The Particle Swarm Optimization Against the Runge’s Phenomenon: Application to the Generalized Integral Quadrature Method
In the present work, we introduce the particle swarm optimization called (PSO in short) to avoid the Runge-s phenomenon occurring in many numerical problems. This new approach is tested with some numerical examples including the generalized integral quadrature method in order to solve the Volterra-s integral equations
[1] Brunner H. and van der Houwen P. J. The Numerical Solution of Volterra
Equations, CWI Monographs, North-Holland, Amsterdam (1986).
[2] Baker C. T. H. The Numerical Treatment of Integral Equations, Oxford
University Press (1977).
[3] Delves L. M. and Walsh J. Numerical Solution of Integral Equations,
Clarendon Press, Oxford (1974).
[4] Cochran J. A. The Analysis of Linear Integral Equations, McGraw-Hill
(1972).
[5] Atkinson K. E. A Survey of Numerical Methods for the Solution of
Fredholm Integral Equations of the Second Kind SIAM, Philadelphia
(1976).
[6] Wolkenfelt P. H. M. The construction of reducible quadrature rules for
Volterra integral and integro-differential equations, IMA J. Numer. Anal.
2 (1982) 131-152.
[7] Dobner H.-J., Bounds of high quality for first kind Volterra integral
equations. Reliable Computing 2(1), (1996) 35-45,.
[8] Bugajewski D. On the Volterra integral equation in locally convex
spaces, Demonstratio Math., 25, (1992) 747-754.
[9] Delves L.M. and Mohamed J.L. Computational methods for integral
equations. Cambridge University Press, Cambridge (1985).
[10] H. Brunner, A. Pedas, and G. Vainikko. The piecewise polynomial
collocation method for nonlinear weakly singular Volterra equations.
Mathematics of Computation Volume 68 (1999) 1079 - 1095
[11] J. P. Kauthen, H. Brunner. Continuous collocation approximations to
solutions of first kind Volterra equations. Mathematics of Computation
Volume 66 (1997) 1441 - 1459
[12] Q. Hu . Multilevel correction for discrete collocation solutions of
Volterra integral equations with delay arguments Applied Numerical
Mathematics Volume 31 (1999) 159 - 171
[13] H. Brunner, Q. Hu and Q. Lin . Geometric meshes in collocation
methods for Volterra integral equations with proportional delays. IMA
Journal of Numerical Analysis Volume 21 (2001) 783-798.
[14] V. Karlin, V. G. Maz-ya, A. B. Movchan , J. R. Willis, and R. Bullough
. Numerical solution of nonlinear hypersingular integral equations of the
Peierls type in dislocation theory. SIAM Journal on Applied Mathematics,
60 (2000) 664-678.
[15] M. H. Fahmy, M. A. Abdou and M. A. Darwish. Integral equations and
potential-theoretic type integrals of orthogonal polynomials. J. Comput.
Appl. Math. 106 (2) (1999) 245-254.
[16] T. Diogo and P. M. Lima. Numerical solution of a non-uniquely
solvable Volterra integral equation using extrapolation methods, Journal
of Computational and Applied Mathematics, 140 (2002) 537-557.
[17] Fabio Fagnani and Luciano Pandolfi. A recursive algorithm for the
approximate solution of Volterra integral equations of the first kind of
convolution type. Inverse Problems 19 (2003) 23-47
[18] Peeter Oja, Darja Saveljeva. Cubic spline collocation for Volterra integral
equations. Computing Volume 69 ( 2002) 319 - 337
[19] Igor Bock, Jan Lovisek. On a reliable solution of a Volterra integral
equation in a Hilbert space . APPLICATIONS OF MATHEMATICS,
Vol. 48, No. 6 (2003) 469-486,
[20] M. Federson, R. Bianconi and L. Barbanti, Linear Volterra integral
equations, Acta Math Appl. Sinica (English Series), 18(4), (2002) 553-
560.
