ORIGINAL_ARTICLE
Global existence of solutions for an m-component reaction-diffusion system with a tridiagonal 2-Toeplitz diffusion matrix and polynomially growing reaction terms
This paper is concerned with the local and global existence of solutions for a generalized m-componenta reaction-diffusion system with a tridiagonal 2-Toeplitz diffusion matrix and polynomial growth. We derivethe eigenvalues and eigenvectors and determine the parabolicity conditions in order to diagonalize theproposed system. We, then, determine the invariant regions and utilize a Lyapunov functional to establishthe global existence of solutions for the proposed system. A numerical example is used to illustrate andconrm the findings of the study.
https://www.cna-journal.com/article_89518_1b7526e07a8e04529e0d1c831bbb8163.pdf
2017-11-01
1
14
Reaction-diffusion systems
invariant regions
diagonalization
global existence
Lyapunov functional
Salem
Abdelmalek
salem.abdelmalek@univ-tebessa.dz
1
Department of mathematics, University of Tebessa 12002 Algeria.
LEAD_AUTHOR
Samir
Bendoukha
sbendoukha@taibahu.edu.sa
2
Electrical Engineering Department, College of Engineering at Yanbu, Taibah University, Saudi Arabia.
AUTHOR
[1] S. Abdelmalek, Existence of global solutions via invariant regions for a generalized reaction-diffusion system with
1
a tridiagonal Toeplitz matrix of diffusion coefficients, accepted for publication in Funct. Anal. Theory Method
2
[2] M. Andelic, C. M. da Fonseca, Sufficient conditions for positive definiteness of tridiagonal matrices revisited,
3
Positivity, 15 (2011), 155-159.
4
[3] A. Friedman, Partial Differential Equations of Parabolic Type. Prentice Hall Englewood Chis. N. J., (1964).
5
[4] M. J. C. Gover, The eigenproblem of a tridiagonal 2- Toeplitz matrix, Linear Algebra, and its Apps., 197 (1994),
6
[5] D. Henry, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Springer-Verlag,
7
New-York, 1984.
8
[6] S. L. Hollis, J. J. Morgan, On the blow-up of solutions to some semilinear and quasilinear reaction-diffusion
9
systems, Rocky Mountain J. Math., 14 (1994), 1447-1465.
10
[7] S. Kouachi, B. Rebiai, Invariant regions and the global existence for reaction-diffusion systems with a tridiagonal
11
matrix of diffusion coefficients, Memoirs on Diff. Eqs. and Math. Phys., 51 (2010), 93-108.
12
[8] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences
13
44, Springer-Verlag, New York (1983).
14
ORIGINAL_ARTICLE
Existence results for a coupled systems of Chandrasekhar quadratic integral equations
In this article, we study a coupled systems of generalized Chandrasekhar quadratic integral equations,which is widely applicable in various disciplines of science and technology. By using the contraction mappingprinciple and successive approximation, we develop suffcient conditions for existence and uniqueness ofthe solution. Also, an example is provided to illustrate our main results.
https://www.cna-journal.com/article_89746_9ba22fb87047d31efabf168f6246b9cc.pdf
2017-11-01
15
22
Chandrasekhar quadratic integral equations
coupled system
contraction mapping principle
successive approximation method
Fazal
Haq
fazalhaqphd@gmail.com
1
Department of Mathematics, Hazara University Mansehra, Khyber Pakhtunkhwa, Pakistan.
LEAD_AUTHOR
Kamal
Shah
kamalshah408@gmail.com
2
Department of Mathematics, University of Malakand Dir(L), Khyber Pakhtunkhwa, Pakistan.
AUTHOR
Ghaus
UR-Rahman)
dr.ghaus@uswat.edu.pk
3
Department of Mathematics, University of Swat, Khyber Pakhtunkhwa, Pakistan.
AUTHOR
Muhammad
Shahzad
shahzad-maths@hu.edu.pk
4
Department of Mathematics, Hazara University Mansehra, Khyber Pakhtunkhwa, Pakistan.
AUTHOR
[1] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with
1
threepoint boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843.
