Solving Linear System of Fredholm Integral Equations with Homotopy Analysis and Genetic Algorithms
DOI:
https://doi.org/10.29304/jqcm.2023.15.2.1250Keywords:
System of Fredholm Integral Equations (SFIEs),, Homotopy Analysis Method (HAM),, h-curves,, Genetic Algorithm (GA).Abstract
In this paper, we propose a technique called Homotopy Analysis Method (HAM) for solving linear systems of Fredholm integral equations to find relatively close solutions. The HAM approach involves an auxiliary parameter that offers a straightforward method for adjusting and managing the region where the series of solutions converge. We demonstrate the effectiveness of the HAM approach through our experimental results. Additionally, we improve the HAM approach by incorporating a genetic algorithm (HAM-GA) to further optimize the solutions. The performance of HAM-GA is evaluated by comparing its results to those obtained by HAM, using the residual error function as a fitness function for the genetic algorithm.
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References
[1] E. Babolian, J. Biazar, and A. Vahidi, "The decomposition method applied to systems of Fredholm integral equations of the second kind," Applied Mathematics Computation, vol. 148, no. 2, pp. 443-452, 2004.
[2] A. Vahidi and M. Mokhtari, "On the decomposition method for system of linear Fredholm integral equations of the Second Kind," J. Appl. Math. Scie, vol. 2, no. 2, pp. 57-62, 2008.
[3] J. Rashidinia and M. Zarebnia, "Convergence of approximate solution of system of Fredholm integral equations," Journal of Mathematical Analysis Applications, vol. 333, no. 2, pp. 1216-1227, 2007.
[4] K. Maleknejad, N. Aghazadeh, and M. Rabbani, "Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method," Applied Mathematics Computation, vol. 175, no. 2, pp. 1229-1234, 2006.
[5] M. J. M. Javidi and c. modelling, "Modified homotopy perturbation method for solving system of linear Fredholm integral equations," vol. 50, no. 1-2, pp. 159-165, 2009.
[6] Y. Khan, K. Sayevand, M. Fardi, and M. Ghasemi, "A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations," Applied Mathematics Computation, vol. 249, pp. 229-236, 2014.
[7] M. S. Muthuvalu and J. Sulaiman, "Half-Sweep Arithmetic Mean method with composite trapezoidal scheme for solving linear Fredholm integral equations," Applied Mathematics Computation, vol. 217, no. 12, pp. 5442-5448, 2011.
[8] F. Khan, G. Mustafa, M. Omar, and H. Komal, "Numerical approach based on Bernstein polynomials for solving mixed Volterra-Fredholm integral equations," AIP Advances, vol. 7, no. 12, p. 125123, 2017.
[9] F. Khan, M. Omar, and Z. Ullah, "Discretization method for the numerical solution of 2D Volterra integral equation based on two-dimensional Bernstein polynomial," AIP Advances, vol. 8, no. 12, p. 125209, 2018.
[10] A. Jafarian and S. M. Nia, "Utilizing feed-back neural network approach for solving linear Fredholm integral equations system," Applied Mathematical Modelling, vol. 37, no. 7, pp. 5027-5038, 2013.
[11] P. Huabsomboon, B. Novaprateep, and H. Kaneko, "On Taylor-series expansion methods for the second kind integral equations," Journal of computational applied Mathematics Computation, vol. 234, no. 5, pp. 1466-1472, 2010.
[12] S. S. Ray and P. Sahu, "Numerical methods for solving Fredholm integral equations of second kind," in Abstract and Applied Analysis, 2013, vol. 2013: Hindawi.
[13] K. Maleknejad and A. Arzhang, "Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method," Applied Mathematics Computation, vol. 182, no. 1, pp. 888-897, 2006.
[14] D. Baleanu, A. Jajarmi, H. Mohammadi, and S. Rezapour, "A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative," Chaos, Solitons Fractals, vol. 134, p. 109705, 2020.
