A fractional calculus of variations for multiple integrals with application to vibrating string
نویسندگان
چکیده
We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string. © 2010 American Institute of Physics. doi:10.1063/1.3319559
منابع مشابه
Some new results using Hadamard fractional integral
Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The purpose of this work is to use Hadamard fractional integral to establish some new integral inequalities of Gruss type by using one or two parameters which ensues four main results . Furthermore, other integral inequalities of reverse ...
متن کاملSome Weighted Integral Inequalities for Generalized Conformable Fractional Calculus
In this paper, we have obtained weighted versions of Ostrowski, Čebysev and Grüss type inequalities for conformable fractional integrals which is given by Katugompola. By using the Katugampola definition for conformable calculus, the present study confirms previous findings and contributes additional evidence that provide the bounds for more general functions.
متن کاملFractional order Adaptive Terminal Sliding Mode Controller Design for MPPT in a Solar Cell under Normal and Partial Shading Condition
In this paper, by combining fractional calculus and sliding mode control theory, a new fractional order adaptive terminal sliding mode controller is proposed for the maximum power point tracking in a solar cell. To find the maximum power point, the incremental conductance method has been used. First, a fractional order terminal sliding mode controller is designed in which the control law depend...
متن کاملCalculus of variations with fractional derivatives and fractional integrals
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.
متن کاملStability and Robust Performance Analysis of Fractional Order Controller over Conventional Controller Design
In this paper, a new comparative approach has been proposed for reliable controller design. Scientists and engineers are often confronted with the analysis, design, and synthesis of real-life problems. The first step in such studies is the development of a 'mathematical model' which can be considered as a substitute for the real problem. The mathematical model is used here as a plant. Fractiona...
متن کامل