An efficacious improvement of the Anuj transformation method for solving higher-order fractional differential equations with constant coefficients

Adnan Ahmad Mahmud; Hozan Hilmi; Kalsum Abdulrahman Muhamad; Adnan Ahmad Mahmud; Hozan Hilmi; Kalsum Abdulrahman Muhamad

doi:10.3934/mmc.2026011

Mathematical Modelling and Control

2026, Volume 6, Issue 2: 140-150. doi: 10.3934/mmc.2026011

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Research article

An efficacious improvement of the Anuj transformation method for solving higher-order fractional differential equations with constant coefficients

1.
Harran University, Faculty of Arts and Sciences, Department of Mathematics, 63290, Şanlıurfa, Türkiye
2.
Information Technology Department, Tishk International University, Kurdistan Region, Iraq
3.
Department of Mathematics, College of Science, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq

Received: 03 July 2024 Revised: 08 April 2025 Accepted: 11 July 2025 Published: 20 May 2026

In this study, we obtained the fractional formula of the Anuj transformation, which is utilized to acquire an accurate outcome for linear fractional differential equations (LFDEs). It is employed for Riemann-Liouville's and Caputo's fractional derivatives. To do this, we started with developing the Anuj transform of basic functions in mathematics and then examined its primary properties, which may be used in solving different mathematical models, especially fractional differential equations. We then proceeded to present the exact solution for a particular case of a fractional differential equation. We explored four numerical challenges and presented a thorough solution for each to illustrate how the studied transform may be useful. The findings revealed that the newly recommended transformation and the specific solutions that have been supplied are more effective and straightforward in solving mathematical models. The obtained formula has been utilized to solve different cases of fractional differential equations and reach a precise solution. The outcomes have been expressed using two-dimensional graphs.
- fractional derivatives,
- Anuj transform,
- inverse Anuj transform,
- linear fractional differential equations
Citation: Adnan Ahmad Mahmud, Hozan Hilmi, Kalsum Abdulrahman Muhamad. An efficacious improvement of the Anuj transformation method for solving higher-order fractional differential equations with constant coefficients[J]. Mathematical Modelling and Control, 2026, 6(2): 140-150. doi: 10.3934/mmc.2026011

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Abstract

In this study, we obtained the fractional formula of the Anuj transformation, which is utilized to acquire an accurate outcome for linear fractional differential equations (LFDEs). It is employed for Riemann-Liouville's and Caputo's fractional derivatives. To do this, we started with developing the Anuj transform of basic functions in mathematics and then examined its primary properties, which may be used in solving different mathematical models, especially fractional differential equations. We then proceeded to present the exact solution for a particular case of a fractional differential equation. We explored four numerical challenges and presented a thorough solution for each to illustrate how the studied transform may be useful. The findings revealed that the newly recommended transformation and the specific solutions that have been supplied are more effective and straightforward in solving mathematical models. The obtained formula has been utilized to solve different cases of fractional differential equations and reach a precise solution. The outcomes have been expressed using two-dimensional graphs.

References

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