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All Journal Mikailalsys Journal of Advanced Engineering International International Journal of Education, Management, and Technology African Multidisciplinary Journal of Sciences and Artificial Intelligence
Hashim, Nura
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Hybrid Integral Transform Techniques for the Solution of Third-Order Nonlinear Ordinary Differential Equations Aliyu, Umar Mujahid; Oyewola, David Opeoluwa; Taura, Joel John; Lukunti, Salisu; Muhammad, Hassan; Adamu, Abubakar Yahya; Ibrahim, Abdulhalim Isah; Muhammad, Mubarak; Ibrahim, Imafidor Hassan; Kolo, Mohammed Abubakar; Adamu, Isah; Piapna'an, Wallen Juliet; Mansur, Mustapha Mohammed; Adamu, Ibrahim Abubakar; Marafa, Mohammed Yusuf; Umar, Abdulwasiu; Ahmad, Sulaiman; Hashim, Nura
Mikailalsys Journal of Advanced Engineering International Vol 3 No 2 (2026): Mikailalsys Journal of Advanced Engineering International
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjaei.v3i2.9236

Abstract

Third-order nonlinear ordinary differential equations frequently arise in the mathematical modeling of complex engineering and physical phenomena; however, exact analytical solutions remain difficult to obtain because of strong nonlinearities and higher-order derivative effects. Classical integral transform techniques, including the Laplace and Fourier transforms, are widely used for solving differential equations but often have limitations when extended to nonlinear systems. Although modern integral transforms such as the Sumudu, Mahgoub, and Elzaki transforms offer computational advantages, their applicability is generally restricted to linear models. This study introduces a hybrid analytical approach that integrates the Mahgoub transform with the Variational Iteration Method (VIM) to solve third-order nonlinear ordinary differential equations more effectively. The proposed method converts the governing equation into the transform domain and applies an iterative correction functional to address nonlinear terms without linearization or discretization. The resulting solutions are expressed in rapidly convergent series form. Numerical validation demonstrates strong agreement with exact solutions, confirming the efficiency, accuracy, and stability of the hybrid Mahgoub–VIM approach. The study concludes that this hybrid semi-analytical method provides a reliable framework for solving higher-order nonlinear differential equations in applied mathematics and engineering analysis. These findings contribute to the development of transform-based analytical methods by extending the applicability of the Mahgoub transform to nonlinear differential equation models through variational iteration.
A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics Yahaya, Isa; Lukunti, Salisu; Aliyu, Umar Mujahid; Ibrahim, Imafidor Hassan; Kolo, Mohammed Abubakar; Ahmad, Sulaiman; Hashim, Nura; Marafa, Mohammed Yusuf
International Journal of Education, Management, and Technology Vol 4 No 1 (2026): International Journal of Education, Management, and Technology
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ijemt.v4i1.8715

Abstract

Building on earlier developments in generalized matrix theory, this paper advances the mathematical framework of paraletrix calculus as an extension of rhotrix mathematics. Previous studies introduced matrix-tertions, matrix-ngittrets, and thotrices as intermediary structures between conventional vector and matrix forms, while subsequent work in rhotrix theory established several multiplication techniques and related results. Recognizing the need for a more flexible structure capable of accommodating unequal numbers of rows and columns, this study focuses on the paraletrix as a generalization of the thotrix. The paper aims to extend this framework by introducing the concepts of differentiation and integration within paraletrix calculus and by defining these operations with respect to an independent variable in functional form. Through this theoretical exploration, the study contributes to the further development of generalized matrix theory by broadening the analytical scope of paraletrix structures and opening new possibilities for formal mathematical operations within this extended system.
A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method Lukunti, Salisu; Aliyu, Umar Mujahid; Hussaini, Abubakar Assidiq; Ibrahim, Imafidor Hassan; Kolo, Mohammed Abubakar; Ahmad, Sulaiman; Hashim, Nura; Marafa, Mohammed Yusuf
International Journal of Education, Management, and Technology Vol 4 No 1 (2026): International Journal of Education, Management, and Technology
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ijemt.v4i1.8717

