Abstract:
If be a monotically increasing function on [a, b] and (a), (b) are Finite and f be a Bounded function defined on [a, b] such that f R ( ), m f M, is continuous on [m, M] And h(x) = f(x) on [a, b], then h R ( ) . In this study, we generalize the above theorem in a suitable forms for both integrand and integrator

Abstract:
If M be closed subspace of a Hilbert space H, then there exist a unique pair of Linear mappings P and Q such that P maps H onto M,Q maps H onto M , And x = Px + Qx for all x H. In this study we prove some similar results For any Hilbert space H, the identity map I and all skew Hermitian maps. Some applications also followed.

Abstract:
In this paper we introduce a new numerical method to computation of definite integrals. This method is based on Legendre wavelets. In usual numerical rules such as Simpson`s rule, we needs the values of integrand on the various points(at least three points of interval). However, in wavelet method we needs the integrand and first derivative of it at one point. In general, the wavelets rule is an accurate rule for any polynomials of degree n if we have the values of integrand and derivatives at one point of interval.

Abstract:
A direct method for solving linear differential equation under initial values using Legendre function is presented. An operational matrix introduces for operator of differential equation and it reduces into a set of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Abstract:
It is well known that the shape of liquid drop on flat surface is dependent to several energy. One of these energy is surface energy. In this paper to attempt calculate the Free energy of liquid drop on the flat surface and then with extermomize of it, suggest a method for measure surface tension or contact liquid angle with flat surface.

Abstract:
Wavelets constitute a family of functions that constructed from dilation and translation of a single function. They are suitable tools for solving variational problems. In this study, we want to extremum the Hamiltonian of hydrogen atom using Legendre wavelets. Legendre wavelets are defined on the domain [0,1]. For solving this problem, we represent a generalized Legendre functions and generalized Legendre wavelets on the [-s, s] and [0, s], respectively. We start from the radial equations of hydrogen atom like and represent the wave function in term of generalized Legendre function and then convert the redial equation of hydrogen atom like to a polynomial in term of coefficients of wave function. The eigenstate will be minimize provided that, the derivative of it respect to the all of coefficients of wave function to be equal zero. The last equation is a algebraic equation and the solutions are the energy states of hydrogen atom like.

Abstract:
In this research, we present a numerical solution for schrodinger equation. This method is based on generalized Legendre wavelets and generalized operational matrices. Generalized Legendre wavelets are a complete orthogonal set on the interval [-s, s] (s is a real large positive number.) The mother function of generalized Legendre wavelets are generalized legendre functions. Generalized Legendre functions are an orthogonal set on the interval [-s, s]. The schrodinger equation is equal to a variational problem and we convert the variational problem to a non linear algebraic equations. From the solving of algebraic equation to get the eigen-states of schrodinger equation. We applied this method to one dimension nonlinear oscillator (V(x) = 1/2kxn, - < x < ) and to get the eigen-states of oscillator for various n. For n = 2, the oscillator is linear and there is an exact solution for its. The results for n = 2 demonstrate the validity of this solution.

Abstract:
A direct method for solving integral equations using Legendre function is presented. An operational matrix introduce for operator of integral equation and it reduce into a set of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.