ORIGINAL_ARTICLE
Transformation to a fixed domain in LP modelling for a class of optimal shape design problems
A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.
https://ijnao.um.ac.ir/article_24736_778b8f32dcb1d6def0bc85c6d947f67f.pdf
2019-03-01
1
16
10.22067/ijnao.v9i1.53910
Approximation
optimal shape design
Linear programming
measure theory
H.H.
Mehne
hmehne@ari.ac.ir
1
Aerospace Research Institute, Tehran,
LEAD_AUTHOR
M. H.
Farahi
farahi@math.um.ac.ir
2
Ferdowsi University of Mashhad, Mashhad, Iran.
AUTHOR
References
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ORIGINAL_ARTICLE
Approximation of the Huxley equation with nonstandard finite-difference scheme
In this paper, an explicit exact finite-difference scheme for the Huxley equation is presented based on the nonstandard finite-difference (NSFD) scheme. Afterwards, an NSFD scheme is proposed for the numerical solution of the Huxley equation. The positivity and boundedness of the scheme is discussed. It is shown through analysis that the proposed scheme is consistent, stable, and convergence. The numerical results obtained by the NSFD scheme is compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD scheme.
https://ijnao.um.ac.ir/article_24741_907d5ba0db6001acea655faa0bb7141d.pdf
2019-03-01
17
35
10.22067/ijnao.v9i1.61735
The Huxley equation
Nonstandard finite-difference scheme
Positivity and boundedness
Consistency
Stability
Convergence
M.
Namjoo
namjoo@vru.ac.ir
1
Department of Mathematics, Vali{e{Asr University of Rafsanjan, Rafsanjan, Iran.
LEAD_AUTHOR
S.
Zibaei
z.sadegh133@gmail.com
2
Department of Mathematics, Vali{e{Asr University of Rafsanjan, Rafsanjan, Iran.
AUTHOR
1. Batiha, B., Noorani, M.S.M. and Hashim, I. Application of variational it eration method to the generalized Burgers–Huxley equation, Chaos Solitons Fractals, 36(3) (2008), 660–663.
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2. Batiha, B., Noorani, M.S.M. and Hashim, I. Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Appl. Math. Comput. 186(2) (2007), 1322–1325.
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3. Biazar, J. and Mohammadi, F. Application of differential transform method to the generalized Burgers–Huxley equation, Appl. Appl. Math. 5(10) (2010), 1726–1740.
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4. Erdogan, U. and Ozis, T. A smart nonstandard finite difference scheme for second order nonlinear boundary value problems, J. Comput. Phys. 230(17) (2011), 6464–6474.
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5. Gonzalez–Parra, G., Arenas, A.J. and Chen–Charpentier, B.M. Combination of nonstandard schemes and Richardsons extrapolation to improve the numerical solution of population models, Math. Comput. Modelling, 52(7-8) (2010), 1030–1036.
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8. Hashim, I., Noorani, M.S.M. and Said Al–Hadidi, M.R. Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Modelling, 43(11-12) (2006),1404–1411.
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15. Namjoo, M. and Zibaei, S. Numerical solutions of FitzHugh–Nagumoequation by exact finite-difference and NSFD schemes, Comp. Appl. Math., (2016) 1–17.
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19. Zeinadini, M. and Namjoo, M. A Numerical Method for Discrete Fractional–Order Chemostat Model Derived from Nonstandard NumericalScheme, Bull. Iranian Math. Soc., (2016), in press.
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20. Zibaei, S. and Namjoo, M. A Nonstandard Finite Difference Scheme for Solving Fractional–Order Model of HIV–1 Infection of CD 4+ T–cells,Iran. J. Math. Chem., vol 6(2) (2015), 145–160.
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21. Zibaei, S. and Namjoo, M. A Nonstandard Finite Difference Scheme for Solving Three–Species Food Chain with Fractional–Order Lotka–Volterra Model, Iran. J. Numer. Anal. Optim., vol 6(1) (2016), 53–78.
