Anew Technique to Obtain Analytical Solutions for The Fractional Foam Drainage Formula
DOI:
https://doi.org/10.29304/jqcm.2023.15.3.1279Keywords:
Caputo operator, fractional foam drainage formula, LRPSAbstract
In the present study, an innovative approach is proposed to solve a special case of fractional foam drainage using the Laplace residual power series technique (LRPS) in conjunction with the Caputo operator for determining the fractional derivative. The study provides extensive guidelines for utilizing this approach to solve time-fractional nonlinear formulas. The effectiveness and validity of the proposed method are investigated and established by comparing the obtained results with the accurate responses using graphs. The study also confirms that the accuracy of the proposed technique increases with the number of items in the combined solution of the problems, as demonstrated by the convergence of the correlation between the obtained solutions and the actual solutions for the special case of fractional foam drainage formula. The research findings suggest that the proposed technique is not only accurate and uncomplicated but also highly adaptable, making it suitable for addressing both linear and nonlinear situations.
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References
[1] K. Oldham and J. Spanier, "The fractional calculus theory and applications of differentiation and integration to arbitrary order," Elsevier, 1974.
[2] K. S. Miller and B. Ross, "An introduction to the fractional calculus and fractional differential equations," John Wiley & Sons, 1993.
[3] S. S. Ray, "Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method," Commun. Nonlinear Sci. Numer. Simulat., vol. 14, no. 4, pp. 1295-1306, Apr. 2009.
[4] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. V. Jara, "Matrix approach to discrete fractional calculus II: partial fractional differential equations," J. Comput. Phys., vol. 228, no. 8, pp. 3137-3153, Apr. 2009.
[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, "Theory and applications of fractional differential equations," Elsevier, vol. 204, 2006.
[6] K. Diethelm, "The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type," Springer, vol. 2004, 2010.
[7] R. Caponetto, "Fractional order systems: modeling and control applications," World Scientific, vol. 72, 2010.
[8] E. Hernández, D. O'Regan, and K. Balachandran, "On recent developments in the theory of abstract differential equations with fractional derivatives," Nonlinear Anal. Theory Methods Appl., vol. 73, no. 10, pp. 3462-3471, Nov. 2010.
[9] R. Magin, M. D. Ortigueira, I. Podlubny, and J. Trujillo, "On the fractional signals and systems," Signal Process., vol. 91, no. 3, pp. 350-371, Mar. 2011.
[10] F. Mainardi, "An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity," Imperial College Press, London, 2010.
[11] V. Turut and N. Güzel, "On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximation," in Proceedings of the 2013 International Conference on Mathematical Sciences and Statistics, Kuala Lumpur, Malaysia, June 2013, pp. 26-34.
[12] K. Al-Khaled and S. Momani, "An approximate solution for a fractional diffusion-wave equation using the decomposition method," Appl. Math. Comput., vol. 165, no. 2, pp. 473-483, Mar. 2005.
[13] Y. Hu, Y. Luo, and Z. Lu, "Analytical solution of the linear fractional differential equation by Adomian decomposition method," J. Comput. Appl. Math., vol. 215, no. 1, pp. 220-229, Jan. 2008.
[14] V. Parthiban and K. Balachandran, "Solutions of system of fractional partial differential equations," Appl. Appl. Math. Int. J. (AAM), vol. 8, no. 1, pp. 17-32, 2013.
[15] K. Al-Khaled and M. Alquran, "An approximate solution for a fractional model of generalized Harry Dym equation," Math. Sci., vol. 9, no. 2, pp. 75-85, 2015.
[16] M. Dehghan, J. Manafian, and A. Saadatmandi, "Solving nonlinear fractional partial differential equations using the homotopy analysis method," Numer. Methods Partial Differ. Equ., vol. 26, no. 2, pp. 448-479, Mar. 2010.
[17] J. Jafaria, M. Nazarib, D. Baleanuc, and C. M. Khalique, "A new approach for solving a system of fractional partial differential equations," Comput. Math. Appl., vol. 66, no. 5, pp. 838-843, Sep. 2013.
[18] A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, and W. Weera, "Evaluation of time-fractional Fisher's equations with the help of analytical methods," AIMS Math., vol. 7, no. 10, pp. 18746-18766, 2022.
