This paper studies impulsive dynamic Sturm-Liouville boundary value problems. The existence of a countably infinite set of eigenvalues and Eigenfunctions is proved and a uniformly convergent expansion formula in the Eigenfunctions is established.

In this paper we consider the regular Sturm-Liouville problem -rg”+(q-rl)g=0On the interval [0,p] with initial conditionsg’1(0)= p  وg’1(0)=0, و g’2(0) وg’(0)= p,where  and are positive functions, we will determine the asymptotics form of the Eigenfunctions of Sturm-Liouville problem in a particular case.

An axisymmetric cup-cored coil placed above a layered conductive half-space is analyzed by using the truncated region Eigenfunction expansion method. Closed-form expressions are presented for the coil impedance as well as the induced eddy current density. The results are in very good agreement with results from 2D-FEM and with experimental ones.

Radial basis functions (RBFs) are a powerful method for obtaining the numerical solution of high-dimensional problems. They are often referred to as a meshfree method and the spectrally accurate can be achieved by them. In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the Eigenfunction expansion. We develop our approach in two-dimensional spaces for solving Helmholtz equations. In this paper, the Eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. Numerical examples are presented to demonstrate the effectiveness and robustness of the proposed method for solving two-dimensional Helmholtz equations.

In this paper we study some properties of the Eigenfunctions corresponding to the eigenvalues of Sturm-Liouville Problem in two turning points case. There is connection between the eigenvalues of Sturm-Liouville Problem and infinite product representation. Before representing the solution in the form of an infinite product, we make a slight digression into an asymptotic solution of these problems.First, in this case we discuss the asymptotic form of the eigenvalues of Sturm-Liouville Problems. Then by making use of the eigenvalues and by substituting them in solution and infinite product representation of solution we get the asymptotic form of Eigenfunction and the asymptotic form of the derivative of Eigenfunction.

This paper aims to investigate the stability and numerical approximation of the Sivashinsky equations. We apply the Galerkin meshfree method based on the radial basis functions (RBFs) to discretize the spatial variables and use a group presenting scheme for the time discretization. Because the RBFs do not generally vanish on the boundary, they can not directly approximate a Dirichlet boundary problem by Galerkin method. To avoid this di culty, an auxiliary parametrized technique is used to convert a Dirichlet boundary condition to a Robin one. In addition, we extend a stability theorem on the higher order elliptic equations such as the bihar-monic equation by the Eigenfunction expansion. Some experimental results will be presented to show the performance of the proposed method.

We investigate the boundary-value problem generated by the Sturm-Liouville equation with discontinuous coefficients, eigenparameter dependent boundary conditions and transmission conditions at the point of discontinuity. With a different approach we introduce an adequate Hilbert space formulation, investigate some properties of eigenvalues, Green’s function and resolvent operator, and find simple conditions on the coefficients which guarantee the self-adjointness of the considered problem.

Dynamic analysis of cracked thin rectangular plates subjected to a moving mass first is investigated in this paper. To this end, the Eigenfunction expansion method is utilized to solve the governing equation. For the first time, the intact plate orthogonal polynomials in combination with admissible crack functions as a composition, are employed in the Eigenfunction expansion method formulation, required professional computer programming to solve the equation. The proposed approach guarantees upper bound of the true solution, which is the property of an appropriate numerical solution. Parametric investigation is performed to determine the effect of the moving mass weight, the moving mass velocity, the crack length, and the crack angular orientation as well as the plate’s aspect ratio, on the dynamic response of cracked plates. The results confirm, that the moving mass has a greater impact than the moving load on the dynamic responses of cracked plates. Furthermore, there is nonlinear relation among enhancing the dynamic responses of the cracked plates with various boundary conditions, and magnifying the moving mass weights, elevating the moving mass velocities, lengthening the crack length, raising the inclined crack angels as well as augmenting the plates aspect ratios.