Modified Fatima method to solve non-linear differential equations with mix derivative

Hussein, Noor Ali; Neamah, Hayder Ismael

doi:10.47974/JIM-2290

Open Access Research Article

Modified Fatima method to solve non-linear differential equations with mix derivative

^*Noor Ali HusseinCorresponding authornoor.ali@qu.edu.iqDepartment of Mathematics College of Education University of Al-Qadisiyah Al-Diwaniyah, Iraq, Al-QadisiyahView full profile → , Hayder Ismael Neamahhayder.neamah@qu.edu.iqDepartment of Mathematics College of Education University of Al-Qadisiyah Al-Diwaniyah, 58002, Al-Qadisiyah0009-0007-6185-3188View full profile →

* Corresponding author · click or hover a name for details

pp. 1107–1117Vol. 29Issue 5-AMay 2026DOI: 10.47974/JIM-2290XML

Received:: 01 May 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2290
Pages:: 1107–1117

Abstract

This research presents an effective approach for resolving nonlinear differential equations with mixed derivatives, focusing on the Ito problem. Unlike traditional methods that require decomposing or transforming the problem into a system of differential equations, our proposed approach directly tackles the problem’s complexity while ensuring computational efficiency and precision. The Ito equation, recognized for its importance in simulating many physical and stochastic phenomena, poses considerable challenges owing to its nonlinearity and mixed-derivative terms. Our methodology uses this technique, which decomposes the mixed derivative to derive partial solutions for the other derivative of the equation, then leads to partial solutions for the equation itself, ultimately representing the entire or approximate solution to the equation. This approach guarantees reliable, accurate solutions while preserving the structural integrity of the original equation.

Keywords

Differential equations Mixed derivatives Modified Fatima method (MFM)Ito equations

Subject Classifications

Primary 93A30Secondary 49K15

References

[1] N. A. Hussein and L. N. M. Tawfiq, “New approach for solving (1+ 1)-dimensional differential equation,” J. Phys. Conf. Ser., vol. 1530, no. 1, p. 12098 (2020).
[2] A. M. Wazwaz, Partial Differential Equations: Methods and Applications. Lisse, The Netherlands: A. A. Balkema Publishers, (2002).
[3] L. N. M. Tawfiq and A. K. Jabber, “Steady State Radial Flow in Anisotropic and Homogenous in Confined Aquifers,” J. Phys. Conf. Ser., vol. 1003, no. 1, p. 12056 (2020).
[4] N. A. Hussein and F. J. Obaid, “Suggested approach to solve nonlinear ordinary differential equations,” Journal of Advances in Mathematics, vol. 22, pp. 72–74 (2023), doi: 10.24297/jam.v22i.9543.
[5] N. A. Hussein and N. N. Hani, “Using proposed approach to solve nonlinear partial differential equations,” Journal of Advances in Mathematics, vol. 22, pp. 101–105 (2023), doi: 10.24297/jam.v22i.9552.
[6] X. Niu, M. Zhang, and S. Lv, “A Darboux Transformation for the Ito Equation,” Journal of Natural Research A: A Journal of Physical Sciences, vol. 71, no. 5, pp. 427–431 (2023).
[7] A. R. Alharbi and M. B. Almatrafi, “New exact and numerical solutions with their stability for Ito integro-differential equation via Riccati–Bernoulli sub-ODE method,” Journal of Taibah University for Science, vol. 14, no. 1, pp. 1447–1456.
[8] A. H. Bhrawy, M. S. Alhuthali, and M. A. Abdelkawy, “New Solutions for (1+ 1)-Dimensional and (2+ 1)-Dimensional Ito Equations,” Math. Probl. Eng., vol. 2012, no. 1, p. 53793 (2012).
[9] A. M. Samoilenko, O. M. Stanzhyts’kyj, and A. M. Ateiwi, “On Invariant Tori for a Stochastic Ito System,” J. Dyn. Differ. Equ., vol. 17, no. 4 (2005).
[10] A. M. Kawala, “Numerical Solutions for Ito Coupled System,” Acta Appl Math, vol. 106, pp. 325–335 (2009), doi: 10.1007/s10440-008-9300-9.
[11] N. Yoshida, “Remarks on the transformation of Ito’s formula for jump-diffusion processes,” JSIAM Letters, vol. 7, pp. 29–32 (2015).
[12] W. X. Ma, “Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions,” Front. Math. China, vol. 14, no. 3, pp. 619–629 (2019), doi: 10.1007/s11464-019-0771-y.
[13] E. J. Parkes and V. O. Vakhnenko, “Explicit solutions of the Camassa–Holm equation,” Chaos Solitons Fractals, vol. 26, no. 5, pp. 1309–1316 (2005).
[14] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations. Boca Raton, FL, USA: Chapman & Hall/CRC Press, (2004).
[15] A. C. Scott, F. Y. Chu, and D. W. McLaughlin, “The soliton: a new concept in applied science,” Proceedings of the IEEE, vol. 61, no. 10, pp. 1443–1483 (1973).
[16] Y.-H. Tian and Z.-D. Dai, “Rogue Waves and New Multi-Wave Solutions of the (2+1)-Dimensional Ito Equation,” Zeitschrift für Naturforschung A: A Journal of Physical Sciences, vol. 70, no. 6, pp. 437–443 (2015).
[17] M. O. Enadi and L. N. M. Tawfiq, “New Approach for Solving Three-Dimensional Space Partial Differential Equation,” Baghdad Science Journal, vol. 16, no. 3, pp. 786–792 (2019).
[18] K. Khan and M. A. Akbar, “Exact traveling wave solutions of Kadomtsev–Petviashvili equation,” Journal of the Egyptian Mathematical Society, vol. 23, no. 2, pp. 278–281 (2015).
[19] P. R. Kundu, M. R. A. Fahim, M. E. Islam, and M. A. Akbar, “The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis,” Heliyon, vol. 7, no. 3 (2021).
[20] A. Soufyane and M. Boulmalf, “Solution of linear and nonlinear parabolic equations by the decomposition method,” Appl. Math. Comput., vol. 162, no. 2, pp. 687–693 (2005), doi: 10.1016/j.amc.2004.01.005.
[21] M. A. Mohammed and L. N. M. Tawfiq, “New accurate method for solving fourth order two-dimension evolution equations,” Journal of Interdisciplinary Mathematics, vol. 28, no. 3-B, pp. 1269–1276 (2025), doi: 10.47974/JIM-2217.
[22] Q. Wang and F. Fu, “Variational iteration method for solving differential equations with piecewise constant arguments,” IJ Engineering and Manufacturing, vol. 2, pp. 36–43 (2012).

Views: 26Downloads: 6Citations: 0