[21] M. Federson, R. Bianconi and L. Barbanti. Linear Volterra integral equations
as the limit of discrete systems. CADERNOS DE MATEMA' TICA
4, (2003) 331-352
[22] Linz, P., Analytical and Numerical Methods for Volterra Equations.
SIAM, Philadelphia, (1985).
[23] Kulisch, U., Miranker, W.L., A new Approach to Scientific Computation.
Academic Press, New York, (1983).
[24] Hammer, R., Hocks, M., Kulisch, U., Ratz, D., Numerical Toolbox for
Verified Computing I. Springer Verlag, Berlin, (1995).
[25] Bugajewski D., On differential and integral equations in locally convex
spaces, Demonstratio Math., 28, (1995) 961-966.
[26] Dobner, H.-J., Bounds for the Solution of Hyperbolic Problems. Computing
38, (1987) 209-218, .
[27] Copson, E.T., Partial Differential Equations. Cambridge University
Press, Cambridge (1975).
[28] Bugajewska D., Topological properties of solution sets of some problems
for differential equations, Ph. D.Thesis, Pozna'n, (1999).
[29] Constantin A., On the unicity of solution for the differential equation
x(n) = f(t, x), Rend. Circ. Mat. Palermo, Serie II, 42, (1991) 59-64.
[30] Bugajewski D., Szu.a S., Kneser-s theorem for weak solutions of the
Darboux problem in Banach spaces, Nonlinear Analysis, 20, No 2,
(1993) 169-173.
[31] Maron, M. and Lopez, R., Numerical Analysis. Wadsworth Publishing
Company, Belmont California, (1991).
[32] Zerarka A. and Soukeur A. A generalized integral quadratic method:
I. an efficient solution for one-dimensional Volterra integral. Communication
in Nonlinear Science and Numerical Simulation, 10 (2005)653
-663.
[33] Zerarka A, Hassouni S, Saidi H and Boumedjane Y. Energy spectra
of the Schr¨odinger equation and the differential quadrature method.
Communication in Nonlinear Science and Numerical Simulation, 10
(2005)737 -745.
[34] J. Kennedy, R.C. Eberhart, Particle Swarm Optimization, Proc. IEEE
Int. Conf. Neural Networks, Piscataway, NJ, USA, 1942-1948, (1995).
[35] Eberhart, R. C. and Kennedy, J. A New Optimizer Using Particles Swarm
Theory. Sixth International Symposium on Micro Machine and Human
Science, Nagoya, Japan, (1995), pp. 39-43.
[36] K.E. Parsopoulos, V.P. Plagianakos, G.D. Magoulas, M.N. Vrahatis,
Objective function stretching to alleviate convergence to local minima,
Nonlinear Analysis TMA 47, 3419-3424, (2001).
[37] K.E. Parsopoulos, V.P. Plagianakos, G.D. Magoulas, M.N. Vrahatis,
Stretching technique for obtaining global minimizers through Particle
Swarm Optimization, Proceedings of the PSO Workshop, Indianapolis,
USA, 22-29, (2001).
[38] K.E. Parsopoulos, M.N. Vrahatis, Modification of the Particle Swarm
Optimizer for locating all the global minima, Artificial Neural Networks
and Genetic Algorithms, V. Kurkova et al. (Eds.), Springer, 324-327,
(2001).
[39] Fourie, P. C. and Groenwold, A. A. Particle Swarms in Topology
Optimization. Extended Abstracts of the Fourth World Congress of
Structural and Multidisciplinary Optimization, Dalian, China, June 4-
8, (2001), pp. 52-53.
[40] Fourie, P. C. and Groenwold, A. A. Particle Swarms in Size and
Shape Optimization. Proceedings of the International Workshop on
Multidisciplinary Design Optimization, Pretoria, South Africa, August
7-10 (2000), pp. 97-106.
[41] Eberhart, R.C., Simpson, P.K., Dobbins, R.W.: Computational Intelligence
PC Tools. Academic Press Professional, Boston (1996).
[42] Kennedy, J. The Behavior of Particles. Evol. Progr. VII (1998) 581-587.
[43] Kennedy, J., Eberhart, R.C. Swarm Intelligence. Morgan Kaufmann
(2001).
