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Fractional calculus is a branch of mathematics , the integer order calculus extended to non- integer order . Fractional variational problems in the disciplines of engineering , mechanics , chemistry , biology , economics , and control of a wide range of applications , and therefore become a hot topic in recent years . Riewe was first proposed in 1996 , Fractional variational problems , his article introduced the Hamiltonian equations and describes some of the principles of classical mechanics using fractional calculus . The Klimek utilize balanced fractional differential described Fractional ordered mechanical model , with a coefficient of fractional order differential equations to describe the stability of conservative rule . Indian mathematician the Agrawal variational method and fractional differential principles to study Fractional variational problem . And the fixed boundary fractional variational problem , the article extremal curve satisfies the boundary conditions : y (a ) = y_a and the first to reach the termination time happens to fall on a fixed curve z = c (x) . Mohamed discussed shaped such as J (y ) = I_a ~ γ L ( x , y ( x) , to RD_a ~ α y , y, ( A )) Functional BEST necessary and optimal fully conditions . In the third chapter will be discussed below in the form of fractional variational problem of optimal necessary and optimal sufficient condition : J (y ) = ∫ _a ~~ b L ( x , y ( x) , C D_a ~ α y ( x), y (a), y (b)) dx → min we discuss six different boundary conditions , wherein the initial moment x = a fixed , but the initial point of y (a), termination timing b and the termination point y (b) are uncertain . In some cases, when the termination time , y (x) falls exactly on a fixed curve . In the fourth chapter , we will discuss the broader fractional optimal necessary conditions for variational problems ( the fractional integral number of fractional variational problem ) and optimal sufficient condition . We will consider the following two forms of Functional : J ( y ) = I_a ~ γ L [ the x, y (x ) , to RDaα Y , CD_a to β y , y, ( A ) ] → min ( y ) = J I_b ~ γ-L [x, y (x), ~ RD ~ b ~ α-y, CD_b ~ β-y, y (b)] → min Lagrangian function and Riemann-Liouville fractional differential , Caputo fractional differential and free boundary conditions .
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