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22. Zibaei, S. and Namjoo, M. A NSFD scheme for Lotka–Volterra food web model, Iran. J. Sci. Technol. Trans. A Sci., vol 38(4) (2014), 399–414.
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24
ORIGINAL_ARTICLE
An operational approach for solving fractional pantograph differential equation
The aim of the current paper is to construct the shifted fractional-order Jacobi functions (SFJFs) based on the Jacobi polynomials to numerically solve the fractional-order pantograph differential equations. To achieve this purpose, first the operational matrices of integration, product, and pantograph, related to the fractional-order basis, are derived (operational matrix of integration is derived in Riemann–Liouville fractional sense). Then, these matrices are utilized to reduce the main problem to a set of algebraic equations. Finally, the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some theorems are presented on existence of solution of the problem under study and convergence of our method.
https://ijnao.um.ac.ir/article_24747_db1042259b54988756efed54c7331caa.pdf
2019-03-01
37
68
10.22067/ijnao.v9i1.66730
Fractional pantograph differential equation
Fractional-order Jacobi functions
Operational matrices
Caputo derivative
Riemann-Liouville integral
H.
Ebrahimi
h.ebrahimi@iaurasht.ac.ir
1
Rasht branch, Islamic Azad University, Rasht, Iran,
LEAD_AUTHOR
K.
Sadri
kh.sadri@uma.ac.ir
2
Rasht Branch, Islamic Azad University, Rasht, Iran.
AUTHOR
1. Bagley, R.L. and Torvik, P.J. Fractional calculus in the transient analysis of viscoelastically damped structures, J. AIAA., 23 (1985), 918–925.
1
2. Bhrawy, A.H. and Zaky, M.A. Shifted fractional–order Jacobi orthogonal functions: Application to a system of fractional differential equations,Appl. Math. Model. 40 (2016), 832–845.
2
3. Borhanifar, A. and Sadri, K. A new operational approach for numerical solution of generalized functional integro-differential equations, J. Comput. Appl. Math. 279 (2015), 80–96.
3
4. Dehghan, M., Hamedi, E. A. and Khosravian-Arab, H. A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials, J.Vib. .Control, 22 (6) (2014), 1547–1559.
4
5. Doha, E.H., Bhrawy, A.H., and Ezz-Eldien, S.S. A new Jacobi operational matrix: An application for solving fractional differential equations, Appl. Math. Model. 36 (10) (2012), 4931–4943.
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6. Duan, B., Zheng, Z. and Cao, W. Spectral approximation methods and error estimates for Caputo fractional derivative with applications to initial value problems, J. Comput. Phys. 319 (2016), 108–128.
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18. Rahimkhani, P., Ordokhani, Y. and Babolian, E. Numerical solution of fractional pantograph differential equations by using generalized fractionalorder Bernoulli wavelet, J. Comput. Appl. Math.309 (2017), 493–510.
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20. Saeed, U. and Rehman, M. Hermite wavelet method for fractional delay differential equations, J. Difference Equ., doi: 10.1155/2014/359093, (2014), 8 pp.
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21. Sedaghat, S., Ordokhani, Y., and Dehghan, M. Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4815–4830.
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22. Szego, G. Orthogonal polynomials, American Mathematical Society. Providence, Rhodes Island, 1939.
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23. Tohidi, E., Bhrawy, A.H. and Erfani, K.A. Collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model. 37 (2012), 4283–4294.
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24. Yang, Y. and Huang, Y. Spectral-collocation methods for fractional pantograph delay integro-differential equations, Adv. Math. Phys., doi: 10.1155/2013/821327, (2013) 14 pp.
24
25. Yousefi, S.A. and Lotfi, A. Legendre multiwavelet collocation method for solving the linear fractional time delay systems, Cent. Eur. J. Phys. 11(10) (2013), 1463–1469.
