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Approximate Solution by Ant Colony Programming to Symmetrical Drop Suspended Equilibrium Equation

Amel Serrat(1*), Bachir Djebbar(2)

(1) Université des Sciences et de la Technologie d’Oran Mohamed Boudiaf, USTO MB, BP. 1505 El M’naouer 31000 Oran, Algeria
(2) Université des Sciences et de la Technologie d’Oran Mohamed Boudiaf, USTO MB, BP. 1505 El M’naouer 31000 Oran,
(*) Corresponding author



The paper presents the Ant Colony Programming (ACP) method as a solver for Differential Equations of a suspended symmetric drop. The Ant Colony method developed by Gambardella Dorigo in 1997 has been applied in Routing, Scheduling, Constraint Satisfaction, Graph Coloring, and such other optimization problems. The Differential Equation Solution can be expressed as an expression containing Terminals and Functions and both are represented as a graph with nodes and edges. In this graph, each node (city) represents a function or terminal, the information flow represents pheromone quantity disposed by ants while walking. An initial quantity of pheromone is disposed on links between nodes. After that and until termination criteria, ants construct solutions (paths) from the source to the destination by visiting nodes, and deposing (updating) pheromones quantity at links of the path. This is one of the advantages of Ant Colony Programming, i.e. the diversification by local pheromone update. The select step is fundamental, too. In this step, the ant reflects the performance of the link in the cost path according to fitness value. The simulation experiments show that Ant Colony Programming method can obtain very good results in finding solution.
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Ant Colony Programming; Differential Equation; Symmetrical Drop Suspended Equilibrium Equation

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M. Hatami, and J. Dengwei. Nanofluids Mathematical, Numerical and Experimental Analysis. Academic Press, an Imprint of Elsevier, 2020.

A. Bényi and A. Okoudjou. Modulation Spaces: With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations. Springer New York, 2020.

A. Borzì. Modelling with Ordinary Differential Equations: a Comprehensive Approach. CRC Press, 2020.

D. Xue. Differential Equation Solutions with MATLAB. De Gruyter, 2020.

S. Chakraverty. Advanced Numerical and Semi Analytical Methods for Differential Equations. John Wiley & Sons, Inc., 2019.

B. C. Mohan and R. Baskaran, A survey: Ant Colony Optimization based recent research and implementation on several engineering domain, Expert Systems with Applications, vol. 39, no. 4, pp. 4618–4627, 2012.

F. Zhao, Z. Yao, J. Luan, and X. Song,“A Novel Fused Optimization Algorithm of Genetic Algorithm and Ant Colony Optimization, Mathematical Problems in Engineering, vol. 2016, pp. 1–10, 2016.

M. Z. M. Kamali, N. Kumaresan, and K. Ratnavelu, Solving differential equations with ant colony programming, Applied Mathematical Modelling, vol. 39, no. 10–11, pp. 3150–3163, Jun. 2015.

S. Mall and S. Chakraverty, Application of Legendre Neural Network for solving ordinary differential equations, Applied Soft Computing, vol. 43, pp. 347–356, Jun. 2016.

T. Dockhorn, A Discussion on Solving Partial Differential Equations using Neural Networks. 2019. arXiv:1904.07200.

S. Panghal and M. Kumar, Optimization free neural network approach for solving ordinary and partial differential equations, Engineering with Computers, 2020.

S. Panghal and M. Kumar, Optimization free neural network approach for solving ordinary and partial differential equations, Engineering with Computers, Feb. 2020.

M. Magill, F. Qureshi, H. Haan. Neural Networks Trained to Solve Differential Equations Learn General Representations. 2018. arXiv:1807.00042.

C. Michoski, M. Milosavljević, T. Oliver, and D. R. Hatch, Solving differential equations using deep neural networks, Neurocomputing, vol. 399, pp. 193–212, Jul. 2020.

H. S. Yazdi and R. Pourreza, Unsupervised adaptive neural-fuzzy inference system for solving differential equations, Applied Soft Computing, vol. 10, no. 1, pp. 267–275, Jan. 2010.

K. Parand, M. Razzaghi, R. Sahleh, and M. Jani, Least squares support vector regression for solving Volterra integral equations, Engineering with Computers, Oct. 2020.

N. Panagant and S. Bureerat, Solving Partial Differential Equations Using a New Differential Evolution Algorithm, Mathematical Problems in Engineering, vol. 2014, pp. 1–10, 2014.

A. Bilesanmi, A. S. Wusu, and A. L. Olutimo, Solution of Second-Order Ordinary Differential Equations via Simulated Annealing, OJOp, vol. 08, no. 01, pp. 32–37, 2019.

A. Ullah, S. A. Malik, and K. S. Alimgeer, Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations, Plos One, vol. 13, no. 1, 2018.

D. Gutierrez-Navarro and S. Lopez-Aguayo, Solving ordinary differential equations using genetic algorithms and the Taylor series matrix method, J. Phys. Commun., vol. 2, no. 11, p. 115010, Nov. 2018.

Y. Boudouaoui, H. Habbi, C. Ozturk, and D. Karaboga, Solving differential equations with artificial bee colony programming, Soft Comput, vol. 24, no. 23, pp. 17991–18007, May 2020.

L. Pan, J. Liang, and B. Qu, Bio-inspired computing: theories and applications 14th International Conference, BIC-TA 2019, Zhengzhou, China, November 22-25, 2019: revised selected papers. Singapore: Springer, 2020.

