Spectral Approximation of -Fractional Differential Equation Based on Mapped Jacobi Functions

In this article, we explore a new method for approximating solutions to fractional differential equations using spectral analysis. The approach is based on the idea of mapping the original equation onto an orthogonal function system, which allows for efficient numerical computation. By exploiting the properties of these functions, we can derive accurate spectral approximations that capture the behavior of the solution over long times or distances.
To begin with, let’s define what fractional differential equations are. Unlike their integer-valued counterparts, these equations involve a power of the derivative that is not necessarily an integer. For instance, the classical heat equation is a second-order integer-differential equation, while its fractional version has a power of 2 in the derivative. This difference leads to distinct mathematical properties and practical applications.
The key insight behind our approach is the use of log orthogonal functions, which are similar to regular orthogonal polynomials but have additional properties that make them ideal for fractional differential equations. By expanding the solution in terms of these functions, we can reduce the computation complexity from an exponential function to a polynomial one, making it more feasible to tackle complex problems.
The article presents several theoretical results that demonstrate the accuracy and effectiveness of our method. These include mathematical derivations, numerical experiments, and comparisons with existing methods. The authors also provide a detailed analysis of the convergence properties of their approximation scheme, which is essential for validating its reliability.
In summary, this work presents a powerful new tool for solving fractional differential equations using spectral approximation. By leveraging the properties of log orthogonal functions, we can efficiently compute solutions over long times or distances with high accuracy. This approach has broad applications in various fields, including physics, engineering, finance, and biology, where fractional differential equations play a central role in modeling complex phenomena.