2
[2] I. K. Argyros, Quadratic equations and applications to Chandrasekhars and related equations, Bull. Austral. Math.
3
Soc., 32 (1985), 275-292.
4
[3] J. Banas, B. Rzepka, Monotonic solutions of a quadratic integral equations of fractional order, J. Math. Anal.
5
Appl., 332 (2007), 1370-1378.
6
[4] J. Banas, A. Martinon, Monotonic solutions of a quadratic integral equation of volterra type, Comput. Math.
7
Appl., 47 (2004), 271-279.
8
[5] J. Banas, J. Caballero, J. Rocha, K. Sadarangani, Monotonic solutions of a class of quadratic integral equations
9
of volterra type, Comput. Math. Appl., 49(2005), 943-952.
10
[6] J. Banas, J. Rocha Martin, K. Sadarangani, On the solution of a quadratic integral equation of Hammerstein
11
type, Math. Comput. Modelling, 43 (2006), 97-104.
12
[7] J. Banas, B. Rzepka, Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. Comput.
13
Modeling, 49 (2009), 488-496.
14
[8] C. Bai, J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential
15
equations, Appl. Math. Comput., 150, (2004) 611-621.
16
[9] J. Caballero, A. B. Mingarelli, K. Sadarangani, Existence of solutions of an integral equation of chandrasekhar
17
type in the theory of radiative, Electron. J. Di. Equns., 57 (2006), 1-11.
18
[10] S. Chandrasekhar, Radiative transfer, Courier Corporation, USA, (1960).
19
[11] Y. Chen, H. An, Numerical solutions of coupled Burgers equations with time and space fractional derivatives,
20
Appl. Math. Comput., 200 (2008), 87-95.
21
[12] W. G. El-Sayed, B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comput.
22
Math. Appl., 67 (51) (2006), 1065-1074.
23
[13] A. M. A. El-Sayed, H. H. G. Hashem, E. A. A. Ziada, Picard and Adomian methods for coupled systems of
24
quadratic integral equations of fractional order, J. Nonlinear Anal. Optim., 3 (2) (2012), 171-183.
25
[14] A. M. A. El-Sayed, H. H. G. Hashem, Existence results for coupled systems of quadratic integral equations of
26
fractional orders, Optim. Lett., 7 (2013), 1251-1260.
27
[15] V. Gaychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J.
28
Math. Appl., 220,(2008), 215-225. 1
29
[16] V. D. Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal.
30
Appl., 302 (2005), 56-64.
31
[17] V. Gaychuk, B. Datsko, V. Meleshko, D. Blackmore, Analysis of the solutions of coupled nonlinear fractional
32
reaction-diffusion equations, Chaos Solitons, and Fract., 41 (2009), 1095-1104.
33
[18] H. A. H. Salem, On the quadratic integral equations and their applications, Comput. Math. Appl., 62 (2011),
34
2931-2943.
35
[19] H. H. G. Hashim, On successive approximation method for coupled systems of Chandrasekhar quadratic integral
36
equations, J. Egyptian Math. Soc., 23 (2015), 108-112.
37
[20] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math.
38
Lett., 22 (2009), 64-69.
39
ORIGINAL_ARTICLE
Some Gamidov like integral inequalities on time scales and applications
In the present paper, we establish some Gamidov like integral inequalities on time scales, the obtainedresults can be used as tools for the study of certain qualitative properties of solutions for differential anddifference equations.
https://www.cna-journal.com/article_89747_beaa5f8791cc7ff79ad61632e1b21644.pdf
2017-11-01
23
33
Dynamic equations
time scale
integral inequality
Badreddine
Meftah
badrimeftah@yahoo.fr
1
Laboratoire des telecommunications, Faculte des Sciences et de la Technologie, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria.
LEAD_AUTHOR
[1] A. Abdeldaim, M. Yakout, On some new integral inequalities of Gronwall-Bellman-Pachpatte type, Appl. Math. Comput.,
1
217 (2011), 7887-7899.
2
[2] R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl., 4 (2001), 535-557.
3
[3] D. R. Anderson, Time-scale integral inequalities. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), 15 pages.