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26
27. Yuzbasi, S. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials, Appl. Math. Comput. 219 (2013), 6328–6343.
27
ORIGINAL_ARTICLE
The network 1-median problem with discrete demand weights and traveling times
In this paper, the 1-median location problem on an undirected network with discrete random demand weights and traveling times is investigated. The objective function is to maximize the probability that the expected sum of weighted distances from the existing nodes to the selected median does not exceed a prespecified given threshold. An analytical algorithm is proposed to get the optimal solution in small-sized networks. Then, by using the centrallimit theorem, the problem is studied in large-sized networks and reduced to a nonlinear problem. The numerical examples are given to illustrate the efficiency of the proposed methods.
https://ijnao.um.ac.ir/article_24768_d4a28f72aa5c65dc11a7dd8078be1eff.pdf
2019-03-01
69
92
10.22067/ijnao.v9i1.68851
Facility location, 1-median problem
probabilistic weights
probabilistic traveling times
M.
Abareshi
abareshi66@gmail.com
1
Hakim Sabzevari University, Sabzevar, Iran
AUTHOR
M.
Zaferanieh
m.zaferanieh@hsu.ac.ir
2
Hakim Sabzevari University, Sabzevar, Iran.
LEAD_AUTHOR
1. Berman, O., Hajizadeh, I. and Krass, D. The maximum covering problem with travel time uncertainty, IIE Transactions, 45 (2013), 81–96.
1
2. Berman, O. and Wang, J. Probabilistic location problems with discrete demand weights, Networks, 44 (2004), 47–57.
2
3. Berman, O. and Wang, J. The network p-median problem with discrete probabilistic demand weights, Computers and Operations Research, 37(2010), 1455–1463.
3
4. Bieniek, M. A note on the facility location problem with stochastic demands, Omega, 55 (2015), 53–60.
4
5. Frank, H. Optimum locations on a graph with probabilistic demands, Operations Research, 14 (1966), 409–421.
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6. Frank, H. Optimum locations on graphs with correlated normal demands, Operations Research, 15 (1967), 552–557.
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7. Gavish, B. and Sridhar, S. Computing the 2-median on tree networks in O(n log n) time, Networks, 26 (1995), 305–317.
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8. Gray, M.R. and Johnson D.S. Computers and intractability: A guide to the theory of Np-completeness W. H. Freeman. New York, 1979.
8
9. Goldman, A.J. Optimal center location in simple networks, Transportation Science, 5 (1971), 212–221.
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10. Hakimi, S.L. Optimal locations of switching centers and the absolute centers and medians of a graph, Operations Research, 12 (1964), 450–459.
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11. Handler, G.Y. and Mirchandani, P.B. Location on networks: theory and algorithm, The MIT Press, Cambridge, 1979.
11
12. Kariv, O. and Hakimi, S.L. An algorithmic approach to network location problems. Part II: p-medians, SIAM Journal on Applied Mathematics, 37 (1979), 539–560.
12
13. Mirchandani, P.B. and Odoni, A.R. Location of medians on stochastic networks, Transportation Science, 13 (1979), 85–97.
13
14. Minieka, E. Anticenters and Antimedians of a Network, Networks, 13(1983), 359–365.
14
15. Nocedal, J. and Wright, S.J. Numerical Optimization, Springer-Verlag, New York, 2006.
15
16. Puga, M.S and Tancrez, J.S. A heuristic algorithm for solving large location-inventory problems with demand uncertainty, European Journal of Operational Research, 259 (2017), 413–423.
16
17. Stancu-Minasian, I.M. Fractional Programming: Theory, Methods and Applications, Springer, Netherlands, 1997.
17
18. Tamir, A. An O(pn2) algorithm for the p-median and related problems on tree graphs, Operations Research Letters, 19 (1996), 59–64.
