Spectral Method for Fractional Integral Equations using Jacobi Fractional Polynomials

Основные понятия

The authors present a spectral method for solving one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including those with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis of Jacobi fractional polynomials, which incorporate the algebraic singularities of the solution into the basis functions.

Аннотация

The content discusses the numerical solution of fractional integral equations (FIEs) and fractional differential equations (FDEs) using a spectral method based on Jacobi fractional polynomials (JFPs). Key highlights:

Fractional integral operators can introduce algebraic singularities that are not well approximated by standard polynomial bases, leading to only algebraic convergence rates for traditional numerical methods.

The JFP basis functions incorporate the algebraic singularities of the solution, allowing for exponentially fast convergence rates.

New algorithms are presented for building the matrices that represent fractional integration operators acting on the JFP basis. These algorithms are unstable and require high-precision computations, but the resulting spectral method is stable and efficient.

The JFP method is compared to the "sum space" method, which also uses a basis that incorporates algebraic singularities. It is shown that the JFP method outperforms the sum space method for an important family of FIEs that arise in the solution of time-fractional heat/wave equations.

The performance of the JFP method is investigated for a wider range of parameters than previously considered, and a pseudo-stabilization technique is developed to ensure the scalability of the method.

The JFP method is applied to solve a variety of FIEs, including FDEs and fractional PDEs reformulated as FIEs, demonstrating its effectiveness and flexibility.

Статистика

The fractional integral of order μ > 0 is a singular integral defined as:
Iμ
a+φ(x) := 1/Γ(μ) ∫_a^x φ(t) / (x-t)^(1-μ) dt
Iμ
b-φ(x) := 1/Γ(μ) ∫_x^b φ(t) / (t-x)^(1-μ) dt
The Riemann-Liouville and Caputo fractional derivatives are defined as:
Dν
RLφ(x) := Dm Iμ φ(x),
Dν
Cφ(x) := Iμ [Dm φ] (x),
where m = ⌈ν⌉, μ = m - ν, and Dm is the m-th order standard derivative.

Цитаты

"Fractional differential equations (FDEs) and fractional integral equations (FIEs) have received a great deal of attention in the literature, not only because they appear in many scientific fields, but also because their solutions are difficult to compute with traditional approaches."
"Numerous standard numerical methods have been applied to FDEs and FIEs, however they only achieve algebraic convergence. This is because the fractional integral operators can introduce algebraic singularities that are not well approximated by polynomials."

Ключевые выводы из

The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

by Tianyi Pu (1... в arxiv.org 04-10-2024

https://arxiv.org/pdf/2206.14280.pdf

Дополнительные вопросы

How can the JFP method be extended to handle fractional differential equations directly, without the need for integral reformulation

The JFP method can be extended to handle fractional differential equations directly by developing differentiation matrices for the JFP basis. This extension would involve constructing matrices that represent fractional differentiation operators acting on the JFP basis functions. By doing so, the JFP method can be applied directly to fractional differential equations without the need for integral reformulation. These differentiation matrices would allow for the computation of fractional derivatives of functions represented in the JFP basis, enabling the solution of fractional differential equations in a spectral framework.

What are the theoretical limits of the JFP method in terms of the range of fractional orders and other problem parameters it can handle effectively

The JFP method has theoretical limits in terms of the range of fractional orders and other problem parameters it can effectively handle. The method is particularly well-suited for problems involving fractional integral equations with various orders, including irrational orders, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. However, the method's effectiveness may be limited by the stability and accuracy of the algorithms used to compute the fractional integration matrices. As the order of the fractional integral increases, the condition numbers of the matrices may grow exponentially, requiring high-precision computations for accurate results. Additionally, the method's performance may be influenced by the choice of parameters such as the basis functions' parameters and the precision of the computations.

Can the pseudo-stabilization techniques developed for the JFP method be applied to other spectral methods for fractional problems to improve their scalability and efficiency

The pseudo-stabilization techniques developed for the JFP method, such as integrating high-precision computations to stabilize the algorithms for computing fractional integration matrices, can potentially be applied to other spectral methods for fractional problems. By incorporating high-precision arithmetic and pseudo-stabilization methods, other spectral methods can improve their scalability and efficiency, particularly when dealing with unstable algorithms or matrices with exponentially growing condition numbers. These techniques can help enhance the numerical stability and accuracy of spectral methods for fractional problems, making them more robust and reliable for a wider range of applications.