Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Numerical solution of nth‐order integro‐differential equations using trigonometric wavelets

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro‐differential equations of nth‐order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro‐differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd.     

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.     

An Ab Initio Molecular Orbital Study of Isophosphinines

Ab initio molecular orbital calculation at HF/6-31G*, HF/6-31G**, HF/6-311G**, HF/6-311++G**, RMP2-FC/6-31G*, and B3LYP/6-31G* levels of theory for geometry optimization and MP4(SDQ)/6-31G* for a single-point total energy calculation are reported for phosphinine and 13 isophosphinines 7-19 . Isomers 7-11 with an allenic system are calculated to be 8-18 kcal mol m 1 more stable than structures 12-17 with an acetylenic moiety. The calculated energy difference (66.19 kcal mol m 1 ) between phosphinine and the most stable isophosphinine (1-phospha-1,2,4-cyclohexatriene, 10 ) is smaller than the difference (78.96 kcal mol m 1 ) between benzene and the most stable isobenzene (cyclohexa-1,2,4-triene, 2 ). The isophosphinines 18 and 19 , with a butatriene moiety, are calculated to be the least stable isomers.     

Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline functions

Three numerical techniques based on cubic Hermite spline functions are presented for the solution of Lane–Emden equation. Some properties of Hermite splines are presented and are utilized to reduce the solution of Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of these techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.     

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

In this study, we derive optimal uniform error bounds for moving least‐squares (MLS) mesh‐free point collocation (also called finite point method) when applied to solve second‐order elliptic partial integro‐differential equations (PIDEs). In the special case of elliptic partial differential equations (PDEs), we show that our estimate improves the results of Cheng and Cheng (Appl. Numer. Math. 58 (2008), no. 6, 884–898) both in terms of the used error norm (here the uniform norm and there the discrete vector norm) and the obtained order of convergence. We then present optimal convergence rate estimates for second‐order elliptic PIDEs. We proceed by some numerical experiments dealing with elliptic PDEs that confirm the obtained theoretical results. The article concludes with numerical approximation of the linear parabolic PIDE arising from European option pricing problem under Merton's and Kou's jump‐diffusion models. The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the MLS collocation scheme is an efficient and reliable numerical method to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.     

Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions

A numerical technique based on the finite difference and collocation methods is presented for the solution of generalized Kuramoto-Sivashinsky (GKS) equation. The derivative matrices between any two families of B-spline functions are presented and are utilized to reduce the solution of GKS equation to the solution of linear algebraic equations. Numerical simulations for five test examples have been demonstrated to validate the technique proposed in the current paper. It is found that the simulating results are in good agreement with the exact solutions.