💻 computer science

Relation between two Sinc-collocation methods for Volterra integral equations of the second kind and further improvement

This paper theoretically clarifies the relationship between Stenger's and Rashidinia--Zarebnia's Sinc-collocation methods for Volterra integral equations of the second kind, justifies the applicability of Stenger's method to general two-variable kernels, proves both methods achieve root-exponential convergence, and proposes an improved version of Stenger's method that attains almost exponential convergence.

Original authors: Tomoaki Okayama

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomoaki Okayama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for the next month. You have a complex set of rules (equations) that tell you how today's temperature affects tomorrow's, and tomorrow's affects the day after, and so on. This is a Volterra Integral Equation. It's a mathematical way of describing a system where the present depends on its entire past history.

Solving these equations exactly is like trying to calculate the path of every single raindrop in a storm—it's impossible to do by hand. So, mathematicians use numerical methods (computer algorithms) to get a very good approximation.

This paper is about comparing and improving three different "recipes" for solving these equations using a specific, powerful tool called the Sinc method. Think of the Sinc method as a super-precise ruler that can measure curves with incredible accuracy.

Here is the breakdown of the paper's story, using simple analogies:

1. The Three Competitors

The paper looks at three different ways to use this Sinc ruler:

The "Nyström" Method (The Heavyweight Champion): This method is mathematically the most accurate. It's like using a high-end, custom-built 3D printer to create your solution. It produces a result that is almost perfect. However, it's slow and expensive because it requires calculating a very complex, special function (called the "sine integral") for every single step. It's like using a Ferrari to drive to the grocery store; it gets you there fast, but the engine is too loud and complicated for the job.
The "Rashidinia-Zarebnia" Method (The Complex Architect): This is a "collocation" method. Imagine trying to build a bridge by only checking if the pillars touch the ground at specific points. This method is clever but requires a very complicated setup with many different cases depending on the edges of your problem. It's like a bridge blueprint that changes depending on whether the river is wide or narrow, making it hard to build.
The "Stenger" Method (The Simple Craftsman): This is the oldest method. It also uses the "check the pillars" approach, but it's much simpler to build. The paper reveals a surprising secret: Stenger's method and the Rashidinia-Zarebnia method actually produce the exact same result at the specific points where they check (the collocation points). They are like two different maps that show the exact same landmarks, even if the roads drawn between them look different.

The Verdict: Since Stenger's method is simpler to code and run, but gives the same accuracy as the complex Rashidinia-Zarebnia method, the paper concludes that Stenger's method is the winner among the original three. It's the "Goldilocks" choice: not too complicated, but accurate enough.

2. The Big Improvement: The "Double-Exponential" Boost

The authors weren't satisfied with just picking the best of the old methods. They wanted to make Stenger's method even faster.

The Old Way (The Tanh Transformation): To use the Sinc ruler on a finite interval (like a specific time period), the old methods used a "Tanh transformation." Imagine this as stretching a rubber band to fit a specific shape. It works well, but the rubber band has limits on how smoothly it stretches.
The New Way (The Double-Exponential Transformation): The authors swapped the rubber band for a Double-Exponential (DE) transformation.
- The Analogy: If the old method was like stretching a rubber band, the new method is like using a magic elastic that stretches infinitely smoothly. It allows the Sinc ruler to "see" the curve much more clearly, especially near the edges where things get tricky.

The Result: By swapping this "magic elastic" into Stenger's simple method, they created a new super-method.

It is significantly faster (converges much quicker) than the old Stenger method.
It is almost as accurate as the heavy-duty Nyström method.
Crucially, it is much faster to compute than the Nyström method because it doesn't need that complicated "sine integral" calculation.

3. The Final Race (Numerical Experiments)

The authors ran computer tests to see how these methods performed in the real world.

Accuracy: The new "DE-Stenger" method was a clear winner, reaching high accuracy with very few steps.
Speed: When they measured how long the computer took to solve the problem, the new method was the fastest. The heavy-duty Nyström method was accurate but took a long time to run because of its complex math. The old Stenger method was fast but needed more steps to get the same accuracy.

Summary: What's the Takeaway?

Think of this paper as a car review:

They compared three existing car models (Nyström, Rashidinia-Zarebnia, and Stenger).
They discovered that the two complex models (Rashidinia-Zarebnia and Stenger) actually drive the same route at the same speed, but Stenger's car is easier to drive and cheaper to maintain.
They then took the Stenger car and installed a turbocharger (the Double-Exponential transformation).
The Result: This new "Turbo-Stenger" car is the best of all worlds. It drives as smoothly as the luxury model (Nyström) but is much cheaper and faster to run than the others.

In short: The paper proves that a simple, well-known method can be upgraded with a clever mathematical trick to become the most efficient tool for solving these specific types of equations.

1. Problem Statement

The paper addresses the numerical solution of Volterra integral equations of the second kind, given by:
$u(t) - \int_a^t k(t, s)u(s) \, ds = g(t), \quad a \leq t \leq b$
where $k(t, s)$ is the kernel and $g(t)$ is a known function. The goal is to determine the unknown solution $u(t)$ with high accuracy and efficiency.

The study focuses on Sinc numerical methods, which are known for their exponential convergence rates for analytic functions. Specifically, it investigates the relationship and performance differences between three existing Sinc-based approaches:

Sinc-Nyström Method: Based on Sinc indefinite integration.
Stenger's Sinc-collocation Method: Originally derived for single-variable kernels but applicable to general cases.
Rashidinia–Zarebnia's Sinc-collocation Method: A standard collocation approach with specific handling of boundary conditions.

