ORIGINAL_ARTICLE
Analysing panel flutter in supersonic flow by Hopf bifurcation
This paper is devoted to study of a partial differential equation governing panel motion in supersonic flow. This PDE can be transformed to an ODE by means of a Galerkin method. Here by using a criterion which is closely related to the Routh-Hurwitz criterion, we investigate the mentioned transformed ODE from Hopf bifurcation point of view. In fact we obtain a region for existence of simple Hopf bifurcation for it. With the aid of computer language Matlab and Hopf bifurcation tool, flutter and limit cycle oscillations of panel are verified. Moreover, Hopf bifurcation theory is used to analyse the flutter speed of the system.
https://ijnao.um.ac.ir/article_24431_dac176d3aaf67e4c385927ea1a7ebb2f.pdf
2014-11-01
1
14
10.22067/ijnao.v4i2.32430
Panel fllutter
Limit cycle
Hopf bifurcation
Routh-Hurwitz criterion
Vibrations
z.
Monfared
monfared.zahra@gmail.com
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
LEAD_AUTHOR
Z.
Dadi
z.dadi@ub.ac.ir
2
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran.
AUTHOR
[1] Clancy, L.J., Aerodynamics, Pitman Publishing Limited, London. ISBN 0-273 01120-0, 1975.
1
[2] Eftekhari, S.A., Bakhtiari-Nejad, F. and Dowell, E.H. Damage detection of an aeroelastic panel using limit cycle oscillation analysis, International Journal of Non-Linear Mechanics, Vol. 58, 99-110, 2014.
2
[3] Feng, Z. C. and Sethna, P. R. Global bifurcations in the motion of parametri cally excited thin plate, Nonlinear Dynamics, 4, 389408, 1993.
3
[4] Guckenheimer, J. and Holmes, P.J. Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, 1993.
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[5] Holmes, P. Bifurcations to divergence and flutter in flow-induced oscillations: A finite-dimensional analysis, Journal of Sound and Vibration, Vol. 53(4), 161174, 1977.
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[6] Holmes, P. Center manifolds, normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions, Physica D2, 449481, 1981.
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[7] Holmes, P. and Marseden, J. Bifurcation to divergence and flutter in flow-induced oscillations: An infinite dimensional Analysis, International Federation of Automatic Control, Vol. 14, pp. 367 384, 1978.
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[8] Kovacic, G. and Wiggins, S. Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D57, 185225, 1992.
8
[9] Kuznetsov, Y.A. Elements of Applied Bifurcation Theory, New York: SpringerVerlag, 2004.
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[10] Liu, W. Criterion of Hopf bifurcation without using eigenvalues, Journal of mathe-matical analysis and applications, 182, 250-256, 1994.
10
[11] Marsden, J. and McCracken, M., Hopf Bifurcation and Its Applications,Springer, 1976.
11
[12] Peng, L. and Yiren, y. On the stability and chaos of a plate with motion constraints subjected to subsonicflow, International Journal of Non-Linear Mechanics, Vol. 59, 28-36, 2014.
12
[13] Perco, L. Differential equations and dynamical systems, Springer- Verlag, 1991.
13
[14] Pourtakdoust, S. H. and Fazelzadeh, S. A., Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow, Nonlinear Dynamics, Vol. 32: 387404, 2003.
14
[15] Song, Z. G. and Li, F. M. Vibration and aeroelastic properties of ordered and disordered two-span panels in supersonic airflow, International Journal of Mechanical Sciences, , Vol. 81, 65-72, 2014.
15
[16] S¨uli, E. Numerical solution of ordinary differential equations, Lecture note, University of Oxford, 2013.
16
[17] Vedeneev, V. Panel flutter at low supersonic speeds, Journal of Fluids and Structures Vol. 29 , 79-96, 2012.
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[18] Yang, X. L. and Sethna, P. R., Local and global bifurcations in parametrically excited vibrations nearly square plates, International Journal of Non-Linear Mechanics 26(2), 199220, 1990.
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[19] Zhang, W. and Zhaomiao, L., Global Dynamics of a Parametrically and Externally Excited Thin Plate, Nonlinear Dynamics 24: 245268, 2001.