[44] Shi Y H, Eberhart R C. Fuzzy adaptive particle swarm optimization.
IEEE Int. Conf. on Evolutionary Computation, (2001) 101-106.
[45] Kennedy, J. and Spears, W. M. Matching Algorithms to Problems: An
Experimental Test of the Particle Swarm and Some Genetic Algorithms
on the Multimodal Problem Generator. Proceedings of the (1998) IEEE
International Conference on Evolutionary Computation, Anchorage,
Alaska, May 4-9 (1998).
[46] Shi, Y. and Eberhart, R. C. A Modified Particle Swarm Optimizer.
Proceedings of the 1998 IEEE International Conference on Evolutionary
Computation, Anchorage, Alaska, May 4-9 (1998).
[47] Shi, Y. H. and Eberhart, R. C. Parameter Selection in Particle Swarm
Optimization. Evolutionary Programming VII, Lecture Notes in Computer
Science, (1998), pp. 591-600.
[48] Clerc, M. The Swarm and the Queen: Towards a Deterministic and
Adaptive Particle Swarm Optimization. Proceedings of the 1999 IEEE
Congress on Evolutionary Computation, Washington D.C., (1999), pp.
1951-1957.
[49] Eberhart, R. C., Shi, Y. Parameter Selection in Particle Swarm Optimization.
Lecture Notes in Computer Science-Evolutionary Programming
VII, Porto, V. W., Saravanan, N. Waagen, D., Eiben, A. E., 1447, 591-
600, Springer, (1998).
[50] Cristian T. I. The particle swarm optimization algorithm: convergence
analysis and parameter selection. Information Processing Letters, (2003),
85(6): 317-325.
[51] R. K. Miller and A. Feldstein. Smoothness of solutions of Volterra
integral equations with weakly singular kernels. SIAM J. Math. Anal. 2
(1971), 242-258.
[1] Brunner H. and van der Houwen P. J. The Numerical Solution of Volterra
Equations, CWI Monographs, North-Holland, Amsterdam (1986).
[2] Baker C. T. H. The Numerical Treatment of Integral Equations, Oxford
University Press (1977).
[3] Delves L. M. and Walsh J. Numerical Solution of Integral Equations,
Clarendon Press, Oxford (1974).
[4] Cochran J. A. The Analysis of Linear Integral Equations, McGraw-Hill
(1972).
[5] Atkinson K. E. A Survey of Numerical Methods for the Solution of
Fredholm Integral Equations of the Second Kind SIAM, Philadelphia
(1976).
[6] Wolkenfelt P. H. M. The construction of reducible quadrature rules for
Volterra integral and integro-differential equations, IMA J. Numer. Anal.
2 (1982) 131-152.
[7] Dobner H.-J., Bounds of high quality for first kind Volterra integral
equations. Reliable Computing 2(1), (1996) 35-45,.
[8] Bugajewski D. On the Volterra integral equation in locally convex
spaces, Demonstratio Math., 25, (1992) 747-754.
[9] Delves L.M. and Mohamed J.L. Computational methods for integral
equations. Cambridge University Press, Cambridge (1985).
[10] H. Brunner, A. Pedas, and G. Vainikko. The piecewise polynomial
collocation method for nonlinear weakly singular Volterra equations.
Mathematics of Computation Volume 68 (1999) 1079 - 1095
[11] J. P. Kauthen, H. Brunner. Continuous collocation approximations to
solutions of first kind Volterra equations. Mathematics of Computation
Volume 66 (1997) 1441 - 1459
[12] Q. Hu . Multilevel correction for discrete collocation solutions of
Volterra integral equations with delay arguments Applied Numerical
Mathematics Volume 31 (1999) 159 - 171
[13] H. Brunner, Q. Hu and Q. Lin . Geometric meshes in collocation
methods for Volterra integral equations with proportional delays. IMA
Journal of Numerical Analysis Volume 21 (2001) 783-798.