4
[4] D. Bainov, P. Simeonov, Integral inequalities and applications, Kluwer Academic Publishers Group, Dordrecht, (1992).
5
[5] P. R. Beesack, Gronwall inequalities, Carleton University, Ottawa, (1975).
6
[6] M. Bohner, A. Peterson, Dynamic equations on time scales. An introduction with applications, Birkhauser Boston, Inc., Boston, MA, (2001).
7
[7] K. Cheng, C. Guo, M. Tang, Some nonlinear Gronwall-Bellman-Gamidov integral inequalities and their weakly singular
8
analogues with applications, Abstr. Appl. Anal., 2014 (2014), 9 pages.
9
[8] H. El-Owaidy, A. Ragab, A. Abdeldaim, On some new integral inequalities of Growall-Bellman [Gronwall-Bellman] type,
10
Appl. Math. Comput., 106 (1999), 289-303.
11
[9] J. Gu, F. Meng, Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math.
12
Comput., 245 (2014), 235-242.
13
[10] S. Hilger, Analysis on measure chains|a unied approach to continuous and discrete calculus, Results Math., 18 (1990),18-56.
14
[11] Y. Huang, W.-S. Wang, Y. Huang, A class of Volterra-Fredholm type weakly singular difference inequalities with power
15
functions and their applications, J. Appl. Math., 2014 (2014), 9 pages.
16
[12] F. Jiang, F. Meng, Explicit bounds on some new nonlinear integral inequalities with delay, J. Comput. Appl. Math., 205
17
(2007), 479-486.
18
[13] W. N. Li, Some new dynamic inequalities on time scales, J. Math. Anal. Appl., 319 (2006), 802-814.
19
[14] O. Lipovan, A retarded Gronwall-like inequality, and its applications, J. Math. Anal. Appl. 252 (2000), 389-401.
20
[15] F. W. Meng, W. N. Li, On some new integral inequalities and their applications, Appl. Math. Comput., 148 (2004),
21
[16] F. Meng, J. Shao, Some new Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 223 (2013), 444-451.
22
[17] B. G. Pachpatte, Inequalities for differential and integral equations, Mathematics in Science and Engineering, 197. Academic Press, Inc., San Diego, CA, (1998).
23
[18] B. G. Pachpatte, Explicit bounds on Gamidov type integral inequalities, Tamkang J. Math., 37 (2006), 1-9.
24
[19] D. B. Pachpatte, Estimates of certain integral inequalities on time scales, J. Math., 2013 (2013), 5 pages.
25
[20] Y. Tian, Y. Cai, L. Li and T. Li, Some dynamic integral inequalities with mixed nonlinearities on time scales, J. Inequal.
26
Appl., 2015 (2015), 10 pages.
27
[21] Y. Tian, M. Fan, Y. Sun, Certain nonlinear integral inequalities and their applications, Discrete Dyn. Nat. Soc., 2017
28
(2017), 8 pages.
29
ORIGINAL_ARTICLE
Existence of positive solutions to a coupled system with threepoint boundary conditions via degree theory
In this paper, we study the existence of solutions of nonlinear fractional hybrid differential equations. Byusing the topological degree theory, some results on the existence of solutions are obtained. The results aredemonstrated by a proper example.
https://www.cna-journal.com/article_89748_41f1eed2eb95b25a10bd51f449e6ab53.pdf
2017-11-01
34
43
coupled system
Fractional derivative
green function
growth condition
topological degree theory
fixed-point theorem
S.
Samina
saminakhanqau@yahoo.com
1
Department of Mathematics, University of Malakand Dir(L), Khyber Pakhtunkhwa, Pakistan.
LEAD_AUTHOR
Kamal
Shah
kamalshah408@gmail.com
2
Department of Mathematics, University of Malakand Dir(L), Khyber Pakhtunkhwa, Pakistan.
AUTHOR
Rahmat
Ali Khan
rahmat_alipk@yahoo.com
3
Department of Mathematics, University of Malakand Dir(L), Khyber Pakhtunkhwa, Pakistan.
AUTHOR
[1] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear dierential equations and inclusions
1
involving Riemann-Liouville fractional derivative, Adv. Difference Equ., 2009 (2009), 47 pages.