[15] A. H. Ali, G. A. Meften, O. Bazighifan, M. Iqbal, S. Elaskar, and J. Awrejcewicz, “A study of continuous dependence and symmetric properties of double diffusive convection: forchheimer model,” Symmetry, vol. 14, no. 4, p. 682, 2022.
[16] G. A. Meften, A. H. Ali, K. S. Al-Ghafri, J. Awrejcewicz, and O. Bazighifan, “Nonlinear stability and linear instability of double-diffusive convection in a rotating with LTNE effects and symmetric properties: brinkmann-forchheimer model,” Symmetry, vol. 14, no. 3, p. 565, 2022.
[17] G. A. Meften, A. H. Ali, and M. T. Yaseen, “Continuous dependence for thermal convection in a Forchheimer-Brinkman model with variable viscosity,” AIP Conference Proceedings, vol. 2457, no. 1, p. 020005, 2022.
[18] T. Omay and D. Baleanu, "Fractional unit-root tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid-19," Advances in Difference Equations, vol. 2021, no. 1, p. 167, 2021.
[19] R. T. Alqahtani, A. Yusuf, and R. P. Agarwal, "Mathematical analysis of oxygen uptake rate in continuous process under caputo derivative," Mathematics, vol. 9, no. 6, p. 675, 2021.
[20] D. Baleanu, B. Ghanbari, J. H. Asad, A. Jajarmi, and H. M. Pirouz, "Planar system-masses in an equilateral triangle: numerical study within fractional calculus," Computer Modeling in Engineering Sciences, vol. 124, no. 3, pp. 953-968, 2020.
[21] I. Jaradat, M. Alquran, S. Sivasundaram, and D. Baleanu, "Simulating the joint impact of temporal and spatial memory indices via a novel analytical scheme," Nonlinear Dynamics, vol. 103, no. 3, pp. 2509-2524, 2021.
[22] N. H. Tuan, D. Baleanu, T. N. Thach, D. O’Regan, and N. H. Can, "Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data," Journal of Computational Applied Mathematics Computation, vol. 376, p. 112883, 2020.
[23] S. Qureshi and A. Yusuf, "A new third order convergent numerical solver for continuous dynamical systems," Journal of King Saud University-Science, vol. 32, no. 2, pp. 1409-1416, 2020.
[24] A. Georgieva and S. Hristova, "Homotopy analysis method to solve two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations," Fractal Fractional, vol. 4, no. 1, p. 9, 2020.
[25] S. Liao, Beyond perturbation: introduction to the homotopy analysis method. CRC press, 2003.
[26] S. Liao, "Comparison between the homotopy analysis method and homotopy perturbation method," vol. 169, no. 2, pp. 1186-1194, 2005.
[27] S. Liao, Homotopy analysis method in nonlinear differential equations. Springer, 2012.
[28] S. Liao, Advances in the homotopy analysis method. World Scientific, 2013.
[29] A. Jafarian, P. Ghaderi, A. K. Golmankhaneh, and D. Baleanu, "Analytical treatment of system of Abel integral equations by homotopy analysis method," Rom. Rep. Phys, vol. 66, no. 3, pp. 603-611, 2014.
[30] E. Hetmaniok, D. Słota, T. Trawiński, and R. Wituła, "Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind," Numerical Algorithms, vol. 67, pp. 163-185, 2014.
[31] O. Kramer, Genetic algorithm essentials Springer, 2017.
[32] K.-F. Man, K.-S. Tang, and S. Kwong, "Genetic algorithms: concepts and applications [in engineering design]," IEEE transactions on Industrial Electronics, vol. 43, no. 5, pp. 519-534, 1996.
[33] A. Jayachitra and R. Vinodha, "Genetic algorithm based PID controller tuning approach for continuous stirred tank reactor," Advances in Artificial Intelligence, vol. 2014, pp. 9-9, 2015.
[34] J.-H. He, M. H. Taha, M. A. Ramadan, and G. M. Moatimid, "A combination of bernstein and improved block-pulse functions for solving a system of linear fredholm integral equations," Mathematical Problems in Engineering, vol. 2022, pp. 1-12, 2022.
