In this paper, the nonlinear Schrödinger equation with power law nonlinearity is studied. The first integral method, the Riccati sub - ODE method are efficient methods to construct the exact solutions of nonlinear partial differential equations.By means of these methods, the periodic and solitary wave solutions of the nonlinear Schrödinger equation with power law nonlinearity are obtained.

In this paper we introduced the (E.A.)-property and weak compatibility of mappings in G-metric spaces. We have utilized these concepts to deduce certain common fixed point theorems in G-metric space

In this paper a Legendre integral method is proposed to solve integral and integro - differential problems and optimal co ntrol problems governed by integral and integro - differential equations. Galerkin method is used to reformulate the problem as constrained optimization problem. The resulting constrained optimization problem is solved by Hybrid penalty partial quadratic interpolation technique. Numerical results are included to confirm the efficiency and accuracy of the method

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We have derived an analytical solution of the 3D SchrÄodinger equation where the mass varies with position. Our solution is a general solution for any massive particle with a position dependent mass m(r) scattered by an arbitrary potential V(r)

In this paper a new matrix approach for solving l inear hyperbolic partial differential equations (HPDEs) is presented. The method is based on the Bernoulli expansion of two - variable functions, which consists of the matrix representation of expressions in the considered HPDE. Also, a new operational matri x of differentiation is introduced, which consists of nonzero elements under its diagonal, meanwhile the operational matrix of differentiation of other polynomial bases (such as Chebyshev, Legendre, etc.) is a strictly (upper or lower) triangular matrix. I n the proposed method, HPDE together with the initial and boundary conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Bernoulli coefficients. Combining these matrix equations a nd then solving the system yields the Bernoulli coefficients of the approximated solution. Illustrative examples are included to demonstrate the validity and applicability of the technique. All of computations are performed on a PC using several programs w ritten in MATLAB 7.12.0.

In this paper, we consider the approximate solution of the par tial integro - differential e q uation. To solve this problem, we introduce a new nonstandard time discretization scheme. Then the fourth order finite difference and collocation method is pr e sented for the numerical solution of this type of partial integro - dif ferential equation (PIDE). A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed - form difference scheme. The efficiency and accuracy of the scheme is validated by its application to one test pro blem which have exact solutions. Numerical results show that this fourth - order scheme has the expected accuracy. The most advantages of compact finite difference method for PIDE are that it obtains high order of accuracy, while the time complexity to solve the matrix equations after we use compact finite difference method on PIDE is O(N), and it can solve very ge n eral case of PIDE.