18
ORIGINAL_ARTICLE
On the finding 2-(k,l)-core of a tree with arbitrary real weight
Let T = (V, E) be a tree with | V |= n. A 2-(k, l)-core of T is two subtrees with at most k leaves and with a diameter of at most l, which the sum of the distances from all vertices to these subtrees is minimized. In this paper, we first investigate the problem of finding 2-(k, l)-core on an unweighted tree and show that there exists a solution that none of (k, l)-cores is a vertex. Also in the case that the sum of the weights of vertices is negative, we show that one of (k, l)-cores is a single vertex. Then an algorithm for finding the 2-(k, l)-core of a tree with the pos/neg weight is presented.
https://ijnao.um.ac.ir/article_24787_186bfec924d7eddc3d86bc2cfda26488.pdf
2019-03-01
93
104
10.22067/ijnao.v9i1.65852
core
Facility location
Median subtree
Semi-obnoxious
S. M.
Ashkezari
samane.motevalli@gmail.com
1
Shahrood University of Technology, University Blvd., Shahrood, Iran.
AUTHOR
J.
Fathali
fathali@shahroodut.ac.ir
2
Shahrood University of Technology, University Blvd., Shahrood, Iran.
LEAD_AUTHOR
1. Becker R.I. Inductive algorithms on finite trees, Quaest Math., 13, (1990), 165–181.
1
2. Becker R.I., Lari I., Storchi G. and Scozzari A. Efficient algorithms for finding the (k, l)-core of tree networks, Networks, 40, (2002), 208–215.
2
3. Becker R.I. and Perl Y. Finding the two-core of a tree, Discrete Applied Mathematics, 11, (1985), 103–113.
3
4. Minieka E. and Patel N.H. On finding the core of a tree with a specified length, J. Alg., 4, (1983), 345–352.
4
5. Morgan C.A. and Slater P.J. A linear algorithm for a core of a tree, Journal of Algorithms, 1, (1980), 247–258.
5
6. Motevalli S. and Fathali J. A linear algorithm for finding core of weighted interval trees, Journal of Operational Research and Its Applications, 13, (2016), 101–111.
6
7. Motevalli S., Fathali J. and Zaferanieh M. An efficient algorithm for finding the semi-obnoxious (k,l)-core of a tree, Journal of Mathematical Modeling, 3, (2015), 129–144.
7
8. Peng S. and Lo W. Efficient algorithms for finding a core of a tree with a specified length, Journal of Algorithms 20, (1996), 445–458.
8
9. Peng S., Stephens A.B. and Yesha Y. Algorithms for a core and a k-tree core of a tree, J. Alg., 15, (1993), 143–159.
9
10. Rahbari M., Fthali J. and Mortazavi R. A hybrid algorithm for the path center problem, Global Analysis and Discrete Mathematics, 1, (2016), 83–92.
10
11. Shioura A. and Uno T. A linear time algorithm for finding a k-tree core, J. Alg., 23, (1997), 281–290.
11
12. Wang B.F. Finding a k-tree core and a k-tree center of a tree network in parallel, IEEE Transactions on Parallel and Distributed Systems, 9, (1998), 186–191.
12
13. Wang B.F. Finding a 2-core of a tree in linear time, SIAM Journal on Discrete Mathematics, 15, (2002), 193–210.
13
14. Wang Y., Wang D.Q., Liu W. and Tian B.Y. Efficient parallel algorithms for constructing a k-tree center and a k-tree core of a tree network, Lecture Notes in Computer Science 3827, Springer-Verlag Berlin Heidelberg, (2005), 553–562.
14
15. Wang Y. and Wang Y. Efficient algorithms for constructing a (k,l)-center and a (k,l)-core in a tree network, Fourth International Conference on Innovative Computing, Information and Control, (ICICIC), 2009.
15
16. Zaferanieh M. and Fathali J. Ant colony and simulated annealing algorithms for finding the core of a graph, World Applied Science Journal, 7, (2009), 1335–1341.
16
17. Zaferanieh M. and Fathali J. Finding a core of a tree with pos/neg weight, Math. Meth. Oper. Res., 76, (2012), 147–160.
17
18. Zelinka B. Medians and peripherians of trees, Archvum Mathematicum, 4, (1968), 87–95.
18
19. Zhou J., Kang L. and Shan E. Two paths location of a tree with positive or negative weights, Theoretical Computer Science, 607, (2015), 296–305.
