Basic properties of incomplete Macdonald function with applications

What is it about?

This article explores a special mathematical function called the incomplete Macdonald function, a variant of a well-known tool for solving complex equations in physics and engineering. Unlike the standard version, the incomplete Macdonald function offers greater flexibility by adjusting the limits of integration, making it more suitable for modeling real-world systems whose conditions vary over time or space. The authors examine its fundamental properties, such as its behavior under different conditions (recurrence and differential relations), how to expand it into simpler forms (series and asymptotic expansions), and how to solve certain types of equations (parabolic partial differential equations). They also show how the function naturally arises in many fields, such as viscous fluid flow, heat conduction, groundwater movement, electromagnetism, astrophysics, and nuclear reactor modeling.

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Photo by Logan Voss on Unsplash

Photo by Logan Voss on Unsplash

Why is it important?

This function can help scientists and engineers more accurately and efficiently solve equations arising in systems that are intractable using traditional methods. It is particularly useful in cases involving changing boundaries or conditions, transient (time-dependent) phenomena, and diffusive processes such as heat diffusion or fluid mixing. By understanding and formalizing the properties of this function, this paper provides a unifying framework applicable across multiple disciplines, bridging the gap between previously unrelated research areas.

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