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Approximate Analytical Solutions (approximate + analytical_solution)
Selected AbstractsA new perturbation solution for systems with strong quadratic and cubic nonlinearitiesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2010Mehmet Pakdemirli Abstract The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt,Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd. [source] Approximate analytical solutions of the pseudospin symmetric Dirac equation for exponential-type potentialsANNALEN DER PHYSIK, Issue 10-11 2009A. Arda Abstract The solvability of The Dirac equation is studied for the exponential-type potentials with the pseudospin symmetry by using the parametric generalization of the Nikiforov,Uvarov method. The energy eigenvalue equation, and the corresponding Dirac spinors for Morse, Hulthen, and q -deformed Rosen,Morse potentials are obtained within the framework of an approximation to the spin-orbit coupling term, so the solutions are given for any value of the spin-orbit quantum number , = 0, or , , 0. [source] Approximate analytical solutions of the pseudospin symmetric Dirac equation for exponential-type potentialsANNALEN DER PHYSIK, Issue 10-11 2009A. Arda Abstract The solvability of The Dirac equation is studied for the exponential-type potentials with the pseudospin symmetry by using the parametric generalization of the Nikiforov,Uvarov method. The energy eigenvalue equation, and the corresponding Dirac spinors for Morse, Hulthen, and q -deformed Rosen,Morse potentials are obtained within the framework of an approximation to the spin-orbit coupling term, so the solutions are given for any value of the spin-orbit quantum number , = 0, or , , 0. [source] Analytical Methods for Transient Flow to a Well in a Confined-Unconfined AquiferGROUND WATER, Issue 4 2008Li-Tang Hu Concurrent existence of confined and unconfined zones of an aquifer can arise owing to ground water withdrawal by pumping. Using Girinskii's potential function, Chen (1974, 1983) developed an approximate analytical solution to analyze transient ground water flow to a pumping well in an aquifer that changes from an initially confined system to a system with both unconfined and confined regimes. This article presents the details of the Chen model and then compares it with the analytical model developed by Moench and Prickett (1972) for the same problem. Hypothetical pumping test examples in which the aquifer undergoes conversion from confined to water table conditions are solved by the two analytical models and also a numerical model based on MODFLOW. Comparison of the results suggests that the solutions of the Chen model give better results than the Moench and Prickett model except when the radial distance is very large or aquifer thickness is large compared with drawdown. [source] An anisotropic strength criterion for jointed rock masses and its application in wellbore stability analysesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2008X. Chen Abstract In this paper, an anisotropic strength criterion is established for jointed rock masses. An orientation distribution function (ODF) of joint connectivity, is introduced to characterize the anisotropic strength of jointed rock masses related to directional distributed joint sets. Coulomb failure condition is formulated for each plane of jointed rock masses by joint connectivity, where the friction coefficient and cohesion of the jointed rock mass are related to those of the intact rock and joint and become orientation dependent. When approximating joint connectivity by its second-order fabric tensor, an anisotropic strength criterion is derived through an approximate analytical solution to the critical plane problem. To demonstrate the effects of joint distribution on the anisotropic strength of jointed rock masses, the failure envelopes are worked out for different relative orientations of material anisotropy and principal stress axes. The anisotropic strength criterion is also applied to wellbore stability analyses. It is shown that a borehole drilled in the direction of the maximum principal in situ stress is not always the safest due to the anisotropic strength of the jointed rock mass. Copyright © 2007 John Wiley & Sons, Ltd. [source] A unified formulation for continuum mechanics applied to fluid,structure interaction in flexible tubesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2005C. J. Greenshields Abstract This paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid,structure interaction in flexible tubes. A review of current methods of fluid,structure coupling highlights common limitations of high computational cost and solution instability. It is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain. A single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes. The method is validated first by simulating dynamic oscillation of a clamped elastic beam. It is then applied to study the case of interest,wave propagation in highly flexible tubes,in which a predicted wave speed of 8.58 m/s falls within 2% of an approximate analytical solution. The method shows further good agreement with analytical solutions for tubes of increasing rigidity, covering a range of wave speeds from those found in arteries to that in the undisturbed fluid. Copyright © 2005 John Wiley & Sons, Ltd. [source] Kinematic response functions and dynamic stiffnesses of bridge embankmentsEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 11 2002Jian Zhang Abstract Recognizing that soil,structure interaction affects appreciably the earthquake response of highway overcrossings, this paper compares approximate analytical solutions and finite element results to conclude on a simple procedure that allows for the estimation of the kinematic response functions and dynamic stiffnesses of approach embankments. It is shown that the shear-wedge model yields realistic estimates for the amplification functions of typical embankments and reveals the appropriate levels of dynamic strains which are subsequently used to estimate the stiffness and damping coefficients of embankments. The shear-wedge model is extended to a two-dimensional model in order to calculate the transverse static stiffness of an approach embankment loaded at one end. The formulation leads to a sound closed-form expression for the critical length, Lc, that is the ratio of the transverse static stiffness of an approach embankment and the transverse static stiffness of a unit-width wedge. It is shown through two case studies that the transverse dynamic stiffness (,spring' and ,dashpot') of the approach embankment can be estimated with confidence by multiplying the dynamic stiffness of the unit-width wedge with the critical length, Lc. The paper concludes that the values obtained for the transverse kinematic response function and dynamic stiffness can also be used with confidence to represent the longitudinal kinematic response function and dynamic stiffness, respectively. Copyright © 2002 John Wiley & Sons, Ltd. [source] A dynamic approach for evaluating parameters in a numerical methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005A. A. Oberai Abstract A new methodology for evaluating unknown parameters in a numerical method for solving a partial differential equation is developed. The main result is the identification of a functional form for the parameters which is derived by requiring the numerical method to yield ,optimal' solutions over a set of finite-dimensional function spaces. The functional depends upon the numerical solution, the forcing function, the set of function spaces, and the definition of the optimal solution. It does not require exact or approximate analytical solutions of the continuous problem, and is derived from an extension of the variational Germano identity. This methodology is applied to the one-dimensional, linear advection,diffusion problem to yield a non-linear dynamic diffusivity method. It is found that this method yields results that are commensurate to the SUPG method. The same methodology is then used to evaluate the Smagorinsky eddy viscosity for the large eddy simulation of the decay of homogeneous isotropic turbulence in three dimensions. In this case the resulting method is found to be more accurate than the constant-coefficient and the traditional dynamic versions of the Smagorinsky model. Copyright © 2004 John Wiley & Sons, Ltd. [source] Application of variational iteration method for modified Camassa-Holm and Degasperis-Procesi equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2010H. Jafari Abstract In this article, the variational iteration method (VIM) is used to obtain approximate analytical solutions of the modified Camassa-Holm and Degasperis-Procesi equations. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The results reveal that the VIM is very effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] | |||||||||||||