19
[20] Zhang, X. Local bifurcations of nonlinear viscoelastic panel in supersonic flow, Commun Nonlinear Sci Numer Simulat, Vol. 18, 1931-1938, 2013.
20
ORIGINAL_ARTICLE
Hopf bifurcation in a general n-neuron ring network with n time delays
In this paper, we consider a general ring network consisting of n neurons and n time delays. By analyzing the associated characteristic equation, a classification according to n is presented. It is investigated that Hopf bifur-cation occurs when the sum of the n delays passes through a critical value.In fact, a family of periodic solutions bifurcate from the origin, while the zero solution loses its asymptotically stability. To illustrate our theoretical results, numerical simulation is given.
https://ijnao.um.ac.ir/article_24432_e3ffe80f49324d2ada232629971141b5.pdf
2014-11-01
15
30
10.22067/ijnao.v4i2.40227
Ring network
Stability
Periodic solution
Hopf bifurcation
Time delay
E.
Javidmanesh
a_javidmanesh@yahoo.com
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
AUTHOR
M.
Khorshidi
mohsen.khorshidi89@gmail.com
2
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
LEAD_AUTHOR
1. Campbell, S.A. Stability and bifurcation of a simple neural network with multiple time delays, Fields Institute Communications 21 (1999), 65-79.
1
2. Campbell, S.A., Ruan, S. and Wei, J. Qualitative analysis of a neural network model with multiple time delays, Int. J. of Bifurcation and Chaos 9 (1999), 1585-1595.
2
3. Cao, J., Yu, W. and Qu, Y. A new complex network model and convergence dynamics for reputation computation in virtual organizations, Phys. Lett.A 356 (2006), 414-425.
3
4. Driver, R.D. Ordinary and delay differential equations, Springer, 1977.
4
5. Eccles, J. C., Ito, M. and Szenfagothai, J. The cerebellum as neuronal machine, Springer, New York, 1967.
5
6. Guckenheimer, J. and Holmes, P. Nonlinear oscillation, dynamical system and bifurcations of vector felds, Springer, 1993.
6
7. Hopfield, J. Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. 79 (1982), 2554-2558.
7
8. Hu, H. and Huang, L. Stability and Hopf bifurcation analysis on a ring of four neurons with delays, Appl.Math. Comput. 213 (2009), 587-599.
8
9. Javidmanesh, E., Afsharnezhad, Z. and Effati, S. Existence and stability analysis of bifurcating periodic solutions in a delayed five-neuron BAM neural network model, Nonlinaer Dyn. 72 (2013), 149-164.
9
10. Lee, S.M., Kwonb, O.M. and Park, J.H. A novel delay-dependent crite-rion for delayed neural networks of neutral type, Phys. Lett. A 374 (2010), 1843-1848.
10
11. Li, X. and Cao, J. Delay-dependent stability of neural networks of neutral type with time delay in the leakage term, Nonlinearity 23 (2010), 1709-1726.
11
12. Marcus, C.M. and Westervelt, R.M. Stability of analog neural network with delay, Phys. Rev. A 39 (1989), 347-359.
12
13. Perko, L. Differential equations and dynamical system, Springer 1991.
13
14. Ruan, S. and Wei, J. On the zeros of transcendental functions with appli-cations to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematics and Analytical 10 (2003), 863-874.
14
ORIGINAL_ARTICLE
Population based algorithms for approximate optimal distributed control of wave equations
In this paper, a novel hybrid iterative scheme to find approximate optimal distributed control governed by wave equations is considered. A partition of the time-control space is considered and the discrete form of the problem is converted to a quasi assignment problem. Then a population based algorithm, with a finite difference method, is applied to extract approximate optimal distributed control as a piecewise linear function. A convergence analysis is proposed for discretized form of the original problem. Numerical computations are given to show the profciency of the proposed algorithm and the obtained results applying two popular evolutionary algorithms, genetic and particle swarm optimization algorithms.
https://ijnao.um.ac.ir/article_24433_b5cf12ddc3b07e03b62c2953a21a1822.pdf
2014-11-01
31
41
10.22067/ijnao.v4i2.40230
Optimal control problem
Evolutionary algorithm
finite difference method
Wave equation
A. H.
Borzabadi
borzabadi@mazust.ac.ir
1
School of Mathematics and Computer Science, Damghan University, Damghan, Iran.