19
ORIGINAL_ARTICLE
New S-ROCK methods for stochastic differential equations with commutative noise
The class of strong stochastic Runge–Kutta (SRK) methods for stochas tic differential equations with a commutative noise proposed by R¨ oßler (2010) is considered. Motivated by Komori and Burrage (2013), we design a class of explicit stochastic orthogonal Runge–Kutta Chebyshev (SROCKC2) meth ods of strong order one for the approximation of the solution of Itˆo SDEs with an m-dimensional commutative noise.The mean-square and asymptotic stability analysis of the newly proposed methods applied to a scalar linear test equation with a multiplicative noise is presented. Finally, some numer ical experiments for stochastic models arising in applications are given that confirm the theoretical discussion.
https://ijnao.um.ac.ir/article_24806_e91d1901ef10e00b53e9ef5e361e2a1b.pdf
2019-03-01
105
126
10.22067/ijnao.v9i1.69454
Stochastic differential equations
Runge-Kutta methods
Stochastic mean square stability
Stiff equations
Commutative noise
A.
Haghighi
a.haghighi@razi.ac.ir
1
Razi University, Kermanshah, Iran.
LEAD_AUTHOR
1. Abdulle, A. and Cirilli, S. S-ROCK: Chebyshev methods for stiff stochastic differential equations, SIAM J. Sci. Comput. 30(2) (2008), 997–1014.
1
2. Abdulle, A. and Li, T. S-ROCK methods for stiff Itˆo SDEs, Commun. Math. Sci. 6(4) (2008), 845–868.
2
3. Abdulle, A. and Medovikov, A. Second order Chebyshev methods based on orthogonal polynomials, Numer. Math. 90 (2001), 1–18.
3
4. Burrage, K., Burrage, P. and Tian, T. Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041) (2004), 373–402.
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5. Burrage, K. and Burrage, P.M. General order conditions for stochastic Runge–Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Appl. Numer. Math. 28 (1998), 161–177.
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6. Falsone, G. Stochastic differential calculus for Gaussian and non-Gaussian noises: A critical review, Commum. Nonlinear. Sci. 56 (2018), 198–216.
6
7. Haghighi, A. and Hosseini, S.M. A class of split-step balanced methods for stiff stochastic differential equations, Numer. Algorithms. 61 (2012), 141–162.
7
8. Haghighi, A., Hosseini, S.M. and R¨ oßler, A. Diagonally drift-implicit Runge–Kutta methods of strong order one for stiff stochastic differential systems, J. Comput. Appl. Math. 293 (2016), 82–93.
8
9. Higham, D. Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38(3) (2000), 753–769.
9
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ORIGINAL_ARTICLE
Regularization technique and numerical analysis of the mixed system of first and second-kind Volterra–Fredholm integral equations
It is important to note that mixed systems of first and second-kind Volterra–Fredholm integral equations are ill-posed problems, so that solving discretized system of such problems has a lot of difficulties. We will apply the regularization method to convert this mixed system (ill-posed problem) to system of the second kind Volterra–Fredholm integral equations (well-posed problem). A numerical method based on Chebyshev wavelets is suggested for solving the obtained well-posed problem, and convergence analysis of the method is discussed. For showing efficiency of the method, some test problems, for which the exact solution is known, are considered.
https://ijnao.um.ac.ir/article_24823_c56ff6c60d32ee8cc0c6bc4a8604afa5.pdf
2019-03-01
127
150
10.22067/ijnao.v9i1.63182
Mixed systems of first and second-kind Volterra-Fredholm integral equations
Regularization method
Chebyshev wavelets
Convergence analysis
S.
Pishbin
s.pishbin@urmia.ac.ir
1
Urmia University, Urmia, Iran.
LEAD_AUTHOR
J.
Shokri
j.shokri@urmia.ac.ir
2
Urmia University, Urmia, Iran.
AUTHOR
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