[35] Z. Elahi, S. S. Siddiqi, and G. Akram, "Laguerre method for solving linear system of Fredholm integral equations," International Journal of Computer Mathematics, vol. 98, no. 11, pp. 2175-2185, 2021.
[2] A. Vahidi and M. Mokhtari, "On the decomposition method for system of linear Fredholm integral equations of the Second Kind," J. Appl. Math. Scie, vol. 2, no. 2, pp. 57-62, 2008.
[3] J. Rashidinia and M. Zarebnia, "Convergence of approximate solution of system of Fredholm integral equations," Journal of Mathematical Analysis Applications, vol. 333, no. 2, pp. 1216-1227, 2007.
[4] K. Maleknejad, N. Aghazadeh, and M. Rabbani, "Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method," Applied Mathematics Computation, vol. 175, no. 2, pp. 1229-1234, 2006.
[5] M. J. M. Javidi and c. modelling, "Modified homotopy perturbation method for solving system of linear Fredholm integral equations," vol. 50, no. 1-2, pp. 159-165, 2009.
[6] Y. Khan, K. Sayevand, M. Fardi, and M. Ghasemi, "A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations," Applied Mathematics Computation, vol. 249, pp. 229-236, 2014.
[7] M. S. Muthuvalu and J. Sulaiman, "Half-Sweep Arithmetic Mean method with composite trapezoidal scheme for solving linear Fredholm integral equations," Applied Mathematics Computation, vol. 217, no. 12, pp. 5442-5448, 2011.
[8] F. Khan, G. Mustafa, M. Omar, and H. Komal, "Numerical approach based on Bernstein polynomials for solving mixed Volterra-Fredholm integral equations," AIP Advances, vol. 7, no. 12, p. 125123, 2017.
[9] F. Khan, M. Omar, and Z. Ullah, "Discretization method for the numerical solution of 2D Volterra integral equation based on two-dimensional Bernstein polynomial," AIP Advances, vol. 8, no. 12, p. 125209, 2018.
[10] A. Jafarian and S. M. Nia, "Utilizing feed-back neural network approach for solving linear Fredholm integral equations system," Applied Mathematical Modelling, vol. 37, no. 7, pp. 5027-5038, 2013.
[11] P. Huabsomboon, B. Novaprateep, and H. Kaneko, "On Taylor-series expansion methods for the second kind integral equations," Journal of computational applied Mathematics Computation, vol. 234, no. 5, pp. 1466-1472, 2010.
[12] S. S. Ray and P. Sahu, "Numerical methods for solving Fredholm integral equations of second kind," in Abstract and Applied Analysis, 2013, vol. 2013: Hindawi.
[13] K. Maleknejad and A. Arzhang, "Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method," Applied Mathematics Computation, vol. 182, no. 1, pp. 888-897, 2006.
[14] D. Baleanu, A. Jajarmi, H. Mohammadi, and S. Rezapour, "A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative," Chaos, Solitons Fractals, vol. 134, p. 109705, 2020.
[15] A. H. Ali, G. A. Meften, O. Bazighifan, M. Iqbal, S. Elaskar, and J. Awrejcewicz, “A study of continuous dependence and symmetric properties of double diffusive convection: forchheimer model,” Symmetry, vol. 14, no. 4, p. 682, 2022.
[16] G. A. Meften, A. H. Ali, K. S. Al-Ghafri, J. Awrejcewicz, and O. Bazighifan, “Nonlinear stability and linear instability of double-diffusive convection in a rotating with LTNE effects and symmetric properties: brinkmann-forchheimer model,” Symmetry, vol. 14, no. 3, p. 565, 2022.
[17] G. A. Meften, A. H. Ali, and M. T. Yaseen, “Continuous dependence for thermal convection in a Forchheimer-Brinkman model with variable viscosity,” AIP Conference Proceedings, vol. 2457, no. 1, p. 020005, 2022.
