In This paper generalized spline method and Caputo differential operator is applied to solve linear fractional integro-differential equations of the second kind. Comparison of the applied method with exact solutions reveals that the method is tremendously effective.

The concern of this article is the calculation of an upper bound of second Hankel determinant for the subclasses of functions defined by Al-Oboudi differential operator in the unit disc. To study special cases of the results of this article, we give particular values to the parameters A, B and λ

    In this paper, we derive some subordination and superordination results for certain subclasses of p− valent analytic functions that defined by generalized Fox-wright functions using the principle of differential subordination, ----------producing best dominant univalent solutions. We have also derived inclusion relations and solved majorization problem.

The main objective of" this paper is to study a subclass of holomrphic and univalent functions with negative coefficients in the open unit disk U= defined by Hadamard Product. We obtain coefficients estimates, distortion theorem , fractional derivatives, fractional integrals, and some results.

In these notes, our goal is to give some results on criterion for complex analytic map-germs by their tangent spaces with respect to -equivalence where is the module of complex analytic vector fields on .In addition, we give some results about -trivial analytic family, the direct product and direct sum of map-germs.

In this paper, we define certain subclasses of analytic univalent function associated with quasi-subordination. Some results such as coefficient bounds and Fekete-Szego bounds for the functions belonging to these subclasses are derived.

     An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integral

In this paper, we define two operators of summation and summation-integral of q-type in two dimensional spaces. Firstly, we study the convergence of these operators and then we prove Voronovskaya- type asymptotic formulas for these operators.

      In this paper, we consider new subclasses     of meromorphic uniformly of multivalent functions in    with fixed second coefficient, we obtain the estimation of coefficients, distortion theorems, closure theorems and some other results.

      This study aims to classify the critical points of functions with 4 variables and 8 parameters, we found the caustic for the certain function with the spreading of the critical points. Finally, as an application, we found the bifurcation solutions for the equation of sixth order with boundary conditions using the Lyapunov-Schmidt method in the variational case.