2. Methodology

The paper employs a comparative theoretical analysis and proposes a new hybrid method. The core methodologies include:

Variable Transformations: To map the finite interval $[a, b]$ $[a, b]$ to the infinite real line $\mathbb{R}$ $R$ (where Sinc functions are defined), the paper utilizes two transformations:
- SE (Single-Exponential) Transformation: $t = \psi_{SE}(x) = \frac{b-a}{2}\tanh(\frac{x}{2}) + \frac{b+a}{2}$ .
- DE (Double-Exponential) Transformation: $t = \psi_{DE}(x) = \frac{b-a}{2}\tanh(\frac{\pi}{2}\sinh x) + \frac{b+a}{2}$ .
Sinc Approximation & Indefinite Integration: The methods rely on approximating functions and their integrals using Sinc basis functions $S(j, h)(x)$ and the Sine Integral function $Si(x)$.
Generalized Sinc Approximation: To handle non-zero boundary conditions ( $u(a) \neq 0$ or $u(b) \neq 0$ ), the authors use a generalized approximation operator ( $P_N$ ) that incorporates linear boundary functions $\omega_a(t)$ and $\omega_b(t)$ .

3. Key Contributions

A. Theoretical Relationship Between Existing Methods

The paper rigorously analyzes the relationship between Stenger's method ( $v_N^{SE}$ ) and Rashidinia–Zarebnia's method ( $w_N^{RZ}$ ).

Coincidence at Collocation Points: The study proves that while the two methods generate different continuous functions globally, they yield identical values at the collocation points ( $t_i$ ).
Non-Equivalence: Despite coinciding at nodes, the methods are not generally equivalent because they use different approximation operators ( $P_N^{SE}$ vs. $P_N^{RZ}$ ) to interpolate between nodes.
Convergence Rates: Both methods are proven to achieve the same root-exponential convergence rate:
$O(\sqrt{N} e^{-\sqrt{\pi d \alpha N}})$
where $d$ is the domain of analyticity and $\alpha$ is the Hölder continuity order.

B. Practical Implementation Analysis

Complexity: Rashidinia–Zarebnia's method requires solving a more complex linear system involving boundary terms explicitly in the matrix structure. Stenger's method is derived by first solving the Sinc-Nyström system and then applying the generalized Sinc approximation, making it simpler to implement.
Efficiency vs. Nyström: While the Sinc-Nyström method has a slightly better theoretical convergence rate (no $\sqrt{N}$ factor), it requires evaluating the Sine Integral (Si) function within the basis functions, which incurs a high computational cost. Stenger's collocation method avoids this, offering a better trade-off between accuracy and computation time.

C. Proposed Improvement: DE-Sinc-Collocation Method

The paper proposes a new method: the DE-Sinc-collocation method.

Mechanism: It combines Stenger's collocation procedure with the Double-Exponential (DE) transformation instead of the standard tanh (SE) transformation.
Theoretical Justification: The authors prove that this new method attains a significantly higher convergence rate:
$O(e^{-\pi d N / \log(2dN/\alpha)})$
This rate is "almost exponential" and significantly superior to the root-exponential rate of the SE-based methods.

4. Results

Theoretical Proofs:
- Proved that Stenger's and Rashidinia–Zarebnia's methods coincide at collocation points but differ globally.
- Established rigorous error bounds for both SE-based collocation methods under Hölder continuity assumptions.
- Derived the error bound for the new DE-based collocation method, showing it outperforms the SE-based collocation methods.
Numerical Experiments:
- Tested on two benchmark Volterra integral equations with known solutions ( $u(t) = e^{-t^2}$ and $u(t) = 1+\sqrt{t}$ ).
- Accuracy: The DE-Sinc-collocation method demonstrated the best performance, achieving the highest accuracy for a given number of nodes $N$ .
- Efficiency: When plotting error against computation time, the DE-Sinc-collocation method significantly outperformed the Sinc-Nyström methods. The Nyström methods were slower due to the computational cost of the Sine Integral function, despite their theoretical convergence advantages.
- Stenger's SE-collocation method performed almost identically to Rashidinia–Zarebnia's method but was easier to implement.

5. Significance

Clarification of Literature: The paper resolves the ambiguity regarding the relationship between two prominent Sinc-collocation methods, proving they are distinct but share the same convergence order and nodal values.
Implementation Guidance: It provides a clear recommendation for practitioners: Stenger's method is preferable to Rashidinia–Zarebnia's due to simplicity, and both are preferable to the Sinc-Nyström method for general applications due to lower computational overhead.
Methodological Advancement: The introduction of the DE-Sinc-collocation method represents a significant improvement. By combining the structural simplicity of Stenger's collocation approach with the superior convergence properties of the Double-Exponential transformation, the authors achieve a method that is both computationally efficient and highly accurate.
Generalization: The study theoretically justifies the use of Stenger's method for general kernels $k(t, s)$ (functions of two variables), extending its applicability beyond its original formulation for single-variable kernels.

In conclusion, the paper establishes that the DE-Sinc-collocation method is the optimal choice among the discussed techniques, offering a robust, fast, and highly accurate solution for Volterra integral equations of the second kind.