Fractional variational calculus in terms of Riesz fractional derivatives

By Om Prakash Agrawal. 2007

394 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/305e6407a28c0a7cbd8a0334b366582888a55c38

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Fractional order models of leaves (2014, background): “, 2011b), to mechanical (Oustaloup et al., 1996; Deng and Li, 2005; Nigmatullin, 2006; Agrawal, 2007; Mainardi, 2010) and electrical systems (Petras, 2008; Radwan et al.”
A class of time‐fractional reaction‐diffusion equation with nonlocal boundary condition (2018, neutral): “t0 (t − s)α−1 ((∇ · d(u)∇u,w) − (∇ · v∇u,w) − (βu(1 − u) − χu,w)) ds (34)”
Fractional Optimal Control Problems with Several State and Control Variables (2010, neutral): “…and Klafter, 2000 Agrawal, 2001 Tenreiro-Machado, 2001 Agrawal, 2002 Agrawal et al., 2004 Agrawal, 2005 Baleanu and Muslih, 2005 Muslih and Baleanu, 2005 Baleanu et al., 2006 Magin, 2006 Agrawal, 2007a,b El-Nabulsi and Torres, 2007 Rabei et al., 2007 Vinagre and Feliu, 2007 Tricaud and Chen, 2008).”
Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost (2020, methodology): “For instance, Agrawal [5] and Agrawal and Baleanu [6] have achieved the necessary conditions of optimality for FOCPs with the Riemann–Liouville derivative and also have solved the problem numerically by solving the necessary condition.”
Fractional order optimal control problems with free terminal time (2013, background): “context of fractional order optimal control. An important tool is the integration by parts formula for Caputo fractional derivatives, which is stated in the following theorem. Theorem 1.1 (cf., e.g., [2]). Let α ∈ (0,1), and x,y : [a,b] → Rbe two functions of class C1. Then the following integration”
A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES (2011, background): “ng equal to zero, a property important in Applied Mathematics and Engineering applications. Another fractional integral that is important in potential theory is the Riesz fractional integral given by [2], RIα t f(t) = 1 2Γ(α) Zb a |t −τ|α−1f(τ)dτ, α > 0. (2.5) Two other variations of this integra”
Fractional calculus of variations for a combined Caputo derivative (2011, background): “Caputo fractional derivative (see, e.g., [2, 10, 27]), the symmetric fractional derivative (see, e.g., [25]), the Jumarie fractional derivative (see, e.g., [4, 7, 22, 23, 31]), and others (see, e.g., [3, 15, 16, 20]). Although the literature of FCV is already vast, much remains to be done. In this p”
Variational problems with fractional derivatives: Euler–Lagrange equations (2008, methodology): “left Caputo fractional derivative, then the equation c aD α t u = u has a solution bounded at zero, which is also a solution of the corresponding fractional variational problem (9). In a recent paper [3], the author tried to overcome the problem of non-solvability of the equation aDα t u = u by usin”
Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives (2010, neutral): “I α x CDαf(x) = f(x) − nX−1 k=0 f(k)(a) k! (x −a)k and xI α b CDαf(x) = f(x) − nX−1 k=0 (−1)kf(k)(b) k! (b−x)k. We also need for our purposes integration by parts formulas. For α > 0, we have (cf. [4]) Z b a g(x) · C a D α xf(x)dx = Z b a f(x) · xDα b g(x)dx + nX−1 j=0 h xD α+j−n b g(x) ·xD n−1−j”
A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials (2016, neutral): “(Agrawal, 2007a,b; Malinowska and Torres, 2010) As the first example, consider the following unconstrained FVPs minimize J½y ¼ 1 2 Z 1 0 ðC0D xyðxÞ f ðxÞÞ 2dx, 05 51 such that yð0Þ ¼ 0, yð1Þ ¼ 1 where f(x) is given f ðxÞ ¼ ð1þ Þ ð1þ Þ x In this case the exact solution is yðxÞ ¼ x .”
Advanced Local Fractional Calculus and Its Applications (2012, neutral): “[239] O.”