Trapezoidal methods for fractional differential equations: theoretical and computational aspects (1912.09878v1)

Abstract: The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.

• The paper analyzes theoretical aspects of trapezoidal methods for fractional differential equations, covering convergence and stability of adapted methods.
• It details computational efficiency strategies using FFT and addresses intricacies like starting weights for implementation.
• Numerical experiments show the fractional trapezoidal rule performs best for α<1, while other methods are superior for α>1.

Analysis of Trapezoidal Methods for Fractional Differential Equations

Roberto Garrappa's paper explores the theoretical and computational aspects of trapezoidal methods designed for fractional differential equations (FDEs). Fractional calculus, an extension of traditional integer-order calculus, is vital for modeling phenomena in diverse fields such as bioengineering and physics. This paper addresses the challenges and potentials of adapting the trapezoidal rule—a staple in numerical analysis for ordinary differential equations (ODEs)—to accommodate the unique characteristics of FDEs.

Theoretical Insights

The adaptation of numerical methods originally devised for ODEs to the context of FDEs is not straightforward due to the non-local and history-dependent nature of fractional derivatives. In this paper, Garrappa outlines two primary approaches: Product Integration (PI) rules and Fractional Linear Multistep Methods (FLMMs). These methods are particularly relevant when dealing with fractional derivatives defined in the Caputo sense.

An essential theoretical contribution of the paper is the examination of convergence and stability properties. For instance, PI rules generally fail to achieve the anticipated order of convergence when applied on uniform grids for fractional cases due to the singularities at the origin. This limitation can be mitigated by employing graded meshes, which concentrate computational effort near the initial singularity.

FLMMs, derived by taking fractional powers of generating functions from classical multistep methods, offer another path forward. The paper discusses the stability sectors of these methods, demonstrating that only certain adaptations (such as fractional trapezoidal rules) consistently achieve $A(\alpha\frac{\pi}{2})$-stability across different fractional orders.

Computational Aspects

Garrappa provides a detailed exposition on computational strategies for implementing these methods efficiently, particularly leveraging Fast Fourier Transforms (FFT) to expedite computation. This approach is crucial to overcoming the computational bottlenecks associated with the lengthy interaction terms characteristic of convolution quadratures.

Starting weights, incorporated to address initial condition singularities, are calculated through linear systems that ensure precision in the early integration phase. The paper discusses this implementation intricacy, emphasizing the need for small step sizes, especially when $\alpha$ is small.

Numerical Experiments and Results

The paper presents a series of numerical experiments that benchmark the discussed methods against standard test equations, including the nonlinear fractional Brusselator system. The findings highlight contradiction wherein, depending on the value of $\alpha$, different methods demonstrate superior numerical accuracy and computational efficiency. Notably, the fractional trapezoidal rule exhibits remarkable performance for $\alpha < 1$, aligning with traditional trapezoidal attributes in the ODE scenario. Conversely, the Newton-Gregory formula surpasses other methods for $\alpha > 1$.

Practical and Theoretical Implications

The implications of this research are twofold. Practically, the insights on graded mesh design and method stability provide valuable guidance for computational scientists dealing with FDEs, ensuring they can select the most appropriate method for specific problem characteristics. Theoretically, it contributes to a deeper comprehension of the numerical behavior of fractional methods, challenging preconceived notions based on integer-order paradigms.

Future Directions

Looking ahead, further exploration is warranted to refine the computational frameworks, particularly in handling multidimensional fractional systems and enhancing parallel computation efficiency. As fractional models gain traction in more intricate biological and physical systems, the demand for robust, scalable numerical methods will drive research efforts in this field, potentially leading to novel algorithmic innovations.