Abstract

Nonlinear differential equations arise widely in applied mathematics, physics, and engineering, yet many conventional analytical and numerical methods remain limited in their ability to handle strong nonlinearities efficiently and accurately. This paper presents a novel computational framework based on the Modified Laplace–Adomian Polynomial Method (LAPM) for solving nonlinear differential equations. The proposed method integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method, enabling complex nonlinear terms to be decomposed into rapidly convergent Adomian polynomials. This integration simplifies the solution procedure, reduces computational complexity, and preserves high accuracy. The performance of LAPM was validated using several benchmark nonlinear and linear differential equations, and the results demonstrated superior convergence speed, precision, and stability compared with traditional methods. The study concludes that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient approach for solving a broad class of nonlinear differential equations. This work contributes to the advancement of computational methods by offering a robust alternative for the analysis of differential equation models encountered in mathematics, physics, and engineering.
A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method Lukunti, Salisu; Aliyu, Umar Mujahid; Hussaini, Abubakar Assidiq; Ibrahim, Imafidor Hassan; Kolo, Mohammed Abubakar; Ahmad, Sulaiman; Hashim, Nura; Marafa, Mohammed Yusuf; Yahaya, Isa
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 1 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i1.9097

Abstract

Nonlinear differential equations pose significant challenges for conventional analytical and numerical techniques, particularly in efficiently handling complex nonlinear terms while maintaining solution accuracy and stability. This paper presents a novel computational framework for solving such equations using the Modified Laplace–Adomian Polynomial Method (LAPM), which integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method. In the proposed approach, nonlinear terms are systematically decomposed into rapidly convergent Adomian polynomials, simplifying the solution process and reducing computational complexity without compromising precision. The performance of LAPM is evaluated using several benchmark nonlinear and linear differential equations, where it exhibits superior convergence speed, accuracy, and stability when compared with traditional methods. These results demonstrate that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient tool for addressing a wide class of nonlinear differential equations in applied mathematics, physics, and engineering, and contributes to the growing repertoire of semi-analytical techniques for nonlinear problem solving.
A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics Yahaya, Isa; Lukunti, Salisu; Aliyu, Umar Mujahid; Ibrahim, Imafidor Hassan; Kolo, Mohammed Abubakar; Ahmad, Sulaiman; Hashim, Nura; Marafa, Mohammed Yusuf
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 1 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i1.9100

Abstract

This paper, titled A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics, builds upon earlier studies in generalized matrix theory by extending the structural and operational framework of non-standard matrix-like objects. Atanassov and Shannon [1] first introduced matrix-tertions and matrix-ngittrets as entities that interpolate between 2-dimensional vectors and 2×2 matrices, thereby enriching the conceptual landscape of generalized matrices. Ajibade [2] subsequently advanced the field by proposing thotrices as intermediates between 2×2 and 3×3 matrices, while further developments in rhotrix theory have established various multiplication techniques, such as heart-oriented and row–column multiplications—and yielded several important results. Recognizing the diversity of both rectangular and square matrices, the paraletrix structure was formulated as a generalization of the thotrix, allowing unequal numbers of rows and columns and thus providing a more flexible algebraic setting. This study extends the mathematical framework by introducing differentiation and integration within paraletrix calculus, defining these operations for paraletrix-valued functions with respect to an independent variable. In doing so, it lays the groundwork for a coherent calculus on paraletrices as a theoretical extension of rhotrix mathematics and generalized matrix theory.
Co-Authors Adamu, Abubakar Yahya Adamu, Ibrahim Abubakar Adamu, Isah Ahmad Sulaiman Aliyu, Umar Mujahid Hussaini, Abubakar Assidiq Ibrahim, Abdulhalim Isah Ibrahim, Imafidor Hassan Kolo, Mohammed Abubakar Lukunti, Salisu Mansur, Mustapha Mohammed Marafa, Mohammed Yusuf Muhammad, Hassan Muhammad, Mubarak Oyewola, David Opeoluwa Piapna'an, Wallen Juliet Taura, Joel John Umar, Abdulwasiu Yahaya, Isa