In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfl...

We have considered flat Friedmann-Robertson-Walker (FRW) model of the universe and reviewed modified Chaplygin gas as fluid source. Associated with scalar field model, we determined Hubble parameter a generating function in terms field. Instead hyperbolic function, taken Jacobi elliptic Abel obtained Chaplygin-Jacobi (MCJG) Chaplygin-Abel (MCAG) equation states, respectively. Next, assumed that...

Integral equations can be defined as in which unknown function to determined appears under the integral sign. These have been used many problems occurring different fields due connection they establish with differential equations. Abel?s equation is an important singular and Abel found this from a problem of mechanics, namely tautochrone problem. This some variants it applications heat transfer...

The morphic Abel-Jacobi map is the analogue of the classical Abel-Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of r-cycles on a complex variety that are algebraically equivalent to zero to a certain “Jacobian” built from the Lawson homology groups viewed as inductive limits of mixed Hodge structur...

We study the rational solutions of Abel equation x′=A(t)x3+B(t)x2 where A and B∈C[t]. prove that if deg⁡(A) is even or deg⁡(B)>(deg⁡(A)−1)/2 then has at most two solutions. For any other case, an upper bound on number obtained. Moreover, we there are more than (deg⁡(A)+1)/2 admits a Darboux first integral.

In this paper,The concept of a (∈,∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ ,∈ ∨q)-fuzzy ideals.

In this paper, we show that there are solutions of degree r the equation Pell–Abel on some real hyperelliptic curve genus g if and only r>g. This result, which is known to experts, has consequences, seem be unknown experts. First, deduce existence a primitive k-differential an with unique zero order k(2g-2) for every (k,g)≠(2,2). Moreover, exists non Weierstrass point n modulo n>2g.

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the d-exact form on one side of the equation. We also derive...

We study the periodic solutions of the generalized Abel equation x′ = a1A1(t)x 1+a2A2(t)x 2+a3A3(t)x n3 , where n1, n2, n3 > 1 are distinct integers, a1, a2, a3 ∈ R, and A1, A2, A3 are 2π-periodic analytic functions such that A1(t) sin t,A2(t) cos t, A3(t) sin t cos t are πperiodic positive even functions. When (n3−n1)(n3−n2) < 0 we prove that the equation has no nontrivial (different from zero...

This paper provides some new a priori choice strategy for regularization parameters in order to obtain convergence rates in Tikhonov regularization for solving ill-posed problems Af0 = g0, f0 ∈ X, g0 ∈ Y , with a linear operator A mapping in Hilbert spaces X and Y. Our choice requires only that the range of the adjoint operator A∗ includes a member of some variable Hilbert scale and is, in prin...