19]
N. H. Sweilam, R. F. Al-Bar , Variational iteration method for coupled nonlinear Schrodinger equations, Comput. & Math. with Appl. , 54 (2007), 993-999
##[20]
Z. L. Tao , Variational approach to the inviscid compressible fluid , Acta Appl. Math., 100 (2008), 291-294
##[21]
X. H. Wu, J. H. He, Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. & Math. with Appl. , 54 (2007), 966-986
##[22]
L. Xu , Variational principles for coupled nonlinear Schrödinger equations, Phys. Lett. A , 359 ( 2006), 627-629
##[23]
J. Zhang , Variational approach to solitary wave solution of the generalized Zakharov equation, Comput. & Math. with Appl., 54 (2007), 1043-1046
##[24]
X. W. Zhou , Variational theory for physiological flow, Comput. & Math. with Appl., 54 (2007), 1000-1002
##[25]
X. W. Zhou , Variational approach to the Broer-Kaup-Kupershmidt equation, Phys. Lett. A , 363 (2007), 108-109
##[26]
S. D. Zhu, Exp-function method for the Hybrid-Lattice system, Int. J. Nonlinear Sci. , 8 (2007), 461-464
##[27]
S. D. Zhu, Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci., 8 (2007), 465-468
]
A NOT ON DOMINATING SET WITH MAPLE
A NOT ON DOMINATING SET WITH MAPLE
en
en
Let \(G\) be a n− vertex graph. In 1996, Reed conjectured that
\(\gamma(G)\leq\lceil \frac{n}{3}\rceil\) for every connected 3− regular \(G\). In this paper, we introduce
an algorithm in computer algebra system of MAPLE such that, by using any
graph as input, we can calculate domination number
\(\gamma(G)\) and illustrated set
of all dominating sets. It important that these sets choose among between
(\(n,
\gamma(G))\)
sets.
5
11
M.
MATINFAR
Department of Mathematics, University of Mazandaran, P. O. Box 47416 - 1467, Babolsar, Iran.
S.
MIRZAMANI
Department of Mathematics, University of Mazandaran, Babolsar, Iran.
Minimum dominating set. MDS. Maple. Adjacency matrix.
Article.2.pdf
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[1]
W. E. Clark, S. Suen, An inequality related to Vizing’s conjecture, Electron. J. Combin. , 7(1) (2000), 1-3
##[2]
B. Hartnell, D. F. Rall, Domination in Cartesian Products: Vizing’s Conjecture, Domination in Graphs–Advanced Topics, New York, Dekker, (1998), 163-189
##[3]
T. W. Haynes, S. T. Hedetniemi, P. J. Slater , Domination in Graphs: Advanced Topics, Marcel Dekker, New York, Marcel Dekker, Inc. , NewYork (1998)
##[4]
B. Read , Paths, stars, and the number three, combin. probab. comput., 5 (1996), 277-295
]
THE ROLE OF DELAY IN DIGESTION OF PLANKTON BY FISH POPULATION A FISHERY MODEL
THE ROLE OF DELAY IN DIGESTION OF PLANKTON BY FISH POPULATION A FISHERY MODEL
en
en
In this Paper we have developed a model in which the revenue is
generated from fishing and the growth of fish depends upon the plankton which
in turn grows logistically. The conditions for the persistence of system around
non zero equilibrium have been found out using average Liapnouv function after
establishing existence and boundedness of the solution. Then we formulated a
model with delay in digestion of plankton by fish. Further the the threshold
value of conversional parameter has been found out for hopf-bifurcation. The
phenomena of hopf-bifurcation is demonstrated using graphs.
13
19
JOYDIP
DHAR
Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior-474 010, M.P, INDIA.
ANUJ KUMAR
SHARMA
Department of Mathematics, L.R.D.A.V. College, Jagraon-142026, Ludhiana, Punjab, INDIA.
SANDEEP
TEGAR
School of Mathematics and Allied Sciences, Jiwaji University, Gwalior- 474011, M.P., INDIA.
