Source-linked AI summary
Calculus of variations with fractional derivatives and fractional integrals
Ricardo Almeida, Delfim F. M. Torres
TL;DR
The paper addresses variational problems whose functionals contain both Riemann-Liouville fractional derivatives and integrals. It derives necessary Euler-Lagrange conditions for fundamental, non-commensurate-order, multivariable, and isoperimetric problems, and proves sufficient conditions under convexity assumptions. The paper concludes with necessary and sufficient optimality conditions while identifying existence and direct minimization methods as future work.
Problem
Existing fractional variational work largely considers Riemann-Liouville derivatives, while this paper addresses functionals containing both fractional derivatives and fractional integrals.
Method
The paper derives Euler-Lagrange equations using fractional variational arguments and establishes sufficient optimality conditions under appropriate convexity assumptions.
Results
Necessary Euler-Lagrange conditions are obtained for fundamental, non-commensurate-order, multivariable, and isoperimetric problems.
Takeaways & Limitations
The paper provides necessary and sufficient optimality conditions for fractional functionals involving Riemann-Liouville derivatives and integrals.
Takeaways & Limitations
The paper focuses on functionals containing left Riemann-Liouville fractional integrals and derivatives; existence of solutions and direct minimization methods remain future work.
Abstract
from arXiv · showhide
We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.
1 Introduction
The paper introduces fractional variational functionals that combine Riemann-Liouville fractional derivatives and integrals, then derives necessary and sufficient optimality conditions.
- The proposed Lagrangian contains both a Riemann-Liouville fractional derivative and a Riemann-Liouville fractional integral.
- Necessary Euler-Lagrange-type conditions are proved for fundamental and fractional isoperimetric variational problems.
- Sufficient optimality conditions are obtained under appropriate convexity assumptions.
2 Fractional Calculus
This section reviews Riemann-Liouville fractional calculus, including fractional integrals, derivatives, integration-by-parts rules, and endpoint regularity constraints.
- Fractional calculus is presented as an interdisciplinary area with applications including engineering, chemistry, electrical systems, viscoplasticity, and physics.
- The section introduces left and right Riemann-Liouville fractional integrals and derivatives for functions on [a, b].
- Fractional integration-by-parts rules are stated under integrability conditions involving p, q, and α.
- For 0 < α < 1, continuous left and right Riemann-Liouville fractional derivatives require f(a) = f(b) = 0.
3 The Euler-Lagrange equation
The paper derives fractional Euler-Lagrange equations for functionals involving fractional integrals and derivatives, extends them to related problems, and distinguishes necessary from sufficient conditions.
- Scope: The paper considers left-sided fractional integrals and derivatives, noting that right-sided operators can also be included by generalization.
- The Euler-Lagrange equation: A local minimizer satisfies the fractional Euler-Lagrange equation for all x ∈ [a, b].
- The Euler-Lagrange equation: The derivation sets the first variation to zero, applies fractional integration by parts, and uses the fundamental lemma for arbitrary variations.
- The Euler-Lagrange equation: As α and β approach 1, the fractional Euler-Lagrange equation becomes the classical Euler-Lagrange equation.
- The Euler-Lagrange equation: The equation includes right Riemann-Liouville fractional integrals and derivatives even when the original functional contains left-sided operators.
4 Some generalizations
The paper generalizes fractional Euler-Lagrange equations to functionals with Riemann-Liouville derivatives and integrals of non-commensurate orders and to multiple unknown functions.
- 4.1 Extension to variational problems of non-commensurate order: The generalized functional contains Riemann-Liouville fractional integrals and derivatives with different orders α_i and β_j.The orders satisfy α_i, β_j ∈(0, 1).
- 4.1 Extension to variational problems of non-commensurate order: A local minimizer of the non-commensurate-order functional satisfies the corresponding Euler-Lagrange equation for all x ∈[a, b].
- 4.2 Extension to several dependent variables: The paper also extends the framework to calculus-of-variations problems with multiple unknown functions y1, . . . , yn.
- 4.2 Extension to several dependent variables: A local minimizer for the multi-function problem satisfies a system of n fractional differential equations throughout [a, b].
- 4.2 Extension to several dependent variables: The proof uses vector-valued variations, differentiation at zero, fractional integration by parts, and the fundamental lemma of the calculus of variations.
5 The fractional isoperimetric problem
The paper studies a fractional variational problem with an integral constraint and derives an Euler-Lagrange condition using Lagrange multipliers.
- 5 The fractional isoperimetric problem: The constrained problem minimizes the functional J over functions satisfying the prescribed integral constraint I(y) = l.
- 5 The fractional isoperimetric problem: A local minimum satisfies a fractional Euler-Lagrange equation for K = λ0L + λg, with λ0 and λ not both zero.
- 5 The fractional isoperimetric problem: If y is not an extremal for I, λ0 can be chosen as 1, yielding an equation for F = L + λg.
6 Sufficient conditions
The section establishes sufficient optimality conditions for fractional variational problems, using convexity and exact fields to prove that selected Euler–Lagrange solutions minimize the functional.
- Convexity conditions are used to establish sufficient conditions guaranteeing the existence of minima.The paper states that convexity is needed analogously to the classical calculus of variations.
- If L is convex in [a, b]×R2 and y0 satisfies the fractional Euler-Lagrange equation, then y0 minimizes the functional.
- An exact field for L is defined through a potential S whose partial derivatives combine L and its third-variable derivative evaluated along the field.
- The exact-field construction is motivated by the classical Euler-Lagrange equation and applies to curves satisfying the associated differential relation.
- For the constrained problem, the proof concludes that J(y0) ≤ J(y) for every admissible curve satisfying the constraint.
7 Conclusions
The paper introduces fractional functionals involving both fractional derivatives and fractional integrals, and gives optimality conditions for fundamental and isoperimetric problems.
- The proposed functionals depend on both fractional derivatives and fractional integrals.
- The paper provides necessary and sufficient optimality conditions for fundamental and integral-constrained calculus-of-variations problems.
- Existence of solutions and direct minimization methods are identified as topics for future work.