LEAD_AUTHOR
S.
Mirassadi
mirassadi@yahoo.com
2
School of Mathematics and Computer Science, Damghan University, Damghan, Iran.
AUTHOR
M.
Heidari
hedari@yahoo.com
3
School of Mathematics and Computer Science, Damghan University, Damghan, Iran.
AUTHOR
1. Alavi, S.A., Kamyad, A.V. and Farahi, M.H. The optimal control of an inhomogeneous wave problem with internal control and their numerical solution, Bulletin of the Iranian Mathematical society 23(2) (1997) 9-36.
1
2. Borzabadi, A.H. and Mehne, H. H. Ant colony optimization for optimal control problems, Journal of Information and Computing Science, 4(4)
2
3. Borzabadi, A.H. and Heidari, M. Comparison of some evolutionary algorithms for approximate solutions of optimal control problems, Australian Journal of Basic and Applied Sciences, 4(8) (2010) 3366-3382.
3
4. Borzabadi, A.H. and Heidari, M. Evolutionary algorithms for approximate optimal control of the heat equation with thermal sources, Journal of Mathematical Modelling and Algorithms, DOI: 10.1007/s10852-011-9166-0.
4
5. Farahi, M.H., Rubio, J.E. and Wilson, D.A. The Optimal control of the linear wave equation, International Journal of control, 63 (1996) 833-848.
5
6. Farahi, M.H., Rubio, J.E. and Wilson, D.A. The global control of a nonlinear wave equation, International Journal of Control 65(1) (1996) 1-15.
6
7. Fard, O.S. and Borzabadi, A.H. Optimal control problem, quasiassignment problem and genetic algorithm, Enformatika, Transaction on Engin. Compu. and Tech., 19 (2007) 422 - 424.
7
8. Gerdts, M., Greif, G. and Pesch, H.J. Numerical optimal control of the wave equation: Optimal boundary control of a string to rest in finite time, Proceedings 5th Mathmod Vienna, February (2006).
8
9. Glowinski, R., Lee, C.H. and Lions, J.L. A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: description of the numerical methods, Japan J. Appl. Math., 7 (1990) 1-76.
9
10. Goldwyn, R.M., Sriram, K.P. and Graham, M.H. Time optimal controlof a linear hyperbolic system, International Journal of Control, 12 (1970)645-656.
10
11. Gugat, M., Keimer, A. and Leugering, G. Optimal distributed control ofthe wave equation subject to state constraints, ZAMM . Z. Angew. Math. Mech. 89(6) (2009) 420-444.
11
12. Gugat, M. Penalty techniques for state constrained optimal control problems with the wave equation, SIAM J. Control Optim. 48 (2009) 3026-3051.
12
13. Hasanov, K.K. and Gasumov, T.M. Minimal energy control for the wave equation with non-classical boundary condition, Appl. Comput. Math.,9(1) (2010) 47-56.
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14. Kim, M. and Erzberger, H. On the design of optimum distributed parameter system with boundary control function, IEEE Transactions on Automatic Control, 12(1) (1967) 22-28.
14
15. Lions, J.L. Optimal control of systems governed by partial differential
15
equations, Springer, Berlin, 1971.
16
ORIGINAL_ARTICLE
Operational Tau method for nonlinear multi-order FDEs
This paper presents an operational formulation of the Tau method based upon orthogonal polynomials by using a reduced set of matrix operations for the numerical solution of nonlinear multi-order fractional differential equations(FDEs). The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of non-linear algebraic equations. Some numerical examples are provided to demonstrate the validity and applicability of the method.
https://ijnao.um.ac.ir/article_24434_e94aaa028fd400b6f520cdb773ae202e.pdf
2014-11-01
43
55
10.22067/ijnao.v4i2.33577
Fractional differential equations(FDEs)
Caputo derivative
Operational Tau method
P.
Mokhtary
mokhtary.payam@gmail.com
1
Department of Mathematics, Sahand University of Technology, Tabriz, Iran.
LEAD_AUTHOR
1. Abbasbandy, S. and Taati, A. Numerical solution of the system of nonlin-ear Volterra integro-differential equations with nonlinear differential partby the operational Tau method and error estimation, J. Comput. Appl.Math., 231 (2009) 106-113.
