Algorithms and applications of the new modified decomposition method to solve initial-boundary value problems for fractional partial differential equations

Mariam Al-Mazmumy; Mona Alsulami; Norah AL-Yazidi; Mariam Al-Mazmumy; Mona Alsulami; Norah AL-Yazidi

doi:10.3934/math.2025267

AIMS Mathematics

2025, Volume 10, Issue 3: 5806-5829. doi: 10.3934/math.2025267

Previous Article Next Article

Research article

Algorithms and applications of the new modified decomposition method to solve initial-boundary value problems for fractional partial differential equations

Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah 23218, Saudi Arabia

Received: 13 January 2025 Revised: 25 February 2025 Accepted: 04 March 2025 Published: 17 March 2025
MSC : 35A25, 35C05, 35G16, 35R11

This study aims to efficiently solve the class of initial-boundary problems for fractional partial differential equations by expanding the modified Adomian decomposition method. Three new algorithms are proposed, two for handling various fractional initial-boundary problems via the $ x $-differential operator, and one for using the $ t $-differential operator. The proposed methods are designed to increase computational efficiency and ensure greater accuracy in the solutions. The effectiveness of the techniques is reviewed by applying them to a variety of cases, demonstrating their ability to address a broad range of fractional calculus problems. The results emphasize the flexibility and adaptability of the developed algorithms as reliable methods.
- fractional differential equations,
- advanced decomposition method,
- initial and boundary value conditions,
- Riemann–Liouville derivative,
- Caputo derivative
Citation: Mariam Al-Mazmumy, Mona Alsulami, Norah AL-Yazidi. Algorithms and applications of the new modified decomposition method to solve initial-boundary value problems for fractional partial differential equations[J]. AIMS Mathematics, 2025, 10(3): 5806-5829. doi: 10.3934/math.2025267

Related Papers:

Abstract

This study aims to efficiently solve the class of initial-boundary problems for fractional partial differential equations by expanding the modified Adomian decomposition method. Three new algorithms are proposed, two for handling various fractional initial-boundary problems via the $ x $-differential operator, and one for using the $ t $-differential operator. The proposed methods are designed to increase computational efficiency and ensure greater accuracy in the solutions. The effectiveness of the techniques is reviewed by applying them to a variety of cases, demonstrating their ability to address a broad range of fractional calculus problems. The results emphasize the flexibility and adaptability of the developed algorithms as reliable methods.

