Riemann-Liouville integrals of fractional order and extended KP hierarchy
نویسنده
چکیده
An attempt is given to formulate the extensions of the KP hierarchy by introducing fractional order pseudo-differential operators. In the case of the extension with the half-order pseudo-differential operators, a system analogous to the supersymmetric extensions of the KP hierarchy is obtained. Unlike the supersymmetric extensions, no Grassmannian variable appears in the hierarchy considered here. More general hierarchies constructed by the 1/N -th order pseudo-differential operators, their integrability and the reduction procedure are also investigated. In addition to finding out the new extensions of the KP hierarchy, brief introduction to the Riemann-Liouville integral is provided to yield a candidate for the fractional order pseudo-differential operators. PACS numbers: 02.30.Ik, 02.30.Jr § e-mail address: [email protected] ‖ e-mail address: [email protected] Riemann-Liouville integrals of fractional order and extended KP hierarchy 2
منابع مشابه
Integral Inequalities for h(x)-Riemann-Liouville Fractional Integrals
In this article, we obtain generalizations for Grüss type integral inequality by using h(x)-Riemann-Liouville fractional integral.
متن کاملOn Generalizations of Hadamard Inequalities for Fractional Integrals
Fej'{e}r Hadamard inequality is generalization of Hadamard inequality. In this paper we prove certain Fej'{e}r Hadamard inequalities for $k$-fractional integrals. We deduce Fej'{e}r Hadamard-type inequalities for Riemann-Liouville fractional integrals. Also as special case Hadamard inequalities for $k$-fractional as well as fractional integrals are given.
متن کاملNew operational matrix for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative
In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractio...
متن کاملThe Stability of Non-standard Finite Difference Scheme for Solution of Partial Differential Equations of Fractional Order
Fractional derivatives and integrals are new concepts of derivatives and integrals of arbitrary order. Partial differential equations whose derivatives can be of fractional order are called fractional partial differential equations (FPDEs). Recently, these equations have received special attention due to their high practical applications. In this paper, we survey a rather general case of FPDE t...
متن کاملNecessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus with Riemann-Liouville Derivatives of Order (α, β)∗
We derive Euler-Lagrange type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order (α, β), α > 0, β > 0, recently introduced by J. Cresson and S. Darses. Some interesting consequences are obtained and discussed. Mathematics Subject Classification 2000: 49K05, 49S05, 70H33, 26A33.
متن کامل