[14] V. Karlin, V. G. Maz-ya, A. B. Movchan , J. R. Willis, and R. Bullough
. Numerical solution of nonlinear hypersingular integral equations of the
Peierls type in dislocation theory. SIAM Journal on Applied Mathematics,
60 (2000) 664-678.
[15] M. H. Fahmy, M. A. Abdou and M. A. Darwish. Integral equations and
potential-theoretic type integrals of orthogonal polynomials. J. Comput.
Appl. Math. 106 (2) (1999) 245-254.
[16] T. Diogo and P. M. Lima. Numerical solution of a non-uniquely
solvable Volterra integral equation using extrapolation methods, Journal
of Computational and Applied Mathematics, 140 (2002) 537-557.
[17] Fabio Fagnani and Luciano Pandolfi. A recursive algorithm for the
approximate solution of Volterra integral equations of the first kind of
convolution type. Inverse Problems 19 (2003) 23-47
[18] Peeter Oja, Darja Saveljeva. Cubic spline collocation for Volterra integral
equations. Computing Volume 69 ( 2002) 319 - 337
[19] Igor Bock, Jan Lovisek. On a reliable solution of a Volterra integral
equation in a Hilbert space . APPLICATIONS OF MATHEMATICS,
Vol. 48, No. 6 (2003) 469-486,
[20] M. Federson, R. Bianconi and L. Barbanti, Linear Volterra integral
equations, Acta Math Appl. Sinica (English Series), 18(4), (2002) 553-
560.
[21] M. Federson, R. Bianconi and L. Barbanti. Linear Volterra integral equations
as the limit of discrete systems. CADERNOS DE MATEMA' TICA
4, (2003) 331-352
[22] Linz, P., Analytical and Numerical Methods for Volterra Equations.
SIAM, Philadelphia, (1985).
[23] Kulisch, U., Miranker, W.L., A new Approach to Scientific Computation.
Academic Press, New York, (1983).
[24] Hammer, R., Hocks, M., Kulisch, U., Ratz, D., Numerical Toolbox for
Verified Computing I. Springer Verlag, Berlin, (1995).
[25] Bugajewski D., On differential and integral equations in locally convex
spaces, Demonstratio Math., 28, (1995) 961-966.
[26] Dobner, H.-J., Bounds for the Solution of Hyperbolic Problems. Computing
38, (1987) 209-218, .
[27] Copson, E.T., Partial Differential Equations. Cambridge University
Press, Cambridge (1975).
[28] Bugajewska D., Topological properties of solution sets of some problems
for differential equations, Ph. D.Thesis, Pozna'n, (1999).
[29] Constantin A., On the unicity of solution for the differential equation
x(n) = f(t, x), Rend. Circ. Mat. Palermo, Serie II, 42, (1991) 59-64.
[30] Bugajewski D., Szu.a S., Kneser-s theorem for weak solutions of the
Darboux problem in Banach spaces, Nonlinear Analysis, 20, No 2,
(1993) 169-173.
[31] Maron, M. and Lopez, R., Numerical Analysis. Wadsworth Publishing
Company, Belmont California, (1991).
[32] Zerarka A. and Soukeur A. A generalized integral quadratic method:
I. an efficient solution for one-dimensional Volterra integral. Communication
in Nonlinear Science and Numerical Simulation, 10 (2005)653
-663.
[33] Zerarka A, Hassouni S, Saidi H and Boumedjane Y. Energy spectra
of the Schr¨odinger equation and the differential quadrature method.
Communication in Nonlinear Science and Numerical Simulation, 10
(2005)737 -745.
[34] J. Kennedy, R.C. Eberhart, Particle Swarm Optimization, Proc. IEEE
Int. Conf. Neural Networks, Piscataway, NJ, USA, 1942-1948, (1995).
[35] Eberhart, R. C. and Kennedy, J. A New Optimizer Using Particles Swarm
Theory. Sixth International Symposium on Micro Machine and Human
Science, Nagoya, Japan, (1995), pp. 39-43.