References

[1]	M. P. Lazarevic, M. R. Rapaić, T. B. Šekara, Introduction to fractional calculus with brief historical background, In: Advanced topics on applications of fractional calculus on control problems, system stability and modeling, 2014, 3–16.
[2]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, 1974.
[3]	R. Herrmann, Fractional calculus within the optical model used in nuclear and particle physics, J. Phys. G Nucl. Part. Phys., 50 (2023), 065102.
[4]	C. M. A. Pinto, A. R. M. Carvalho, A latency fractional order model for HIV dynamics, J. Comput. Appl. Math., 312 (2017), 240–256. https://doi.org/10.1016/j.cam.2016.05.019 doi: 10.1016/j.cam.2016.05.019
[5]	M. Shi, Z. Wang, Abundant bursting patterns of a fractional-order Morris–Lecar neuron model, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1956–1969. https://doi.org/10.1016/j.cnsns.2013.10.032 doi: 10.1016/j.cnsns.2013.10.032
[6]	A. I. K. Butt, W. Ahmad, M. Rafiq, D. Baleanu, Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic, Alexandria Eng. J., 61 (2022), 7007–7027. https://doi.org/10.1016/j.aej.2021.12.042 doi: 10.1016/j.aej.2021.12.042
[7]	E. F. D. Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Modell. Anal., 21 (2016), 188–198. https://doi.org/10.3846/13926292.2016.1145607 doi: 10.3846/13926292.2016.1145607
[8]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, World Scientific, 2010. https://doi.org/10.1142/p614
[9]	J. Singh, J. Y. Hristov, Z. Hammouch, New trends in fractional differential equations with real-world applications in physics, Frontiers Media SA, 2020.
[10]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[11]	M. I. Asjad, R. Karim, A. Hussanan, A. Iqbal, S. M. Eldin, Applications of fractional partial differential equations for MHD Casson fluid flow with innovative ternary nanoparticles, Processes, 11 (2023), 218. https://doi.org/10.3390/pr11010218 doi: 10.3390/pr11010218
[12]	B. Baeumer, M. Kovacs, M. M. Meerschaert, Numerical solutions for fractional reaction–diffusion equations, Comput. Math. Appl., 55 (2008), 2212–2226. https://doi.org/10.1016/j.camwa.2007.11.012 doi: 10.1016/j.camwa.2007.11.012
[13]	E. Firmansah, M. F. Rosyid, Study of heat equations with boundary differential equations, J. Phys. Conf. Ser., 1170 (2019), 012015. https://doi.org/10.1088/1742-6596/1170/1/012015 doi: 10.1088/1742-6596/1170/1/012015
[14]	J. J. Foit, Spreading under variable viscosity and time-dependent boundary conditions: Estimate of viscosity from spreading experiments, Nucl. Eng. Des., 227 (2004), 239–253. https://doi.org/10.1016/j.nucengdes.2003.10.002 doi: 10.1016/j.nucengdes.2003.10.002
[15]	M. D. Vahey, D. A. Fletcher, The biology of boundary conditions: Cellular reconstitution in one, two, and three dimensions, Curr. Opin. Cell Biol., 26 (2014), 60–68. https://doi.org/10.1016/j.ceb.2013.10.001 doi: 10.1016/j.ceb.2013.10.001
[16]	A. Elsaid, Homotopy analysis method for solving a class of fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3655–3664. https://doi.org/10.1016/j.cnsns.2010.12.040 doi: 10.1016/j.cnsns.2010.12.040
[17]	M. Z. Mohamed, T. M. Elzaki, Comparison between the Laplace decomposition method and Adomian decomposition in time-space fractional nonlinear fractional differential equations, Appl. Math., 9 (2018), 448–458. https://doi.org/10.4236/am.2018.94032 doi: 10.4236/am.2018.94032
[18]	Z. Odibat, S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Modell., 32 (2008), 28–39. https://doi.org/10.1016/j.apm.2006.10.025 doi: 10.1016/j.apm.2006.10.025
[19]	S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 355 (2006), 271–279. https://doi.org/10.1016/j.physleta.2006.02.048 doi: 10.1016/j.physleta.2006.02.048
[20]	H. Ahmad, T. A. Khan, I. Ahmad, P. S. Stanimirovi, Y. Chu, A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations, Results Phys., 19 (2020), 103462. https://doi.org/10.1016/j.rinp.2020.103462 doi: 10.1016/j.rinp.2020.103462
[21]	S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006), 264–274. https://doi.org/10.1016/j.jcp.2005.12.006 doi: 10.1016/j.jcp.2005.12.006
[22]	S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365 (2007), 345–350. https://doi.org/10.1016/j.physleta.2007.01.046 doi: 10.1016/j.physleta.2007.01.046
[23]	H. K. Jassim, H. Kadmim, Fractional Sumudu decomposition method for solving PDEs of fractional order, J. Appl. Comput. Mech., 7 (2021), 302–311. https://doi.org/10.22055/jacm.2020.31776.1920 doi: 10.22055/jacm.2020.31776.1920