2
[2] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with
3
three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843.
4
[3] B. Ahmad, J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional
5
differential equations via Leray-Schauder degree theory, Topol. Methods Nonlinear Anal., 35 (2010), 295-304.
6
[4] T. S. Aleroev, The Sturm-Liouville problem for a second-order ordinary differential equation with fractional
7
derivatives in the lower terms, Differentsial'nye Uravneniya, 18 (1982), 341-342.
8
[5] Z. B. Bai, H. S. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation,
9
J. Math. Anal. Appl., 311 (2005), 495-505.
10
[6] D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, NY, USA, (2012).
11
[7] M. Benchohra, J. R. Graef and S. Hamani, Existence results for boundary value problems with nonlinear fractional
12
differential equations, Appl. Anal., 87 (2008), 851-863.
13
[8] L. Cai, J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence rate, Chaos Solitons Fractals,
14
41 (2009), 175-182.
15
[9] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, (1985).
16
[10] K. Diethelm, The Analysis of Fractional Differential Equations, in Lecture Notes in Mathematics, Springer-Verlag,
17
(2010).
18
[11] V. Gaychuk, B. Datsko, V. Meleshko and D. Blackmore , Analysis of the solutions of coupled nonlinear fractional
19
reaction-diffusion equations, Chaos Solitons Fractals, 41 (2009), 1095-1104.
20
[12] F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian., 75 (2006),
21
[13] R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value
22
problems, Commun. Appl. Anal., 19 (2015), 515-526.
23
[14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,
24
Elsevier Science B.V., Amsterdam, (2006).
25
[15] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge
26
Scientic Publishers, (2009).
27
[16] L. Lv, J. Wang and W. Wei, Existence and uniqueness results for fractional differential equations with boundary
28
value conditions, Opuscul. Math., 31 (2011), 629-643.
29
[17] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley and
30
Sons, New York, USA, (1993).
31
[18] M. W. Michalski, Derivatives of Noninteger Order and their Applications: Master dissertation Polish Acad. Sci.,
32
(1993).
33
[19] I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
34
[20] M. Rehman, R. A. Khan, Existence and uniqueness of solutions for multipoint boundary value problems for
35
fractional differential equations, Appl. Math. Lett., 23 (2010), 1038-1044.
36
[21] K. Shah, A. Ali and R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-point
37
boundary value problems, Bound. Value Probl., 2016 (2016), 12 pages.
38
[22] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math.
39
Lett., 22 (2009), 64-69.
40
[23] K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional
41
order differential equations with anti-periodic boundary conditions, Dier. Equ. Appl. 7 (2015), 245-262.
42
[24] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and
43
Media, Springer, (2011).
44
[25] J. Wang, Y. Zhou and W. Wei, Study in fractional differential equations by means of topological degree methods,
45
Numer. Funct. Anal. Optim., 33 (2012), 216-238.
46
[26] J. Wang, H. Xiang and Z. Liu , Positive solution to non zero boundary values problem for a coupled system of
47
nonlinear fractional differential equations, Int. J. Diff. Equ., 2010 (2010), 12 pages.
48
[27] A. Yang, W. Ge , Positive solutions of multi-point boundary value problems of nonlinear fractional differential
49
equation at resonance, J. Korea Sco. Math.,16 (2009), 181-193.
50
[28] W. Yang , Positive solution to non zero boundary values problem for a coupled system of nonlinear fractional
51
differential equations, Comput. Math. Appl., 63 (2012), 288-297.
52
[29] W. Yang, Positive solution to non zero boundary values problem for a coupled system of nonlinear fractional
53
differential equations, Comput. Math. Appl., 63 (2012), 288-297.
54
[30] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation,
55
Comput. Math. Appl., 59 (2010), 1300-1309.
56
[31] S. Q. Zhang, Existence of solution for a boundary value problem of fractional order, Acta Math. Sci., 26B (2006),
57
[32] S. Q. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron.
58
J. Differential Equations, 36 (2006), 1-12.
59
[33] E. Zeidler, Nonlinear Functional Analysis an Its Applications, Springer, NewYork (1986).
60