Abstract

Abstract

This study presents a novel fractional semi-analytical iterative approach for solving nonlinear fractional Fisher’s equations using the Caputo fractional operator. The primary objective is to provide a method that yields exact solutions to nonlinear fractional equations without requiring assumptions about nonlinear terms. By applying the Temimi-Ansari Method (TAM) with fractional calculus, this approach offers a robust solution to the time-fractional nonlinear Fisher’s equation, a model relevant in fields such as population dynamics, tumor growth, and gene propagation. In this work, tables and graphical illustrations show that the proposed method minimizes computational complexity and delivers significant accuracy across multiple cases of Fisher’s equations. The findings indicate that TAM with fractional order derivatives provides accurate, efficient approximations with reduced computational workload, showcasing the technique’s potential for addressing a wide range of nonlinear fractional differential equations.OPEN ACCESS Received: 19/07/2024 Accepted: 01/11/2024

Abstract

This study presents a novel fractional semi-analytical iterative approach for solving nonlinear fractional Fisher’s equations using the Caputo fractional operator. The primary objective is to provide a method that yields exact solutions to nonlinear fractional equations without requiring assumptions about nonlinear terms. By applying the Temimi-Ansari Method (TAM) with fractional calculus, this approach offers a robust solution to the time-fractional nonlinear Fisher’s equation, a model relevant in fields such as population dynamics, tumor growth, and gene propagation. In this work, tables and graphical illustrations show that the proposed method minimizes computational complexity and delivers significant accuracy across multiple cases of Fisher’s equations. The findings indicate that TAM with fractional order derivatives provides accurate, efficient approximations with reduced computational workload, showcasing the technique’s potential for addressing a wide range of nonlinear fractional differential equations.OPEN ACCESS Received: 19/07/2024 Accepted: 01/11/2024

Abstract

This paper presents a rigorous framework for generalizing Laguerre polynomials to the fractional domain using the Caputo derivative. We solve the resulting fractional Laguerre differential equation via the power series method, deriving an explicit form for the fractional Laguerre functions. A key contribution is the identification of a novel weight function, wα(x) = x−(2α−1)e−x, which is essential to prove the orthogonality of these functions over the interval [0,∞). Comprehensive numerical validation is provided, confirming the theoretical orthogonality across a wide range of fractional orders α and demonstrating a clean reduction to the classical polynomials when α = 2. An analysis of computational feasibility confirms the practical applicability of these functions for solving fractional differential equations and other applied problems.

Abstract

This paper presents a rigorous framework for generalizing Laguerre polynomials to the fractional domain using the Caputo derivative. We solve the resulting fractional Laguerre differential equation via the power series method, deriving an explicit formfor the fractional Laguerre functions. A key contribution is the identification of a novel weight function, wα(x) = x−(2α−1)e−x, which is essential to prove the orthogonality of these functions over the interval [0,∞). Comprehensive numerical validation is provided, confirming the theoretical orthogonality across a wide range of fractional orders α and demonstrating a clean reduction to the classical polynomials when α = 2. An analysis of computational feasibility confirms the practical applicability of these functions for solving fractional differential equations and other applied problems.

Abstract

Classical regression can only examine the relation between response and predictor variables based on integer order calculus theory. What happens when non integer order calculus is considered is a field where a vast spectrum of studies can be undertaken. The purpose of this study introduces a novel fractional-order quadratic regression model grounded in the Caputo derivative framework, addressing the limitation and the rigidity of classical polynomial regression in adapting to the intrinsic curvature of data. The core innovation is the use of the fractional order ν as a tunable parameter for curvature-sensitive optimization. Our main contributions are fourfold: First, we establish a fundamental theoretical pillar by proving that the second-order Caputo derivative preserves the curvature direction of quadratic functions, enabling a principled optimization framework. Second, we rigorously demonstrate the model’s robustness by proving the existence and uniqueness of solutions via Banach’s fixed point theorem and establishing stability bounds through a fractional Grönwall inequality. Third, we develop a practical methodology to identify an optimal fractional order ν that minimizes the error-to-explained-variation ratio (SSE/SSR). Finally, we validate the framework on four diverse real-world datasets from air quality, soil science, education, and meteorology. The proposed model consistently outperforms classical quadratic regression, achieving a reduction in the SSE/SSR ratio by up to 21% in specific cases. The proposed method yields more efficient models with either lower estimation error or higher correlation coefficients, positioning Caputo fractional quadratic regression as a powerful and theoretically sound alternative for modeling cases where quadratic regression is considered appropriate.OPEN ACCESS Received: 10/09/2025 Accepted: 05/11/2025