[18] T. Omay and D. Baleanu, "Fractional unit-root tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid-19," Advances in Difference Equations, vol. 2021, no. 1, p. 167, 2021.
[19] R. T. Alqahtani, A. Yusuf, and R. P. Agarwal, "Mathematical analysis of oxygen uptake rate in continuous process under caputo derivative," Mathematics, vol. 9, no. 6, p. 675, 2021.
[20] D. Baleanu, B. Ghanbari, J. H. Asad, A. Jajarmi, and H. M. Pirouz, "Planar system-masses in an equilateral triangle: numerical study within fractional calculus," Computer Modeling in Engineering Sciences, vol. 124, no. 3, pp. 953-968, 2020.
[21] I. Jaradat, M. Alquran, S. Sivasundaram, and D. Baleanu, "Simulating the joint impact of temporal and spatial memory indices via a novel analytical scheme," Nonlinear Dynamics, vol. 103, no. 3, pp. 2509-2524, 2021.
[22] N. H. Tuan, D. Baleanu, T. N. Thach, D. O’Regan, and N. H. Can, "Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data," Journal of Computational Applied Mathematics Computation, vol. 376, p. 112883, 2020.
[23] S. Qureshi and A. Yusuf, "A new third order convergent numerical solver for continuous dynamical systems," Journal of King Saud University-Science, vol. 32, no. 2, pp. 1409-1416, 2020.
[24] A. Georgieva and S. Hristova, "Homotopy analysis method to solve two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations," Fractal Fractional, vol. 4, no. 1, p. 9, 2020.
[25] S. Liao, Beyond perturbation: introduction to the homotopy analysis method. CRC press, 2003.
[26] S. Liao, "Comparison between the homotopy analysis method and homotopy perturbation method," vol. 169, no. 2, pp. 1186-1194, 2005.
[27] S. Liao, Homotopy analysis method in nonlinear differential equations. Springer, 2012.
[28] S. Liao, Advances in the homotopy analysis method. World Scientific, 2013.
[29] A. Jafarian, P. Ghaderi, A. K. Golmankhaneh, and D. Baleanu, "Analytical treatment of system of Abel integral equations by homotopy analysis method," Rom. Rep. Phys, vol. 66, no. 3, pp. 603-611, 2014.
[30] E. Hetmaniok, D. Słota, T. Trawiński, and R. Wituła, "Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind," Numerical Algorithms, vol. 67, pp. 163-185, 2014.
[31] O. Kramer, Genetic algorithm essentials Springer, 2017.
[32] K.-F. Man, K.-S. Tang, and S. Kwong, "Genetic algorithms: concepts and applications [in engineering design]," IEEE transactions on Industrial Electronics, vol. 43, no. 5, pp. 519-534, 1996.
[33] A. Jayachitra and R. Vinodha, "Genetic algorithm based PID controller tuning approach for continuous stirred tank reactor," Advances in Artificial Intelligence, vol. 2014, pp. 9-9, 2015.
[34] J.-H. He, M. H. Taha, M. A. Ramadan, and G. M. Moatimid, "A combination of bernstein and improved block-pulse functions for solving a system of linear fredholm integral equations," Mathematical Problems in Engineering, vol. 2022, pp. 1-12, 2022.
[35] Z. Elahi, S. S. Siddiqi, and G. Akram, "Laguerre method for solving linear system of Fredholm integral equations," International Journal of Computer Mathematics, vol. 98, no. 11, pp. 2175-2185, 2021.
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Published
2023-09-25
How to Cite
Ahmed, R. F., Al-Hayani, W., & Al-Bayati, A. Y. (2023). Solving Linear System of Fredholm Integral Equations with Homotopy Analysis and Genetic Algorithms. Journal of Al-Qadisiyah for Computer Science and Mathematics, 15(2), Math Page. 65–86. https://doi.org/10.29304/jqcm.2023.15.2.1250
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