A. Akhtar, Evolution of Ant Colony Optimization Algorithm - A Brief Literature Review,, 27-Aug-2019. [Online - Accessed: 17-Mar-2021].

A. Nayyar, L. Dac-Nhuong, N. Nhu Gia. Advances in Swarm Intelligence for Optimizing Problems in Computer Science. CRC Press/Taylor & Francis Group, 2019.

E. K. Pakpahan, S. Kristina, and A. Setiawan, Proposed algorithm to improve job shop production scheduling using ant colony optimization method, IOP Conf. Ser.: Mater. Sci. Eng., vol. 277, p. 012050, Dec. 2017.

W. Qin, Z. Zhuang, Y. Liu, and O. Tang, A two-stage ant colony algorithm for hybrid flow shop scheduling with lot sizing and calendar constraints in printed circuit board assembly, Computers & Industrial Engineering, vol. 138, p. 106115, 2019.

Y. Zhang, Y. Yanlin, Z. Shenglan ,L. Yingxiong, Z. Lieping, Ant Colony optimization for Cuckoo Search algorithm for permutation Flow Shop Scheduling Problem, Systems Science & Control Engineering, Vol. 7-1, pp.20-27, 2019.

A. Salmasnia, S. Noori, and H. Mokhtari, A redundancy allocation problem by using utility function method and ant colony optimization: tradeoff between availability and total cost, Int J Syst Assur Eng Manag, vol. 10, no. 3, pp. 416–428, May 2019.

R. Rueda, L. Ruiz, M. Cuéllar, and M. Pegalajar, An Ant Colony Optimization approach for symbolic regression using Straight Line Programs. Application to energy consumption modelling, International Journal of Approximate Reasoning, vol. 121, pp. 23–38, 2020.

L. Wang, J. Kan, J. Guo, and C. Wang, Improved Ant Colony Optimization for Ground Robot 3D Path Planning, presented at the 2018 International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery (CyberC), Oct. 2018.

S. A. Mohsin, A. Younes, and S. M. Darwish, Dynamic Cost Ant Colony Algorithm to Optimize Query for Distributed Database Based on Quantum-Inspired Approach, Symmetry, vol. 13, no. 1, p. 70, 2021.

O. Korb, T. Stützle, and T. E. Exner, Empirical Scoring Functions for Advanced Protein−Ligand Docking with PLANTS, J. Chem. Inf. Model., vol. 49, no. 1, pp. 84–96, Jan. 2009.

M. Ashraf, B. S. Tawfik, S. El Diasty, and M. Hassan, MANET’s Energy Consumption using proposed Ant-Colony Optimization and Integer Linear Programming Algorithms, J. Phys.: Conf. Ser., vol. 1447, p. 012047, Jan. 2020.

L. G. Fahad, S. F. Tahir, W. Shahzad, M. Hassan, H. Alquhayz, and R. Hassan, Ant Colony Optimization-Based Streaming Feature Selection: An Application to the Medical Image Diagnosis, Scientific Programming, vol. 2020, pp. 1–10, Oct. 2020.

W. Gao, Improved Ant Colony Clustering Algorithm and Its Performance Study, Computational Intelligence and Neuroscience, vol. 2016, pp. 1–14, 2016.

N. E. Allali, M. Fariss, H. Asaidi, and M. Bellouki, Semantic Web Services Composition Model Using Ant Colony Optimization, presented at the 2020 Fourth International Conference On Intelligent Computing in Data Sciences (ICDS), Oct. 2020.

M. Benyettou, Contribution to the theoretical approach of the concept of wettability, 3rd cycle Ph.D. dissertation, Dept. Mec. Eng., INPL, France, 1985.

Abidi, M., Fizazi, H., Boudali, N., Clustering of Remote Sensing Data Based on Spherical Evolution Algorithm, (2021) International Review of Aerospace Engineering (IREASE), 14 (2), pp. 72-79.

Lahlouh, I., El Akkary, A., Sefiani, N., PID/Multi-Loop Control Strategy for Poultry House System Using Multi-Objective Ant Colony Optimization, (2018) International Review of Automatic Control (IREACO), 11 (5), pp. 273-280.

Gunantara, N., Nurweda Putra, I., Antara, I., The Characteristics of Multi-Criteria Weight on Ad-Hoc Network with Ant Colony Optimization, (2020) International Journal on Communications Antenna and Propagation (IRECAP), 10 (4), pp. 249-256.

Prasetyono, E., Mohammad, L., Dwi Murdianto, F., Performance of ACO-MPPT and Constant Voltage Method for Street Lighting Charging System, (2020) International Review of Electrical Engineering (IREE), 15 (3), pp. 235-244.

Mesleh, A., Battery Power Clustering Using Ant Colony Optimization, (2018) International Journal on Communications Antenna and Propagation (IRECAP), 8 (1), pp. 62-70.

Khamayseh, Y., Bani Yassein, M., Al-Nassan, H., Highly Improved Artificial Bee Colony Scheme to Enhance Coverage and Fault Tolerance in Sensor Networks, (2019) International Journal on Communications Antenna and Propagation (IRECAP), 9 (3), pp. 207-217.

Anuradha, S., Raghuram, G., Sreenivasa Murthy, K., Gurunath Reddy, B., Fast Transfer of Packets Through DB Routing Using Ant Colony Optimization, (2018) International Journal on Engineering Applications (IREA), 6 (1), pp. 35-41.


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