Abstract

Classical regression can only examine the relation between response and predictor variables based on integer order calculus theory. What happens when non integer order calculus is considered is a field where a vast spectrum of studies can be undertaken. The purpose of this study introduces a novel fractional-order quadratic regression model grounded in the Caputo derivative framework, addressing the limitation and the rigidity of classical polynomial regression in adapting to the intrinsic curvature of data. The core innovation is the use of the fractional order ν as a tunable parameter for curvature-sensitive optimization. Our main contributions are fourfold: First, we establish a fundamental theoretical pillar by proving that the second-order Caputo derivative preserves the curvature direction of quadratic functions, enabling a principled optimization framework. Second, we rigorously demonstrate the model’s robustness by proving the existence and uniqueness of solutions via Banach’s fixed point theorem and establishing stability bounds through a fractional Grönwall inequality. Third, we develop a practical methodology to identify an optimal fractional order ν that minimizes the error-to-explained-variation ratio (SSE/SSR). Finally, we validate the framework on four diverse real-world datasets from air quality, soil science, education, and meteorology. The proposed model consistently outperforms classical quadratic regression, achieving a reduction in the SSE/SSR ratio by up to 21% in specific cases. The proposed method yields more efficient models with either lower estimation error or higher correlation coefficients, positioning Caputo fractional quadratic regression as a powerful and theoretically sound alternative for modeling cases where quadratic regression is considered appropriate.OPEN ACCESS Received: 10/09/2025 Accepted: 05/11/2025 Published: 23/01/2026

Abstract

Classical regression can only examine the relation between response and predictor variables based on integer order calculus theory. What happens when non integer order calculus is considered is a field where a vast spectrum of studies can be undertaken. The purpose of this study introduces a novel fractional-order quadratic regression model grounded in the Caputo derivative framework, addressing the limitation and the rigidity of classical polynomial regression in adapting to the intrinsic curvature of data. The core innovation is the use of the fractional order ν as a tunable parameter for curvature-sensitive optimization. Our main contributions are fourfold: First, we establish a fundamental theoretical pillar by proving that the second-order Caputo derivative preserves the curvature direction of quadratic functions, enabling a principled optimization framework. Second, we rigorously demonstrate the model’s robustness by proving the existence and uniqueness of solutions via Banach’s fixed point theorem and establishing stability bounds through a fractional Grönwall inequality. Third, we develop a practical methodology to identify an optimal fractional order ν that minimizes the error-to-explained-variation ratio (SSE/SSR). Finally, we validate the framework on four diverse real-world datasets from air quality, soil science, education, and meteorology. The proposed model consistently outperforms classical quadratic regression, achieving a reduction in the SSE/SSR ratio by up to 21% in specific cases. The proposed method yields more efficient models with either lower estimation error or higher correlation coefficients, positioning Caputo fractional quadratic regression as a powerful and theoretically sound alternative for modeling cases where quadratic regression is considered appropriate.OPEN ACCESS Received: 10/09/2025 Accepted: 05/11/2025 Published: 23/01/2026

Abstract

This paper presents a rigorous framework for generalizing Laguerre polynomials to the fractional domain using the Caputo derivative. We solve the resulting fractional Laguerre differential equation via the power series method, deriving an explicit formfor the fractional Laguerre functions. A key contribution is the identification of a novel weight function, wα(x) = x−(2α−1)e−x, which is essential to prove the orthogonality of these functions over the interval [0,∞). Comprehensive numerical validation is provided, confirming the theoretical orthogonality across a wide range of fractional orders α and demonstrating a clean reduction to the classical polynomials when α = 2. An analysis of computational feasibility confirms the practical applicability of these functions for solving fractional differential equations and other applied problems.

Abstract