1
2. Arikoglu, A. and Ozkol, I. Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals, 34 (2007) 1473-1481.
2
3. Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T. Spectral methodsfundamentals in single domains,Springer-Verlag, Berlin, 2006.
3
4. Cang, J., Tan, Y., Xu, H. and Liao, S. Series solutions of non-linear Riccatidifferential equations with fractional order, Chaos Solitons Fractals, 40 (2009) 1-9.
4
5. Diethelm, K. and Walz, G. Numerical solution of fractional order differ-ential equations by extrapolation, Numer. Algorithms, 16 (1997) 231-253.
5
6. Freilich, J. and Ortiz, E. Numerical solution of systems of ordinary differ-ential equations with the Tau method, an error analysis, Math. Comp., 39(1982) 467-479.
6
7. Galeone, L. and Garrappa, R. On multistep methods for differential equa-tions of fractional order, Mediterr. J. Math., 3 (2006) 565-580.
7
8. Ghorbani, A. Toward a new analytical method for solving nonlinear frac-tional differential equations,Comput. Methods Appl. Mech. Engrg., 197(2009) 4173-4179.
8
9. Ghoreishi, F. and Yazdani, S. An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. Math. Appl., 61 (2011) 30-43.
9
10. Ghoreishi, F. and Hadizadeh, M. Numerical computation of the Tau ap-proximation for the Volterra-Hammerstein integral equations, Numer. Algorithms,52 (2009) 541-559.
10
11. Gottlieb, D. and Orszag, S. Numerical analysis of spectral methods, the-ory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 26. SIAM, Philadelphia, 1997.
11
12. Jafari, H. and Sei, S. Solving a system of non-linear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 1962-1969.
12
13. Jafari, H. and Daftardar-Gejji, V. Solving a system of nonlinear frac-tional differential equations using Adomian decomposition, J. Comput.Appl. Math., 196 (2006) 644-651.
13
14. Kilbas, A., Srivastava, M. and Trujillo, J. Theory and applications of fractional differential equations, Elsevier, 2006.
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15. Lanzos, C. Trigonometric interpolation of empirical and analytical func-tions, J. Math. Phys., 17 (1983) 123-199.
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16. Lanczos, C. Introduction, tables of Chebyshev polynomials, Appl. Math. Ser. US Bur. Stand., 9, Government Printing Office Washington, 1952.
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17. Lanczos, C. Applied analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956.
17
18. Lin, R. and Liu, F. Fractional high order methods for the nonlinear frac-tional ordinary differential equation, Nonlinear Anal., 66 (2007) 856-869.
18
19. Mokhtary, P. and Ghoreishi, F. The L2-convergence of the legendre spec-tral Tau matrix formulation for onlinear fractional integro differential equations, Numer. lgorithms, 58 (2011) 475-496.
19
20. Momani, S. and Odibat, Z. Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals,31 (2007) 1248-1255.
20
21. Momani, S. and Shawagfeh, N. Decomposition method for solving frac-tional Riccati differential quations, Appl. Math. Comput., 182 (2006)1083-1092.
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22. Odibat, Z. and Momani, S. Modified Homotopy perturbation method: application to quadratic Ricatti differential equation of fractional order,Chaos Solitons Fractals, 36 (2008) 167-174.
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23. Onumanyi, P. and Ortiz, E. Numerical solution of stiff and singularity perturbed boundary value problems with a segmented-adaptive formulation of the Tau Method, Math. Comp., 43 (1984) 189-203.
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24. Ortiz, E. and Dinh, A. An error analysis of the Tau method for a class of singularity perturbed problems for differential equations, Math. Methods Appl. Sci., 6 (1984) 457-466.
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25. Ortiz, E. and Dinh, A. On the convergence of the Tau method for non-linear differential equations of icatti's type, Nonlinear Anal., 9 (1985)53-60.
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26. Ortiz, E. The Tau method, SIAM J. Numer. Anal, 6 (1969) 480-492.
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27. Ortiz, E. and Samara, H. An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27(1981) 15-25.
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28. Ortiz, E. and Samara, H. Numerical solution of differential eigenvalue problems with an operational approach to the Tau method, Computing, 31(1983) 95-103.
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29. Ortiz, E. and Samara, H. Numerical solution of partial differential equa-tions with variable coefficients with an operational approach to the Tau method, Comput. Math. Appl., 10 (1984) 5-13.
