[Paper Review] Calculus of Variations with Classical and Fractional Derivatives

Tatiana Odzijewicz, Delfim F. M. Torres|arXiv (Cornell University)|Jul 4, 2010

Fractional Differential Equations Solutions30 references3 citations

TL;DR

This paper introduces a novel variational framework that unifies classical and fractional derivatives in the calculus of variations by incorporating both $ y' $ and $ {}_aD_t^\alpha y $ in the Lagrangian. It derives Euler–Lagrange-type optimality conditions for problems with arbitrary boundary conditions and isoperimetric constraints, ensuring the classical calculus of variations is recovered exactly at $ k=0 $, unlike prior approaches that only recover it in the limit $ \alpha \to 1 $. The key contribution is a consistent extension that avoids the restrictive $ y(a)=y(b)=0 $ conditions imposed by standard Riemann–Liouville fractional derivatives.

ABSTRACT

We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental problem of the calculus of variations with mixed integer and fractional order derivatives as well as isoperimetric problems are considered.

Motivation & Objective

• To extend the classical calculus of variations to include both integer-order and fractional-order derivatives in a single variational framework.
• To resolve the limitation of prior fractional calculus of variations, where the classical case was only recoverable in the limit $ \alpha \to 1 $, by ensuring exact recovery at $ k=0 $.
• To establish necessary optimality conditions for variational problems involving mixed derivatives $ y' + k\,{}_aD_t^\alpha y $, enabling arbitrary boundary conditions $ y(a)=y_a $, $ y(b)=y_b $.
• To address isoperimetric problems in the fractional setting with general constraints, providing a consistent extension beyond previous works.

Proposed method

• Introduces a generalized Lagrangian depending on $ t $, $ y $, and the combined derivative $ v = y' + k\,{}_aD_t^\alpha y $, where $ k \in \mathbb{R} $ and $ \alpha \in (0,1) $.
• Applies integration by parts for Riemann–Liouville fractional derivatives to derive the Euler–Lagrange equation for the extended functional.
• Derives the necessary optimality condition in the form of a fractional differential equation involving both left and right Riemann–Liouville derivatives.
• Uses the Mittag–Leffler function to express solutions of fractional differential equations arising in the variational problem.
• Applies the method of Lagrange multipliers to handle isoperimetric constraints, leading to an augmented Lagrangian formulation.
• Validates the approach through analytical examples, including the classical case ($ k=0 $), the limit $ \alpha \to 1 $, and a fractional isoperimetric problem with $ \alpha = 1/2 $.

Experimental results

Research questions

• RQ1 How can the classical calculus of variations be consistently extended to include both classical and fractional derivatives in a single variational formulation?
• RQ2 What are the necessary optimality conditions for variational problems that involve mixed derivatives $ y' + k\,{}_aD_t^\alpha y $ with arbitrary boundary conditions?
• RQ3 How can isoperimetric constraints be incorporated into fractional variational problems while preserving consistency with the classical case?
• RQ4 Why do previous fractional variational approaches fail to recover the classical theory exactly, and how can this limitation be overcome?
• RQ5 What is the analytical form of the extremal for a fractional isoperimetric problem with $ \alpha = 1/2 $, $ k=1 $, and given boundary values?

Key findings

• The proposed framework ensures the classical calculus of variations is recovered exactly when $ k=0 $, unlike previous approaches that only recover it in the limit $ \alpha \to 1 $.
• The necessary optimality condition for the fundamental problem is derived as a fractional differential equation involving both left and right Riemann–Liouville derivatives.
• For the isoperimetric problem, the extremal is given by $ y(t) = \int_0^t E_{1-\alpha,1}(-k(t-\tau)^{1-\alpha}) \xi \, d\tau $, which satisfies the constraint and boundary conditions.
• When $ k=0 $, the extremal reduces to $ y(t) = \xi t $, matching the classical minimizer of the standard problem.
• In the limit $ \alpha \to 1 $, the fractional problem reduces to a classical variational problem with a rescaled minimizer $ y(t) = \frac{\xi}{k+1}t $, confirming consistency.
• For $ k=1 $, $ \alpha = 1/2 $, and $ \xi=1 $, the extremal is $ y(t) = -\left(1 - e^t \mathrm{erfc}(\sqrt{t}) + \frac{2\sqrt{t}}{\sqrt{\pi}} \right) $, which satisfies the fractional isoperimetric constraint and boundary conditions.