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<p>This work introduces a novel fractional Gegenbauer-Jaiswal function, based on Gegenbauer and Jaiswal polynomials, to solve multi-term time fractional advection–diffusion equation addressing the challenge of accurately modeling complex, memory-dependent systems. The Caputo sense is employed while discussing the fractional derivative. Operational and pseudo-operational matrices play a crucial part in the case of integration of fractional and integer orders. Through the collocation method, the problem is reduced to a system of algebraic equations, with an upper bound established for the error in the operational matrix of fractional integration. This approach’s novelty lies in achieving high computational efficiency, enhanced accuracy, and stability for fractional models, surpassing existing methods by offering a more robust framework for solving complex time fractional equations. Numerical examples in tabular form confirm its efficacy and broader applicability. Our method enhances accuracy and stability for complex, memory-dependent systems.</p>

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Applications of fractional-order Gegenbauer-Jaiswal functions for solving multi-term advection–diffusion equations

  • 期刊论文

  • 发表日期：

    2026年3月12日

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  • Santoshi Tarei,
  • Ankur Kanaujiya,
  • Jugal Mohapatra

摘要

This work introduces a novel fractional Gegenbauer-Jaiswal function, based on Gegenbauer and Jaiswal polynomials, to solve multi-term time fractional advection–diffusion equation addressing the challenge of accurately modeling complex, memory-dependent systems. The Caputo sense is employed while discussing the fractional derivative. Operational and pseudo-operational matrices play a crucial part in the case of integration of fractional and integer orders. Through the collocation method, the problem is reduced to a system of algebraic equations, with an upper bound established for the error in the operational matrix of fractional integration. This approach’s novelty lies in achieving high computational efficiency, enhanced accuracy, and stability for fractional models, surpassing existing methods by offering a more robust framework for solving complex time fractional equations. Numerical examples in tabular form confirm its efficacy and broader applicability. Our method enhances accuracy and stability for complex, memory-dependent systems.