29
30. Podlubny, I. Fractional differential equations, Academic Press, 1999.
30
ORIGINAL_ARTICLE
A new approach for solving nonlinear system of equations using Newton method and HAM
A new approach utilizing Newton Method and Homotopy Analysis Method (HAM) is proposed for solving nonlinear system of equations. Accelerating the rate of convergence of HAM, and obtaining a global quadratic rate of convergence are the main purposes of this approach. The numerical results demonstrate the efficiency and the performance of proposed approach. The comparison with conventional homotopy method, Newton Method and HAM shows the great freedom of selecting the initial guess, in this approach.
https://ijnao.um.ac.ir/article_24435_a799b8bc98a0f6bcfbb86bf286ec9ca8.pdf
2014-11-01
57
72
10.22067/ijnao.v4i2.32555
Homotopy Analysis Method
Zero order deformation equations
control convergence parameter
Newton's method
Iterative method
multi-step iterative method
order of convergence
J.
Izadian
jalal_izadian32@hotmail.com
1
Department of Mathematics, Faculty of Sciences, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
LEAD_AUTHOR
R.
Abrishami
jalili.maryam56@gmail.com
2
Department of Mathematics, Faculty of Sciences, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
AUTHOR
M.
Jalili
jalili.maryam@yahoo.com
3
Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran.
AUTHOR
[1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations bymodifiedAdomian decompositionmethod, Applied Mathematics and Computation, 145(2-3): 887893 (2003)
1
[2] S. Abbasbandy, M. Jalili, Determination of optimal convergence-control parameter value in homotopy analysis method, Numerical Algorithms 64(4): 593-605 (2013)
2
[3] S. Abbasbandy, Y. Tan and S. J. Liao, Newton-homotopy analysis method for nonlinear equations, Appl. Math. Comput. 188: 1794-1800 (2007)
3
[4] F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numerical Algorithms, 54(3): 395409 (2010)
4
[5] E. Babolian, M. Jalili, Application of the Homotopy− Pad´e technique in the prediction of optimal convergence-control paramete, Computational and Applied Mathematics, article in press. DOI:10.1007/s40314-014-0123-1, (2014)
5
[6] J. Faires, R. Burden, Numerical Methods, Brooks Cole 3 edition, (2002)
6
[7] L. Fang, G. He, Some modifications of Newton’s method with higher order convergence for solving nonlinear equations, J. Comput. Appl. Math. 228: 296-303 (2009)
7
[8] L. Fang, G. He, An efficient Newton-type method with fifth-order convergence for Solving Nonlinear Equations, Comput. App. Math., 27(3): 269-274 (2008)
8
[9] J. Izadian, M. Mohammadzade Attar, M. Jalili, Numerical Solution of Deformation Equations in Homotopy Analysis Method, Applied Mathematical Sciences, 6(8): 357- 367 (2012)
9
[10] S. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation 147: 499-513 (2004)
10
[11] S. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simulat., 14: 983-997 (2009)
11
[12] S. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, PhD thesis, Shanghai Jiao Tong University, (1992)
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[13] S. Liao,Y. Tan, A General Approach to Obtain Series Solutions Of Non-Linear Differential Equations, Stud. Appl. Math 119: 297-354 (2007)
13
[14] S. Liao, Beyond perturbation (Introduction to the homotopy analysis method), CHAPMAN and HALL , (2004)
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[15] C.Y. Ku, W. Yeih, C.S. Liu, Solving Non-Linear Algebraic Equations by a Scalar Newton-homotopy Continuation Method, International Journal of Nonlinear Sciences and Numerical Simulation, 11(6): 435450 (2010)
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[16] J. Stoer, R. Bulrish , Introduction to Numerical Analysis, Springer, (1991)
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[17] Y. Wu, K.F. Cheung, Two-parameter homotopy method for nonlinear equations, Numerical Algorithms, 53(4): 555-572 (2010)
17
ORIGINAL_ARTICLE
Solving nonlinear Volterra integro-differential equation by using Legendre polynomial approximations
In this paper, we construct a new iterative method for solving nonlinear Volterra Integral Equation of second kind, by approximating the Legendre polynomial basis. Error analysis is worked using Banach fixed point theorem. We compute the approximate solution without using numerical method. Finally, some examples are given to compare the results with some of the existing methods.
https://ijnao.um.ac.ir/article_24436_27298a4022482a838049e405114504fa.pdf
2014-11-01
73
83
10.22067/ijnao.v4i2.34304
Nonlinear Volterra integro-differential equation
Legendre poly-nomial
error analysis
M.
