On exact solutions of a class of fractional Euler–Lagrange equations

By Dumitru Băleanu, Juan Trujillo. 2007

151 citations recorded by the source corpus.

Abstract In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where acDtαx(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange 1$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}\bigr)x(t)+b\bigl(t,x(t)\bigr)\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+f\bigl(t,x(t)\bigr)=0.}$$ At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations 2$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)=\lambda x(t)\quad (\lambda\in R),}$$3$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+g(t)_{a}^{c}D_{t}^{\alpha}x(t)=f(t),}$$ where g(t) and f(t) are suitable functions.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3d9ddab65f2a3dabe93af4e0acbc540773174ddb

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Citations in the corpus

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Alternative variational iteration method for solving the time-fractional Fornberg–Whitham equation (2015, unknown): “(2015), http://dx.doi.org/10.1016/j.apm.2014.11.048 Mehmet Giyas Sakar ⇑, Hilmi Ergören Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, Van 65080, Turkey a r t i c l e i n f o Article history: Received 4 February 2013 Received in revised form 3 February 2014 Accepted 20…”
Numerical solution of fractional Sturm-Liouville equation in integral form (2014, unknown): “It is an additional drawback for computation of an exact solution (even with simple Lagrangian, see [5, 16, 17]).”
Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel (2016, methodology): “In the third section we developed the corresponding fractional Euler-Lagrange equations [11] and we give an illustrative example of it.”
Existence of a weak solution for fractional Euler-Lagrange equations (2012, background): “reover, (ELα) presents an asymmetry: left and right fractional derivatives are involved and it is an additional drawback for the explicit computation of a solution (even with simple Lagrangian L, see [7, 24]). Consequently, numerous studies have been devoted to numerical schemes for (ELα) (see [3, 9”
On Delta and Nabla Caputo Fractional Differences and Dual Identities (2011, background): “1 Introduction During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [9, 10, 11, 23, 24, 25].”
Advanced Local Fractional Calculus and Its Applications (2012, unknown): “[240] D.”
Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets (2014, background): “We refer the interested reader to Baleanu et al. (2012), Baleanu and Trujillo (2008), Lotfi et al. (2011), Magin et al. (2009), Kayedi et al. (2012), Saadatmandi and Dehghan (2010, 2011a), Saadatmandi et al. (2012),Dehghan et al. (2010a, 2010b), Saadatmandi and Dehghan (2011b), Wang (2006), Wu…”
Challenge in the variational iteration method – A new approach to identification of the Lagrange multipliers (2013, background): “This idea can also be extended to the fractional calculus of variations (Baleanu and Trujillo, 2008,2010).”
On Riemann and Caputo fractional differences (2011, background): “Very recently some solutions of equations containing left and right fractional derivatives were investigated [12–14] and some exact solutions were found [12,15].”