Garuda - Garba Rujukan Digital

Robust Integral Transform Methods for the Solution of Nonlinear Fractional Ordinary Differential Equations in Viscoelastic and Biological Systems Umar Mujahid Aliyu; David Opeoluwa Oyewola; Joel John Taura; Salisu Lukunti; Hassan Muhammad; Abubakar Yahya Adamu; Abdulhalim Isah Ibrahim; Mubarak Muhammad; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Isah Adamu; Wallen Juliet Piapna'an; Mustapha Mohammed Mansur; Ibrahim Abubakar Adamu; Mohammed Yusuf Marafa; Abdulwasiu Umar; Sulaiman Ahmad; Nura Hashim
Mikailalsys Journal of Mathematics and Statistics Vol 4 No 2 (2026): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v4i2.9237

Nonlinear and fractional-order differential equations frequently arise in viscoelastic and biological systems; however, their solution remains challenging due to the presence of nonlocal operators, memory effects, and complex boundary conditions. Classical integral transforms, including the Laplace and Fourier transforms, often have limitations in addressing these features effectively. This study presents a robust hybrid methodology that combines the Mahgoub Transform with the Variational Iteration Method (VIM) to solve nonlinear and fractional-order ordinary differential equations (ODEs). The proposed approach was systematically applied to linear, nonlinear, and fractional-order ODEs to evaluate its convergence, accuracy, and capacity to handle memory-dependent effects. The findings demonstrate that the Mahgoub–VIM method achieves rapid convergence, high accuracy, and improved performance compared with traditional transforms such as the Sumudu Transform. These results indicate that the proposed method provides a reliable and efficient analytical framework for modeling complex viscoelastic and biological phenomena governed by nonlinear and fractional-order dynamics. This study contributes to the advancement of integral transform-based solution methods and offers practical implications for the mathematical modeling of systems characterized by memory-dependent behavior and nonlinear responses.

An Overview of Integral Transformation Methods for Solving Physical Problems Umar Mujahid Aliyu; A. G. Madaki; A. M. Kwami; M. I. Bello; J. O. Okai; Abubakar Assidiq Hussaini
Mikailalsys Journal of Mathematics and Statistics Vol 4 No 3 (2026): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v4i3.9378

Integral transformation methods are widely used to solve physical problems formulated through differential equations; however, their effectiveness may vary across linear, nonlinear, and fractional systems. This study provides an analytical overview of major integral transforms, with particular emphasis on the Kamal and Laplace transforms and their integration with decomposition-based techniques, including the Adomian decomposition method. Through a critical discussion of relevant analytical procedures and illustrative examples, the study examines the applicability, efficiency, and accuracy of these methods in solving linear, nonlinear, and fractional differential equations arising in physical systems. The analysis indicates that integral transforms offer efficient and accurate solution procedures, particularly for linear differential equations. Nevertheless, their direct application to nonlinear and fractional problems presents computational and analytical challenges, thereby requiring hybrid approaches that combine transformation techniques with decomposition methods. The study concludes that hybrid integral-transform methods can extend the applicability of conventional analytical techniques to more complex differential equations. It contributes to the literature by synthesizing the strengths and limitations of integral-transform approaches and identifying opportunities to improve their computational efficiency and applicability in modeling physical systems.

Integrated Mahgoub–VIM Hybrid Transform Technique for Solving Linear, Nonlinear, and Fractional Differential Equations Umar Mujahid Aliyu; A. M. Kwami; M. I. Bello; A. G. Madaki; J. O. Okai; Abubakar Assidiq Hussaini
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 3 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i3.9234

This study develops an integrated Mahgoub–Variational Iteration Method (VIM) hybrid transform technique for solving linear, nonlinear, and fractional-order ordinary and partial differential equations. The study addresses the limitations of classical integral transforms in handling nonlinearities, fractional derivatives, and memory-dependent effects, while ensuring physically consistent initial conditions through the Caputo fractional derivative. The proposed Mahgoub–VIM framework was applied to higher-order nonlinear ordinary differential equations, fractional ordinary differential equations, time-fractional partial differential equations, and fractional relaxation models. The results demonstrate rapid convergence, high stability, and close agreement with exact solutions. Comparative analysis further indicates that the proposed method consistently outperforms the Sumudu transform in terms of accuracy and error control, particularly for nonlinear and fractional problems. By avoiding linearization and discretization, the technique provides an efficient analytical framework for modeling realistic phenomena, including diffusion, heat transfer, viscoelasticity, and damping. The study contributes to the development of hybrid transform-based methods by offering a robust, accurate, and versatile analytical tool for solving complex differential systems.

Hybrid Integral Transform Techniques for the Solution of Third-Order Nonlinear Ordinary Differential Equations Umar Mujahid Aliyu; David Opeoluwa Oyewola; Joel John Taura; Salisu Lukunti; Hassan Muhammad; Abubakar Yahya Adamu; Abdulhalim Isah Ibrahim; Mubarak Muhammad; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Isah Adamu; Wallen Juliet Piapna'an; Mustapha Mohammed Mansur; Ibrahim Abubakar Adamu; Mohammed Yusuf Marafa; Abdulwasiu Umar; Sulaiman Ahmad; Nura Hashim
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

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 Isa Yahaya; Salisu Lukunti; Umar Mujahid Aliyu; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa
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

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 Salisu Lukunti; Umar Mujahid Aliyu; Abubakar Assidiq Hussaini; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa
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

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 Salisu Lukunti; Umar Mujahid Aliyu; Abubakar Assidiq Hussaini; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa; Isa Yahaya
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

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 Isa Yahaya; Salisu Lukunti; Umar Mujahid Aliyu; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa
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

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.

A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method Salisu Lukunti; Umar Mujahid Aliyu; Abubakar Assidiq Hussaini; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa; Isa Yahaya
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

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 Isa Yahaya; Salisu Lukunti; Umar Mujahid Aliyu; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa
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

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.

Title

Found 10 Documents
Search

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Title Search

Found 10 Documents Search

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Abstract

Title

Found 10 Documents
Search