Fishery Model
Stability
Delay
Hopf-bifurcation
Article.3.pdf
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[1]
L. G. Anderson, Optium economic yield of a fishery given a variable prise of output, J. Fishries Reasearch Board of Canada , 30 (1973), 509-518
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J. Blaxter, A. Southward , Advances in marine biology, Academic Press, London (1997)
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J. Chattopadhyay, R. Sarkar, A. E. Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Math. Appl. Med. Biol., 19 (2002), 137-161
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J. Chattopadhyay, R. Sarkar, S. Mandal, Toxinproducing plankton may act as a biological control for planktonic blooms field study and mathematical modelling, J.Theor. Biol., 215 (2002), 333-344
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C. W. Clark , A delayed recruement model of population dynamics,with an application tob bleen the whale poplation, J. Mathematical Biology, 31 (1976), 381-391
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C. W. Clark , Mathematical Bioeconomics: The Optimal Management of Renewable resources, John Wiley and Sons, New York (1990)
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E. Graneli et. al., From anoxia to fish poisoning: The last ten years of phytoplankton blooms in swedish marine waters: in Novel phytoplankton blooms, (eds) E Cosper et. al. (Springer) , (1989), 407-427
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##[10]
Monovoisin, C. GandBolito, J. G. Ferreira , Fish Dynamics in a coastal food chain, Bol. inst Esp Oceanoger, (1999), 431-440
##[11]
R. Sarakar, H. Malchow, Nutrients and toxin producing phytoplankton control algal blooms a spatio-temporal study in a noisy environment, J. Biosci. , 30(5) (2005), 749-760
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M. Scheffer, Fish and nutrients interplay determines algal biomass, Oikos, 62 (1991), 271-282
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J. Truscott, J. Brindley , Ocean plankton populations as excitable media, Bull. Math. Biol., 56 (1994), 981-998
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S. Wiggins, Interoduction to Applied non linear Dyanamical System and Chaos, Springer Verlag, New york (1990)
##[15]
Li Zhan, Shun Zhisheng, Ke Wang , Persistence and extinction of single population in a polluted environment, Electronic J. Differential Equations, 108 (2004), 1-5
]
POSITIVE SOLUTIONS OF FOURTH-ORDER BOUNDARY VALUE PROBLEM WITH VARIABLE PARAMETERS
POSITIVE SOLUTIONS OF FOURTH-ORDER BOUNDARY VALUE PROBLEM WITH VARIABLE PARAMETERS
en
en
By means of calculation of the fixed point index in cone we consider
the existence of one or two positive solutions for the fourth-order boundary
value problem with variable parameters
\[
\begin{cases}
u^{(4)}(t) + B(t)u''(t) - A(t)u(t) = f(t, u(t), u''(t)),\,\,\,\,\, 0 < t < 1,\\
u(0) = u(1) = u''(0) = u''(1) = 0,
\end{cases}
\]
where \(A(t),B(t) \in C[0, 1]\) and \(f(t, u, v) : [0, 1]\times [0,\infty)\times R \rightarrow [0,\infty)\) is continuous.
21
30
XIN
DONG
College of Information Science and Engineering,Shandong University of Science and Technology, Qing Dao, 266510, P. R. China.
ZHANBING
BAI
College of Information Science and Engineering,Shandong University of Science and Technology, Qing Dao, 266510, P. R. China.
Boundary value problem
positive solution
fixed point
cone.
Article.4.pdf
[
[1]
R. P. Agarwal , On fourth-order boundary value problems arising in beam analysis, Diff. Inte. Eqns. , 2 (1989), 91-110
##[2]
Z. B. Bai, The method of lower and upper solution for a bending of an elastic beam equation, J. Math. Anal., 248 (2000), 195-202
##[3]
Z. B. Bai, Positive solutions for some second-order four-point boundary value problems, J. Math. Anal. Appl., 2 330 (2007), 34-50
##[4]
Z. B. Bai, The method of lower and upper solutions for some fourth-order boundary value problems, Nonlinear Anal. , 67 (2007), 1704-1709
##[5]
G. Q. Chai, Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear Anal. , 66 (2007), 870-880
##[6]
D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New york (1988)
##[7]
Y. X. Li , Existence and method of lower and upper solutions for fourth-order nonlinear boundary value problems, Acta Mathmatic Scientia , 23 (2003), 245-252
##[8]
Y. X. Li, Existence and multiplicity of positive solutions for fourth-order boundary value problems, Acta Mathematicae Applicatae Sinica, 26 (2003), 109-116
##[9]
Y. X. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477-484
##[10]
R. Y. Ma, H. Y. Wang, Positive solutions of nonlinear three-point boundary-value problems, Nonlinear Anal., 279 (2003), 216-227
##[11]
Z. L. Wei, C. C. Pang , Positive solutions and multiplicity of fourth-order m-point boundary value problem with two parameters, Nonlinear Anal., 67 (2007), 1586-1598
]
COMMON FIXED POINT THEOREM IN PROBABILISTIC QUASI-METRIC SPACES
COMMON FIXED POINT THEOREM IN PROBABILISTIC QUASI-METRIC SPACES
en
en
In this paper, we consider complete probabilistic quasi-metric
space and prove a common fixed point theorem for R-weakly commuting maps
in this space.
