Authors:
Amandeep Singh,Sarita Pippal,DOI NO:
https://doi.org/10.26782/jmcms.spl.11/2024.05.00001Keywords:
ZZ Transform,Sumudu Transform,Adomian Polynomials,Caputo's system of Fractional Partial Differential Equations (FPDE),Abstract
By combining the ZZ transform with Adomian polynomials, the semi-analytical solutions to nonlinear Caputo partial fractional differential equations have been derived in this work. The Caputo sense has been applied to the fractional derivative. Using the proposed method, several fractional partial differential equations have been resolved. When compared to other similar procedures, it has been shown that applying the ZZ transform and breaking down the nonlinear components using Adomian polynomials is quite convenient.Refference:
I. Adomian G., : ‘A new approach to nonlinear partial differential equations’. J. Math. Anal. Appl. Vol. 102, pp. 420–434, (1984). 10.1016/0022-247X(84)90182-3
II. Arshad S., A. Sohail, and K. Maqbool. : ‘Nonlinear shallow water waves: a fractional order approach’. Alexandria Engineering Journal. vol. 55(1), pp. 525–532, (2016). 10.1016/j.aej.2015.10.014
III. Arshad S., A. M. Siddiqui, A. Sohail, K. Maqbool, and Z. Li. : ‘Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons’. Advances in Mechanical Engineering. vol. 9(3), (2017) 10.1177/1687814017692
IV. Bhalekar S., and V. Daftardar-Gejji. : ‘Solving evolution equations using a new iterative method’. Numerical Methods for Partial Differential Equations: An International Journal, vol. 26(4), pp. 906–916, (2010). 10.1002/num.20463
V. Fethi et al., : ‘SUMUDU TRANSFORM FUNDAMENTAL PROPERTIES INVESTIGATIONS AND APPLICATIONS’. Journal of Applied Mathematics and Stochastic Analysis. Volume 2006, Article ID 91083, Pages 1–23. 10.1155/JAMSA/2006/91083
VI. Hilfer R., : ‘Applications of Fractional Calculus in Physics’. World Scientific, Singapore, Singapore. 2000. 10.1142/3779
VII. Jassim H. K., and H. A. Kadhim. : ‘Fractional Sumudu decomposition method for solving PDEs of fractional order’. J. Appl. Comput. Mech. Vol. 7, pp. 302–311. (2021). 10.22055/JACM.2020.31776.1920
VIII. Katatbeh Q. D., and F. B. M. Belgacem. : ‘Applications of the Sumudu transform to fractional differential equations’. Non-Linear Studies. vol. 18(1), pp. 99–112, (2011).
IX. Lakhdar R., and Mountassir H. Cherif. : ‘A Precise Analytical Method to Solve the Nonlinear System of Partial Differential Equations With the Caputo Fractional Operator’. Cankaya University Journal of Science and Engineering. vol. 19(1), pp. 29-39, (2022). https://dergipark.org.tr/tr/download/article-file/1953334
X. Lakhdar R., et al., : ‘An efficient approach to solving the Fractional SIR Epidemic Model with the Atangana-Baleanu-Caputo Fractional Operator’. Fractal and Fractional. Vol. 7, pp. 618, (2023). 10.3390/fractalfract7080618
XI. Liu Y., : ‘Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method’. Hindawi Publishing Corporation Abstract and Applied Analysis. Volume 2012, Article ID 752869, 14 pages. 10.1155/2012/752869
XII. Metzler R., and J. Klafter. : ‘The random walk’s guide to anomalous diffusion: a fractional dynamics approach’. Physics Reports, vol. 339(1), pp. 1-77, (2000). 10.1016/S0370-1573(00)00070-3
XIII. Moazzam A., and M. Iqbal. : ‘A NEW INTEGRAL TRANSFORMATION “ALI AND ZAFAR” TRANSFORMATION AND ITS APPLICATIONS IN NUCLEAR PHYSICS’. June 2022.
XIV. Podlubny I. : ‘Fractional Differential Equations’. Academic Press, New York, NY, USA. 1999.
XV. Riabi L. A., et al., : ‘HOMOTOPY PERTURBATION METHOD COMBINED WITH ZZ TRANSFORM TO SOLVE SOME NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS’. International Journal of Analysis and Applications. Volume 17(3), pp. 406-419. (2019). https://etamaths.com/index.php/ijaa/article/view/1875
XVI. Shah R., H. Khan M. Arif and P. Kumam. : ‘Application of Laplace-Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations’. Entropy (Basel). Vol. 21(4), pp. 335, (2019). 10.3390/e21040335
XVII. Singh A., and S. Pippal. : ‘Solution of nonlinear fractional partial differential equations by Shehu transform and Adomian decomposition method (STADM)’. International Journal of Mathematics for Industry. 2350011 (18 pages). 10.1142/S2661335223500119
XVIII. Sontakke R., and R. Pandit. : ‘Convergence analysis and approximate solution of fractional differential equations’. : ‘Malaya Journal of Matematik’. Vol. 7(2), pp. 338-344, (2019). 10.26637/MJM0702/0029
XIX. Wang K., and S. Liu. : ‘A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation’. Springer Plus Vol. 5, 865 (2016). https://springerplus.springeropen.com/articles/10.1186/s40064-016-2426-8
XX. Wu G. C., and D. Baleanu. : ‘Variational iteration method for fractional calculus – a universal approach by Laplace transform’. Adv Differ Equ 2013, 18 (2013). 10.1186/1687-1847-2013-18
XXI. X. J. Yang, H. M. Srivastava, and C. Cattani. : ‘Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics’. Romanian Reports in Physics, vol. 67(3), pp. 752–761, 2015. https://rrp.nipne.ro/2015_67_3/A2.pdf
XXII. Zafar Z. U. A., : ‘Application of ZZ transform method on some fractional differential equations’. Int J Adv Eng Global Technol. Vol. 4(13), pp. 55–63, (2016).