[Paper Review] Uniform approximation of fractional derivatives and integrals with application to fractional differential equations

Hassan Khosravian‐Arab, Delfim F. M. Torres|arXiv (Cornell University)|Aug 2, 2013

Fractional Differential Equations Solutions22 references3 citations

TL;DR

This paper proposes a numerical method based on Bernstein polynomials to uniformly approximate fractional derivatives and integrals in Caputo and Riemann–Liouville senses, enabling accurate and stable solution of fractional differential equations (FODEs). The method achieves uniform convergence of fractional derivatives and integrals of approximating polynomials, with convergence rates and stability rigorously established for both linear and nonlinear FODEs.

ABSTRACT

It is well known that for every $f\in C^m$ there exists a polynomial $p_n$ such that $p^{(k)}_n ightarrow f^{(k)}$, $k=0,\ldots,m$. Here we prove such a result for fractional (non-integer) derivatives. Moreover, a numerical method is proposed for fractional differential equations. The convergence rate and stability of the proposed method are obtained. Illustrative examples are discussed.

Motivation & Objective

• Address the lack of readily usable, robust numerical algorithms for solving fractional differential equations (FODEs) involving non-integer order derivatives and integrals.
• Extend the classical Weierstrass approximation theorem to fractional derivatives by proving uniform convergence of Bernstein polynomial approximations for Caputo and Riemann–Liouville fractional derivatives.
• Develop a general, computationally efficient numerical scheme based on Bernstein polynomials for solving both linear and nonlinear FODEs.
• Establish theoretical convergence rates and stability conditions for the proposed method under standard Lipschitz and Hölder-type assumptions.
• Demonstrate the effectiveness of the method through numerical examples, including cases with known exact solutions and nonlinear problems with unknown exact solutions.

Proposed method

• Use Bernstein polynomials to approximate the solution function and its fractional derivatives, leveraging their property of simultaneously approximating both the function and its derivatives.
• Apply the uniform convergence of Bernstein polynomials to fractional derivatives by proving that $ B_n^{(k)}(f;x) \to f^{(k)}(x) $ uniformly for integer $ k $, and extend this to fractional derivatives via integral representations.
• Formulate the fractional derivative of the Bernstein polynomial approximation using the Riemann–Liouville and Caputo definitions, expressed through integral operators involving the kernel $ (t-z)^{\alpha-1} $.
• Derive error bounds for the approximation of fractional integrals and derivatives by analyzing the difference between the exact fractional operator and its Bernstein polynomial approximation.
• Establish stability by bounding the difference between solutions of perturbed FODEs using Gronwall-type inequalities and the Mittag-Leffler function $ E_{\alpha,\alpha}(\cdot) $.
• Implement the method numerically by discretizing the time interval and solving the resulting system of algebraic equations derived from collocation at Gauss-Legendre points or equivalent quadrature rules.

Experimental results

Research questions

• RQ1 Can Bernstein polynomials be used to achieve uniform approximation of fractional derivatives and integrals in the Caputo and Riemann–Liouville sense?
• RQ2 What is the rate of convergence of the Bernstein polynomial approximation for fractional derivatives, and how does it depend on the smoothness of the underlying function?
• RQ3 How can the proposed approximation be embedded into a numerical scheme for solving fractional differential equations with guaranteed stability?
• RQ4 What is the behavior of the method when applied to nonlinear FODEs with or without known exact solutions?
• RQ5 How do the approximation errors evolve with increasing polynomial degree $ n $, especially when the exact solution is a polynomial of low degree?

Key findings

• The method achieves uniform convergence of the $ m $-th order derivative of the Bernstein polynomial $ B_n(f;x) $ to the $ m $-th order classical derivative $ f^{(m)}(x) $, and this extends to fractional derivatives via integral representations.
• For functions in $ C^p[0,1] $, the $ p $-th derivative of the Bernstein polynomial converges uniformly to the $ p $-th derivative of the function as $ n \to \infty $, with an asymptotic error term governed by $ x(1-x)f''(x) $ via Voronovskaya’s theorem.
• The convergence rate of the fractional derivative approximation is shown to be $ \mathcal{O}(1/n) $ under suitable smoothness conditions, with error bounds derived using integral kernel analysis.
• The numerical method for solving FODEs is proven stable under Lipschitz conditions on the right-hand side, with the solution error bounded by $ C \|f - f'\| $, where $ C $ depends on the Mittag-Leffler function $ E_{\alpha,\alpha}(\Lambda t^\alpha) $.
• In Example 5.7, the method achieves machine-level accuracy for $ n=5 $ when the exact solution is a degree-5 polynomial, and error increases for $ n>5 $, confirming the method's optimality for polynomial solutions.
• Numerical results in Examples 5.6–5.9 confirm the method’s effectiveness, with high accuracy for both linear and nonlinear FODEs, even when the exact solution is unknown, and with consistent convergence as $ n $ increases.