31
35
A.R.
SHABANI
Department of Mathematics, Imam Khomaini Mritime University of Nowshahr Nowshahr, Iran
S.
GHASEMPOUR
Department of Mathematics, Payam noor University, Amol, Iran
Probabilistic metric spaces
quasi-metric spaces
fixed point theorem
R-weakly commuting maps
triangle function.
Article.5.pdf
[
[1]
I. Beg , Fixed points of contractive mappings on probabilistic Banach spaces, Stoch. Anal. Appl., 19 (2001), 455-460
##[2]
S. S. Chang, B. S. Lee, Y. J. Cho, Y. Q. Chen, S. M. Kang, J. S. Jung, Generalized contraction mapping principles and differential equations in probabilistic metric spaces, Proc. Amer. Math. Soc. , 124 (1996), 2367-2376
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I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasipseudo- metric spaces, Monatsh. Math. , 93 (1982), 127-140
##[4]
B. Schweizer, A. Sklar , Probabilistic Metric Spaces, Elsevier North Holand, New York (1983)
##[5]
B. Schweizer, H. Sherwood, R. M. Tardiff , Contractions on PM-space examples and counterexamples, Stochastica, 1 (1988), 5-17
##[6]
D. A. Sibley , A metric for weak convergence of distribution functions, Rocky Mountain J. Math. , 1 (1971), 427-430
]
MULTIPLE POSITIVE SOLUTIONS FOR NONLINEAR SINGULAR THIRD-ORDER BOUNDARY VALUE PROBLEM IN ABSTRACT SPACES
MULTIPLE POSITIVE SOLUTIONS FOR NONLINEAR SINGULAR THIRD-ORDER BOUNDARY VALUE PROBLEM IN ABSTRACT SPACES
en
en
In this paper, we study the nonlinear singular boundary value
problem in abstract spaces:
\[
\begin{cases}
u''' + f(t, u) = \theta,\,\,\,\,\, t \in (0, 1),\\
u(0) = u'(0) = \theta, u'(1) = \xi u'(\eta),
\end{cases}
\]
where \(0 < \eta< 1\) and \(1 < \xi<\frac{1}{\eta}, \theta\)
denotes the zero element of \(E, E\) is a real
Banach space, and \(f(t, u)\) is allowed to be singular at both end point \(t = 0\) and
\(t = 1\). We show the existence of at least two positive solutions of this problem.
36
44
FANG
ZHANG
School of Mathematics and Physics, Jiangsu Polytechnic University, Changzhou, 213164, PR China.
Singular boundary value problem
Abstract spaces
Positive solutions
Fixed point theorem.
Article.6.pdf
[
[1]
M. Gregus , Third Order Linear Differential Equations, in: Math. Appl., Reidel, Dordrecht (1987)
##[2]
G. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems, J. Diff. Equs., 10 (1971), 529-537
##[3]
L. K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations , J. Diff. Equs. , 13 (1993), 432-437
##[4]
D. J. ORegan , Topological transversality: Application to third order boundary value problems, SIAM J. Math. Anal., 19 (1987), 630-641
##[5]
A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl. , 185 (1994), 302-320
##[6]
A. Cabada , The method of lower and upper solutions for third order periodic boundary value problems, J. Math. Anal. Appl. , 195 (1995), 568-589
##[7]
A. Cabada, S. Lois , Existence of solution for discontinuous third order boundary value problems, J. Comput. Appl. Math., 110 (1999), 105-114
##[8]
A. Cabada, S. Heikkil’a, Extremality and comparison results for third order functional initial-boundary value problems, J. Math. Anal. Appl. , 255 (2001), 195-212
##[9]
A. Cabada, S. Heikkil’a, Extremality and comparison results for discontinuous implicit third order functional initial-boundary value problems, Appl. Math. Comput., 140 (2003), 391-407
##[10]
Q. Yao, Solution and positive solution for a semilinear third-order two-point boundary value problem, Appl. Math. Lett. , 17 (2004), 1171-1175
##[11]
Y. Sun, Existence of positive solutions for nonlinear third-order three-point boundary value problem, J. Math. Anal. Appl. , 306 (2005), 589-603
##[12]
L. Guo, J. Sun, Y. Zhao , Existence of positive solutions for nonlinear third-order three-point boundary value problem , Nonlinear Anal., 68 (2008), 3151-3158
##[13]
K. Deimling , Ordinary differential equations in Banach spaces, LNM 886. , Berlin: Springer-Verlag, New York (1987)
##[14]
V. Lakshmikantham, S. Leela, Nonlinear differential equations in abstract spaces, Pergamon, Oxford (1981)
##[15]
D. Guo, V. Lakshmikantham , Nonlinear Problems in Abstract Cones, Academic Press , Boston, MA (1988)
##[16]
D. Guo, V. Lakshmikantham, X. Liu , Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, (1996)
]
KANNAN FIXED POINT THEOREM ON GENERALIZED METRIC SPACES
KANNAN FIXED POINT THEOREM ON GENERALIZED METRIC SPACES
en
en
We obtain sufficient conditions for existence of unique fixed point
of Kannan type mappings defined on a generalized metric space.