[24]	G. Adomian, Nonlinear stochastic systems theory and applications to physics, In: Mathematics and its applications, Dordrecht: Springer, 46 (1989).
[25]	A. Wazwaz, Partial differential equations and solitary waves theory, In: Nonlinear physical science, Heidelberg: Springer Berlin, 2010. https://doi.org/10.1007/978-3-642-00251-9
[26]	I. L. El-Kalla, New results on the analytic summation of Adomian series for some classes of differential and integral equations, Appl. Math. Comput., 217 (2010), 3756–3763. https://doi.org/10.1016/j.amc.2010.09.034 doi: 10.1016/j.amc.2010.09.034
[27]	M. Khan, M. Hussain, H. Jafari, Y. Khan, Application of Laplace decomposition method to solve nonlinear coupled partial differential equations, World Appl. Sci. J., 9 (2010), 13–19.
[28]	R. S. Teppawar, R. N. Ingle, R. A. Muneshwar, Solving nonlinear time-fractional partial differential equations using conformable fractional reduced differential transform with Adomian decomposition method, Contemp. Math., 5 (2024), 853–872. https://doi.org/10.37256/cm.5120242463
[29]	H. Thabet, S. Kendre, New modification of Adomian decomposition method for solving a system of nonlinear fractional partial differential equations, Int. J. Adv. Appl. Math. Mech., 6 (2019), 1–13.
[30]	K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29 (1995), 103–108. https://doi.org/10.1016/0898-1221(95)00022-Q doi: 10.1016/0898-1221(95)00022-Q
[31]	M. M. Hosseini, H. Nasabzadeh, On the convergence of Adomian decomposition method, Appl. Math. Comput., 182 (2006), 536–543. https://doi.org/10.1016/j.amc.2006.04.015 doi: 10.1016/j.amc.2006.04.015
[32]	K. Abbaoui, Y. Cherruault, Convergence of Adomian's method applied to differential equations, Comput. Math. Appl., 28 (1994), 103–109. https://doi.org/10.1016/0898-1221(94)00144-8 doi: 10.1016/0898-1221(94)00144-8
[33]	R. Rajaram, M. Najafi, Analytical treatment and convergence of the Adomian decomposition method for a system of coupled damped wave equations, Appl. Math. Comput., 212 (2009), 72–81. https://doi.org/10.1016/j.amc.2009.02.006 doi: 10.1016/j.amc.2009.02.006
[34]	I. L. El-Kalla, Piece-wise continuous solution to a class of nonlinear boundary value problems, Ain Shams Eng. J., 4 (2013), 325–331. https://doi.org/10.1016/j.asej.2012.08.011 doi: 10.1016/j.asej.2012.08.011
[35]	J. Biazar, K. Hosseini, A modified Adomian decomposition method for singular initial value Emden-Fowler type equations, Int. J. Appl. Math. Res., 5 (2016), 69–72. http://dx.doi.org/10.14419/ijamr.v5i1.5666 doi: 10.14419/ijamr.v5i1.5666
[36]	J. Biazar, K. Hosseini, An effective modification of Adomian decomposition method for solving Emden-Fowler type systems, Natl. Acad. Sci. Lett., 40 (2017), 285–290. https://doi.org/10.1007/s40009-017-0571-4 doi: 10.1007/s40009-017-0571-4
[37]	M. Al-Mazmumy, M. Alsulami, Utilization of the modified Adomian decomposition method on the Bagley-Torvik equation amidst Dirichlet boundary conditions, Eur. J. Pure Appl. Math., 17 (2024), 546–568. https://doi.org/10.29020/nybg.ejpam.v17i1.5050 doi: 10.29020/nybg.ejpam.v17i1.5050
[38]	M. Al-Mazmumy, M. A. Alyami, M. Alsulami, A. S. Alsulami, Efficient modified Adomian decomposition method for solving nonlinear fractional differential equations, Int. J. Anal. Appl., 22 (2024), 76. https://doi.org/10.28924/2291-8639-22-2024-76 doi: 10.28924/2291-8639-22-2024-76
[39]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, In: North-Holland mathematics studies, Elsevier, 2006.
[40]	D. Lesnic, The Cauchy problem for the wave equation using the decomposition method, Appl. Math. Lett., 15 (2002), 697–701. http://dx.doi.org/10.1016/S0893-9659(02)00030-7 doi: 10.1016/S0893-9659(02)00030-7
[41]	M. Entezari, S. Abbasbandy, E. Babolian, Numerical solution of fractional partial differential equations with normalized bernstein wavelet method, Appl. Appl. Math., 14 (2019), 17.
[42]	A. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102 (1999), 77–86. https://doi.org/10.1016/S0096-3003(98)10024-3 doi: 10.1016/S0096-3003(98)10024-3
[43]	Z. Odibat, S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58 (2009), 2199–2208. https://doi.org/10.1016/j.camwa.2009.03.009 doi: 10.1016/j.camwa.2009.03.009
[44]	A. M. A. Sayed, M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A, 359 (2006), 175–182. https://doi.org/10.1016/j.physleta.2006.06.024 doi: 10.1016/j.physleta.2006.06.024

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)