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Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
Guo-cheng Wu
TL;DR
The paper addresses the limitation that prior VIM treatments of fractional differential equations avoid the fractional-derivative term and use restricted variation. It proposes a fractional variational iteration method based on fractional variational theory, enabling generalized Lagrange multipliers and solving fractional nonlinear equations with a modified Riemann-Liouville derivative.
Problem
Prior VIM treatments avoid the fractional-derivative term, handle it as a restricted variation, and cannot identify fractional Lagrange multipliers explicitly.
Method
The paper extends VIM through fractional variational theory and constructs a correction functional with a generalized Lagrange multiplier.
Results
The proposed method solves fractional nonlinear differential equations with a modified Riemann-Liouville derivative, with an exact solution reported for the illustrated case.
Takeaways & Limitations
Fractional variational iteration provides a technique for treating fractional nonlinear differential equations while explicitly identifying the Lagrange multipliers.
Takeaways & Limitations
The paper notes that existing approaches cannot establish a fractional variational functional, motivating the use of the modified Riemann-Liouville framework.
Abstract
from arXiv · showhide
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.
1 Introduction
The paper addresses unresolved shortcomings in applying the variational iteration method to fractional differential equations. It proposes a fractional extension based on fractional variational theory, allowing generalized Lagrange multipliers to be identified explicitly.
- The correction functional is established with a Lagrange multiplier identified optimally through variational theory.
- The approach targets fractional nonlinear differential equations without restrictions such as linearization or small parameters in nonlinear operators.
- The introduction describes fractional differential equations as an active computational problem relevant to fields including rheology, biology, electrochemistry, diffusion, transport, probability, potential theory, and elasticity.
- Previous VIM-based studies avoid the fractional-derivative term, treat it as a restricted variation, and cannot identify fractional Lagrange multipliers explicitly.
- The paper proposes extending VIM to fractional differential equations through fractional variational theory and a generalized Lagrange multiplier.
2 Properties of Modified Riemann-Liouville Derivative
This section introduces the modified Riemann-Liouville derivative and the fractional properties used to construct a fractional variational correction functional. The construction relies on fractional variational theory and explicit fractional Lagrange multipliers.
- The modified Riemann-Liouville derivative differs from the classical Caputo derivative and is presented as avoiding a higher integer-order derivative requirement.
- The derivative is considered for a continuous, not necessarily differentiable function on [0,1], with 0 < α < 1.
- The modified Riemann-Liouville derivative is introduced as the fractional derivative used in the subsequent variational formulation.
- The section states that the fractional framework uses a fractional Leibniz product law, fractional integration by parts, and integration with respect to (dx)^α.
- These fractional properties are used to construct a fractional correction functional and identify fractional Lagrange multipliers.
3 Fractional Variational Theory
The paper develops fractional variational theory using Jumarie’s modified Riemann–Liouville derivative, deriving fractional Euler–Lagrange conditions from extrema of a functional. This addresses the difficulty that earlier fractional variational approaches could not establish a fractional variational functional.
- Earlier fractional variational approaches could not establish a fractional variational functional.
- The theory starts from a functional F(x,y,D_a^αy) and perturbs the desired function as y*(x)=y(x)+εη(x) while preserving boundary conditions.
- Because the functional reaches an extremum at ε=0, its first variation must vanish for every admissible η(x).
- Applying the fractional Leibniz formula and integration by parts yields the fractional Euler–Lagrange equation.
- The same procedure extends to higher-order fractional Euler–Lagrange equations and recovers the classical equation when α=1.
4 Fractional Variational Iteration Method
The fractional variational iteration method replaces the Caputo derivative with the modified Riemann–Liouville derivative and constructs corrected functionals for fractional diffusion and nonlinear equations. Fractional variational theory determines the Lagrange multipliers explicitly, while examples recover exact or matching solutions.
- For α=1/2, the diffusion example gives E_{1/2}(k t) as the exact solution of the fractional diffusion equation.
- For the time-fractional diffusion model, the method replaces the Caputo derivative with the modified Riemann–Liouville derivative and imposes specified initial conditions.
- The corrected functional produces a multiplier satisfying the fractional conditions, yielding λ(t,τ)=−1.
- The method also constructs generalized Lagrange multipliers for higher fractional-order nonlinear ordinary differential equations.
- The second approximate form for the higher-order initial-value problem agrees with the result obtained by the Fractional Decomposition method.
5 Conclusion
The paper proposes a fractional variational iteration method for fractional nonlinear differential equations involving the modified Riemann–Liouville derivative. The technique extends variational iteration beyond integer-order equations.
- Variational iteration is extended from integer-order differential equations to fractional nonlinear equations.
- The paper proposes a fractional variational iteration method for fractional nonlinear differential equations.
- The technique applies to equations with the modified Riemann–Liouville derivative.
List of figure
Figure 1 compares the second approximate solution at fractional orders α = 0.9 and α = 0.99 with the exact solution at α = 1.
- Figure 1 compares the second approximate solution with the exact solution across α = 0.9, α = 0.99, and α = 1.The approximate solutions are shown with discontinuous and dotted lines, while the exact solution uses a continuous line.