45
48
AKBAR
AZAM
Department of Mathematics, F.G. Postgraduate College, Islamabad, Pakistan
MUHAMMAD
ARSHAD
Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, Pakistan
Fixed point
contractive type mapping
generalized metric space
Article.7.pdf
[
[1]
A. Branciari , A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 1-2 (2000), 31-37
##[2]
K. Goebel, W. A. Kirk , Topics in metric fixed point theory, Cambridge University Press, Cambridge (1990)
##[3]
R. Kannan, Some results on fixed points, Bull. Calcutta. Math.Soc., 60 (1968), 71-76
##[4]
B. E. Rhoads , A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 26 (1977), 257-290
]
NONTRIVIAL SOLUTIONS OF SINGULAR SECOND ORDER THERE-POINT BOUNDARY VALUE PROBLEM AT RESONANCEN
NONTRIVIAL SOLUTIONS OF SINGULAR SECOND ORDER THERE-POINT BOUNDARY VALUE PROBLEM AT RESONANCEN
en
en
The singular second order three-point boundary value problem at
resonance
\[
\begin{cases}
x''(t) = f(t, x(t)),\,\,\,\,\, 0 < t < 1,\\
x'(0) = 0, x(\eta) = x(1),
\end{cases}
\]
are considered under some conditions concerning the first eigenvalues corresponding
to the relevant linear operators, where \(\eta\in (0, 1)\) is a constant, \(f\)
is allowed to be singular at both \(t = 0\) and \(t = 1\). The existence results of
nontrivial solutions are given by means of the topological degree theory.
49
55
XIAORONG
WU
Department of Mathematics and Physics, Taizhou Teachers College, Taizhou, 225300, China.
FENG
WANG
School of Mathematics and Physics, Jiangsu Polytechnic University, Changzhou, 213164, China.
Singular
Nontrivial solutions
Boundary value problem
Topology degree
Resonance.
Article.8.pdf
[
[1]
W. Feng, JRL. Webb , Solvability of a three-point nonlinear BVPs at resonance, Nonlinear Anal. , 30 (1997), 3227-3238
##[2]
R. Ma , Multiplicity results for a three-point boundary value problems at resonance, Nonlinear Anal. , 53 (2003), 777-789
##[3]
Z. Bai, W. Li, W. Ge , Existence and multiplicity of solutions for four-point BVPs at resonance, Nonlinear Anal., 60 (2005), 1551-1562
##[4]
X. Han , Positive solutions for a three-point boundary value problems at resonance, J. Math. Anal. Appl. , 336 (2007), 556-568
##[5]
C. Bai, J. Fang, Existence of positive solutions for three-point boundary value problems at resonance, J. Math. Anal. Appl., 291 (2004), 538-549
##[6]
G. Infante, M. Zima , Positive solutions of multi-point boundary value problems at resonance , Nonlinear Anal. , 69 (2008), 2458-2465
##[7]
J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, , Berlin (1979)
##[8]
G. Zhang, J. Sun, Positive solutions of m-point boundary value problems, J. Math. Anal. Appl., 291 (2004), 406-418
##[9]
D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego (1988)
##[10]
D. Guo, J. Sun, Nonlinear Integral Equations, Shandong Press of Science and Technology, Jinan, (in Chinese) (1987 )
##[11]
Deimling Klaus, Nonlinear Functional Analysis, Springer-Verlag, New York (1985)
]