[36] K.E. Parsopoulos, V.P. Plagianakos, G.D. Magoulas, M.N. Vrahatis,
Objective function stretching to alleviate convergence to local minima,
Nonlinear Analysis TMA 47, 3419-3424, (2001).
[37] K.E. Parsopoulos, V.P. Plagianakos, G.D. Magoulas, M.N. Vrahatis,
Stretching technique for obtaining global minimizers through Particle
Swarm Optimization, Proceedings of the PSO Workshop, Indianapolis,
USA, 22-29, (2001).
[38] K.E. Parsopoulos, M.N. Vrahatis, Modification of the Particle Swarm
Optimizer for locating all the global minima, Artificial Neural Networks
and Genetic Algorithms, V. Kurkova et al. (Eds.), Springer, 324-327,
(2001).
[39] Fourie, P. C. and Groenwold, A. A. Particle Swarms in Topology
Optimization. Extended Abstracts of the Fourth World Congress of
Structural and Multidisciplinary Optimization, Dalian, China, June 4-
8, (2001), pp. 52-53.
[40] Fourie, P. C. and Groenwold, A. A. Particle Swarms in Size and
Shape Optimization. Proceedings of the International Workshop on
Multidisciplinary Design Optimization, Pretoria, South Africa, August
7-10 (2000), pp. 97-106.
[41] Eberhart, R.C., Simpson, P.K., Dobbins, R.W.: Computational Intelligence
PC Tools. Academic Press Professional, Boston (1996).
[42] Kennedy, J. The Behavior of Particles. Evol. Progr. VII (1998) 581-587.
[43] Kennedy, J., Eberhart, R.C. Swarm Intelligence. Morgan Kaufmann
(2001).
[44] Shi Y H, Eberhart R C. Fuzzy adaptive particle swarm optimization.
IEEE Int. Conf. on Evolutionary Computation, (2001) 101-106.
[45] Kennedy, J. and Spears, W. M. Matching Algorithms to Problems: An
Experimental Test of the Particle Swarm and Some Genetic Algorithms
on the Multimodal Problem Generator. Proceedings of the (1998) IEEE
International Conference on Evolutionary Computation, Anchorage,
Alaska, May 4-9 (1998).
[46] Shi, Y. and Eberhart, R. C. A Modified Particle Swarm Optimizer.
Proceedings of the 1998 IEEE International Conference on Evolutionary
Computation, Anchorage, Alaska, May 4-9 (1998).
[47] Shi, Y. H. and Eberhart, R. C. Parameter Selection in Particle Swarm
Optimization. Evolutionary Programming VII, Lecture Notes in Computer
Science, (1998), pp. 591-600.
[48] Clerc, M. The Swarm and the Queen: Towards a Deterministic and
Adaptive Particle Swarm Optimization. Proceedings of the 1999 IEEE
Congress on Evolutionary Computation, Washington D.C., (1999), pp.
1951-1957.
[49] Eberhart, R. C., Shi, Y. Parameter Selection in Particle Swarm Optimization.
Lecture Notes in Computer Science-Evolutionary Programming
VII, Porto, V. W., Saravanan, N. Waagen, D., Eiben, A. E., 1447, 591-
600, Springer, (1998).
[50] Cristian T. I. The particle swarm optimization algorithm: convergence
analysis and parameter selection. Information Processing Letters, (2003),
85(6): 317-325.
[51] R. K. Miller and A. Feldstein. Smoothness of solutions of Volterra
integral equations with weakly singular kernels. SIAM J. Math. Anal. 2
(1971), 242-258.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58236", author = "A. Zerarka and A. Soukeur and N. Khelil", title = "The Particle Swarm Optimization Against the Runge’s Phenomenon: Application to the Generalized Integral Quadrature Method", abstract = "In the present work, we introduce the particle swarm optimization called (PSO in short) to avoid the Runge-s phenomenon occurring in many numerical problems. This new approach is tested with some numerical examples including the generalized integral quadrature method in order to solve the Volterra-s integral equations
", keywords = "Integral equation, particle swarm optimization, Runge's phenomenon.", volume = "3", number = "5", pages = "347-6", }
