This paper studies a fractional differential equation combined with Liouville–Caputo operator, namely, LCDηβ,γQ(t)=λϑ(t,Q(t)),t∈[c,d],β,γ∈(0,1],η∈[0,1], where Q(c)=qc is bounded and non-negative initial value. The function ϑ:[c,d]×R→R Lipschitz continuous in the second variable, λ>0 constant operator LCDηβ,γ convex combination of left right derivatives. We study well-posedness using fixed-po...

It is well known that the two spectra {λn} and {μn} uniquely determine the potential function q (r) in a Sturm-Liouville equation defined on the unit interval having the singularity type ( ) 2 1 2 r r + +   (where  is a integer) at the point zero. In this work, we give the solution of the inverse problem on two partially noncoincide spectra for the Sturm-Liouville equation with the peculiari...

To study the problem of the assigned Gauss curvature with conical singularities on Riemanian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce the problem to a Liouville equation on the euclidean plane. We prove new multiplicity results for bounded radial solutions, which improve on earlier results o...

In this paper, the fractional partial differential equations are defined by modified Riemann-Liouville fractional derivative. With the help of fractional derivative and fractional complex transform, these equations can be converted into the nonlinear ordinary differential equations. By using solitay wave ansatz method, we find exact analytical solutions of the space-time fractional Zakharov Kuz...

We study an explicit correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy. A Liouville transformation between the isospectral problems of the two hierarchies also relates their respective recursion operators, and serves to establish the Liouville correspondence between their flows and Hamiltonian conservation laws. In addition, a n...

in this article, we verify existence and uniqueness of positive and nondecreasing solution for nonlinear boundary value problem of fractional differential equation in the form d_{0^{+}}^{alpha}x(t)+f(t,x(t))=0, 0x(0)= x'(0)=0, x'(1)=beta x(xi), where $d_{0^{+}}^{alpha}$ denotes the standard riemann-liouville fractional derivative, 0an illustrative example is also presented.

Two dimensional quantum R 2-gravity and its phase structure are examined in the semiclassical approach and compared with the results of the numerical simulation. Three phases are succinctly characterized by the effective action. A classical solution of R 2-Liouville equation is obtained by use of the solution of the ordinary Liouville equation. The partition function is obtained analytically. A...