Gachpazan
gachpazan@math.um.ac.ir
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
LEAD_AUTHOR
M.
Erfanian
erfaniyan@uoz.ac.ir
2
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
AUTHOR
H.
Beiglo
h.beiglo@gmail.com
3
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
AUTHOR
1] Babolian, E., Masouri, Z. and Hatamzadeh-Varmazyar, S. New direct method to solve nonlinear Volterra-Fredholm integral and integro-differential equations using operational matrix with block-pulse functions, Prog. in Electromag. Research 8 (2008), 59-76.
1
[2] Babolian, E. and Davary, A. Numerical implementation of Adomian decomposition method for linear Volterra integral equations for the second kind, Appl. Math. Comput. 165 (2005) 223-227.
2
[3] Danfu, H. and Xufeng, S. Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comput.194 (2007) 460-466.
3
[4] Darania, P. and Ebadian, A. A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188 (2007) 657-668.
4
[5] Elbarbary, E.M.E. Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Math. Comput. Simul. 59 (2002) 389–399
5
[6] El-Mikkawy, M.E.A. and Cheon, G.S. Combinatorial and hypergeometric identities via the Legendre polynomials- A computational approach, Appl. Math. Comput. 166 (2005) 181-195.
6
[7] Fox, L. and Parker, I. Chebyshev polynomials in Numerical Analysis, Clarendon Press, Oxford, 1968.
7
[8] Ghasemi, M. and Kajani, C.M.T., Application of Hes homotopy perturbation method to nonlinear integrodifferential equations, Applied Mathematics and Computation, vol 188(1)(2007), 538–548
8
[9] Ghasemi, M. and Kajani, C.M.T.,Numerical solutions of the nonlinear integro-differential equations: Wavelet-Galerkin method and homotopy perturbation method, Applied Mathematics and Computation, vol 188(1)(2007), 450–455.
9
[10] Ghasemi, M. and Kajani, C.M.T.,Comparison bet ween the homotopy perturbation method and the sinecosine wavelet method for solving linear integro- differential equations, Computers Mathematics with Applications, vol 54(78)(2007), 1162–1168.
10
[11] Gillis, J., Jedwab J. and Zeilberger, D., A combinatorial interpretation of the integral of the product of Legendre polynomials, SIAM J. Math. Anal. 19 (6) (1988) 1455-1461.
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[12] Gulsu, M. and Sezer, M. The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. Comput. 168 (2005) 76-88.
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[13] Liu, Y. Application of Chebyshev polynomial in solving Fredholm integral equations, Math. Comput. Modelling 50 (2009) 465-469.
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[14] Mahmoudi, Y. Wavelet Galerkin method for numerical solution of nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119-1129.
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[15] Maleknejad,K., Tavassoli Kajani,M. Solving second kind integral equation by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput. 145 (2003) 623–629.
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[16] Marzban, H.R and Razzaghi, M. Optimal control of linear delay systems via hybrid of block-pulse and Legendre polnomials, J. Franklin Inst. 341 (2004) 279-293.
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[17] Rashidinia, J. and Zarebnia, M. Solution of a Volterra integral equation by the Sinc-collocation method, Comput. Appl. Math. 206 (2007) 801-813.
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[18] Streltsov, I.P. Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations, Comput. Phys. Commun. 126 (2000) 178–181.
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[19] U. Lepik, U. and Tamme, E. Solution of nonlinear Fredholm integral equations via the Haar wavelet method, Proc. Estonian Acad. Sci. Phys. Math. 56 (2007) 17-27
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[20] Voigt, R.G., Gottlieb, D. and Hussaini, M.Y. Spectral Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1984.
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[21] Yousefi,S. and Razzaghi, M. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simulat. 70 (1) (2005) 1-8.
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[23] Zhao, J. and Corless, R.M.,Compact finite difference method for integro-differential equations, Applied Mathematics and Computation, vol 177, 271–288.
23