Comparative Analysis of Differential Equation Solutions in Classical and Fractional Calculus

Thesis Abstract

The study investigates the comparative behavior of solutions to differential equations within the frameworks of classical and fractional calculus, addressing the growing need to understand the distinctions and potential advantages of fractional operators in modeling complex dynamical systems. Classical differential equations, rooted in integer-order derivatives, have long been utilized to describe phenomena across physics, engineering, and biological sciences; however, many real-world processes exhibit memory effects and hereditary properties that are inadequately captured by traditional models. Fractional calculus introduces derivatives of non-integer order, providing a more versatile mathematical tool for characterizing such systems. Despite the increasing application of fractional models, systematic comparative analyses between classical and fractional solutions remain limited, particularly concerning their qualitative and quantitative behaviors across diverse problem classes. The primary aim of this research is to conduct a comprehensive comparative analysis of solutions to differential equations in classical and fractional calculus, emphasizing their mathematical properties, stability characteristics, and applicability to real-world phenomena. Specific objectives include (i) to identify and formulate equivalent classical and fractional differential equations representative of physical systems; (ii) to numerically solve these equations using established analytical and computational techniques; (iii) to analyze the stability, boundedness, and long-term behavior of the solutions; (iv) to evaluate the sensitivity of solutions to initial conditions and parameter variations; and (v) to determine conditions under which fractional models outperform classical ones in terms of accuracy and descriptive capability. The methodology adopts a mixed-methods research design, integrating analytical, numerical, and computational approaches. A purposive sampling of three prototypical differential equations—such as the heat conduction equation, the damped harmonic oscillator, and epidemiological models—is employed, with each representing different physical domains. A sample size of 15 distinct equations, developed based on existing literature, will be subjected to solution analysis. The data collection involves deriving exact solutions where feasible, complemented by numerical approximations via methods such as the Grünwald-Letnikov scheme for fractional derivatives and the Runge-Kutta method for classical derivatives. The study utilizes qualitative analytical techniques—including stability analysis using Lyapunov's direct method and fractional Lyapunov exponents—and quantitative measures such as mean squared error (MSE) and convergence rates to compare solution behaviors. Analytical software tools like MATLAB and Wolfram Mathematica facilitate computational modeling, while sensitivity analysis employs parameter variation techniques to assess robustness. Expected findings highlight that fractional solutions demonstrate enhanced memory retention, slower convergence, and increased stability in specific parameter regimes compared to classical solutions. Additionally, the study posits that fractional models can more accurately capture complex, hereditary dynamics in systems with anomalous diffusion, viscoelasticity, or spatial heterogeneity, providing better predictive capabilities. These results are anticipated to challenge conventional assumptions about the sufficiency of classical models, emphasizing the importance of fractional calculus as a complementary or superior modeling paradigm in certain contexts. This study contributes to the existing body of knowledge by systematically delineating the conditions under which fractional differential equations provide significant advantages over classical counterparts, advancing the theoretical understanding of their solution properties and practical applicability. It offers a rigorous comparative framework grounded in computational simulation and stability theory, thereby informing future research and application development in physics, engineering, and biological modeling. The research concludes that fractional calculus methods offer valuable insights into complex systems exhibiting non-locality and memory effects, recommending their integrated application alongside classical models. Further research is suggested to extend the comparative analysis to higher-dimensional problems, explore the impact of varying fractional orders, and develop optimized numerical algorithms for large-scale simulations, thereby reinforcing the foundation for broader adoption of fractional calculus in scientific modeling and analysis.