A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

Abstract

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such property as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

Fractional oscillator equation – Transformation into integral equation and numerical solution

“…One of the new directions in fractional calculus and its applications is to investigate the numerical solutions of fractional differential equations, containing composition of the left and the right derivatives [2,3,5,[7][8][9][10]13,20,27,31]. This problem is an important area of investigations in fractional differential equations theory.…”

Numerical solution of fractional Sturm-Liouville equation in integral form

“…When analytical solution is not available, the rate of convergence p = p i (Δt, α, λ) at nodes t i , for fixed parameters α, λ and variable values of Δt, can be determined from the following formula (see a proposition in [3]) We thus have We present numerical values at three selected nodes and rate of convergence for α ∈ {0.3, 0.5, 0.7} and λ = −3, q(t) = 0 in Table 1. In Table 2 the numerical values at three selected nodes and rates of convergence for α = 0.6, λ ∈ {−5, −7.5, −10} and q(t) = 0 are shown.…”

An efficient computational method based on the hat functions for solving fractional optimal control problems

“…For example in [3,4], the authors have achieved the necessary conditions for optimization of FOCPs with the Caputo fractional derivative. The interested reader can refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] for some recent works on FOCPs. In this paper, we propose a new efficient and accurate computational method based on HFs for solving the following FOCP [6]:…”