[19] A. El-Ajou, "Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach," Eur. Phys. J. Plus, vol. 136, no. 2, 2021.
[20] T. Eriqat, A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, and S. Momani, "A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations," Chaos, Solitons Fractals, vol. 138, p. 109957, Jul. 2020.
[2] K. S. Miller and B. Ross, "An introduction to the fractional calculus and fractional differential equations," John Wiley & Sons, 1993.
[3] S. S. Ray, "Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method," Commun. Nonlinear Sci. Numer. Simulat., vol. 14, no. 4, pp. 1295-1306, Apr. 2009.
[4] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. V. Jara, "Matrix approach to discrete fractional calculus II: partial fractional differential equations," J. Comput. Phys., vol. 228, no. 8, pp. 3137-3153, Apr. 2009.
[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, "Theory and applications of fractional differential equations," Elsevier, vol. 204, 2006.
[6] K. Diethelm, "The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type," Springer, vol. 2004, 2010.
[7] R. Caponetto, "Fractional order systems: modeling and control applications," World Scientific, vol. 72, 2010.
[8] E. Hernández, D. O'Regan, and K. Balachandran, "On recent developments in the theory of abstract differential equations with fractional derivatives," Nonlinear Anal. Theory Methods Appl., vol. 73, no. 10, pp. 3462-3471, Nov. 2010.
[9] R. Magin, M. D. Ortigueira, I. Podlubny, and J. Trujillo, "On the fractional signals and systems," Signal Process., vol. 91, no. 3, pp. 350-371, Mar. 2011.
[10] F. Mainardi, "An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity," Imperial College Press, London, 2010.
[11] V. Turut and N. Güzel, "On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximation," in Proceedings of the 2013 International Conference on Mathematical Sciences and Statistics, Kuala Lumpur, Malaysia, June 2013, pp. 26-34.
[12] K. Al-Khaled and S. Momani, "An approximate solution for a fractional diffusion-wave equation using the decomposition method," Appl. Math. Comput., vol. 165, no. 2, pp. 473-483, Mar. 2005.
[13] Y. Hu, Y. Luo, and Z. Lu, "Analytical solution of the linear fractional differential equation by Adomian decomposition method," J. Comput. Appl. Math., vol. 215, no. 1, pp. 220-229, Jan. 2008.
[14] V. Parthiban and K. Balachandran, "Solutions of system of fractional partial differential equations," Appl. Appl. Math. Int. J. (AAM), vol. 8, no. 1, pp. 17-32, 2013.
[15] K. Al-Khaled and M. Alquran, "An approximate solution for a fractional model of generalized Harry Dym equation," Math. Sci., vol. 9, no. 2, pp. 75-85, 2015.
[16] M. Dehghan, J. Manafian, and A. Saadatmandi, "Solving nonlinear fractional partial differential equations using the homotopy analysis method," Numer. Methods Partial Differ. Equ., vol. 26, no. 2, pp. 448-479, Mar. 2010.
[17] J. Jafaria, M. Nazarib, D. Baleanuc, and C. M. Khalique, "A new approach for solving a system of fractional partial differential equations," Comput. Math. Appl., vol. 66, no. 5, pp. 838-843, Sep. 2013.
[18] A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, and W. Weera, "Evaluation of time-fractional Fisher's equations with the help of analytical methods," AIMS Math., vol. 7, no. 10, pp. 18746-18766, 2022.
[19] A. El-Ajou, "Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach," Eur. Phys. J. Plus, vol. 136, no. 2, 2021.
[20] T. Eriqat, A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, and S. Momani, "A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations," Chaos, Solitons Fractals, vol. 138, p. 109957, Jul. 2020.
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Published
2023-09-30
How to Cite
Ibrahim, B. A. H., Isewd, R. A., & Yaseen, S. R. (2023). Anew Technique to Obtain Analytical Solutions for The Fractional Foam Drainage Formula. Journal of Al-Qadisiyah for Computer Science and Mathematics, 15(3), Math Page 26–33. https://doi.org/10.29304/jqcm.2023.15.3.1279
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Math Articles
