Fractional Variational Study

Fractional calculus is the branch of mathematics that generalizes the derivative and the integr- al of a function to a noninteger order. The study of fractional problems of the calculus of variati- ons is a subject of current strong research due to its many applications in engineering, mechanics, chemistry, biology, economics and control theory.The fractional calculus of variations was introduced by Riewe, where he developed Hamiltoni- an, and other concepts of classical mechanics using fractional calculus.Klimek presented a fractional sequential mechanics model with symmetric fractional derivati- ves and stationary conservation laws for fractional differential equations with variable coefficie- nts.Agrawal combined the calculus of variations and the concepts of fractional derivatives to obta- in the fractional variational problems. He considered the functional which had unspecified end points. The extremal function satisfied the terminal condition y ( a)=ya and intersected the curve z = c(x) for the first time at b , i.e. y (b )= c(b). Here c (x)was the specified curve. Mohamed discussed the necessary optimality conditions for the functionals of the form: J ( y)= I_a~γ+L(x,y(x),~RD_a~α+y,y(a)). In chapter 3, we will develop the theory of fractional variational calculus further by proving the necessary and sufficient optimality conditions for more general problems. The functional that we consider has the form: J ( y)=∫_a ~b L(x,y(x),~C D_a~α+ y(x),y(a),y(b))dxâ†'min The initial time x = a is specified, while the initial point y (a), the terminal time b , and t- he end point y (b) may be not specified. In chapter 4, we will develop the theory of fractional variational calculus further by proving the necessary and sufficient optimality conditions for more general problems. We consider two types of fractional variational problems: J ( y)= I_a~γ+ L[x,y(x),~RDaα+y,CD_a~β+y,y(a)]â†'min J ( y)= I_b~γ- L[x,y(x),~R D~b~α-y,CD_b~β-y,y(b)]â†'min...