Berger Equation Research Articles

An approximate solution of a time fractional Burgers’ equation involving the Caputo-Katugampola fractional derivative

The reduced version of the fractional Laplace transform, called the v-Laplac transform, is used in combination with the Adomian decomposition method to generate approximate solutions of the fractional Berger's equation with the Caputo-Katugampola fractional derivative. The effect of the order of the Caputo-Katugampola fractional derivative in Berger's equation is analyzed. The obtained approximate solutions are displayed graphically. The graphs and numerical solutions have demonstrated a tight correspondence between the exact and v-Laplace DM solutions. It is observed that the solutions for various orders u and v display the same behavior and tend to an integer-order problem's solution, confirming the validity of the provided method.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

Random attractor for stochastic non-autonomous Berger equation with critical exponent

In this paper, we consider the existence of a random attractor for the stochastic non-autonomous Berger equation with additive white noise and critical exponent. Under the assumption that the external term is time-dependent, by splitting the solutions and estimating the bounds of solutions in a higher regular space, we establish the existence of random attractors.

KdV–Berger solution and local cluster effect of two-sided lateral gap continuum traffic flow model

This study proposes a new nonlane-based continuum model derived from a two-sided lateral gap-following theory using the relation between microscopic and macroscopic variables. The model considers the effect of lateral gaps of the leading vehicles available on both sides of the following vehicle in multilane scenario. Linear stability analysis is performed to establish the neutral stability condition for the stable traffic flow. Nonlinear analysis is carried out at neutral stability line to derive the KdV–Berger equation, which describes density wave propagation. For that, one of the traveling wave solutions is also obtained. Numerical simulation results show that the two-sided lateral gap in the model improves the stability of the traffic flow by suppressing the traffic jams even at high-density conditions. The results implies that the proposed model is successful in replicating the properties of actual traffic jams in nonlane-based traffic environment.

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

• Two-dimensional linear and nonlinear Berger equation investigated. • Meshless multiple-scale Pascal polynomials method applied to fourth order equations. • Accurate results on complexly shaped domains with low number of collocation points . • Efficiency of proposed method demonstrated based on six problems. In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.

The method of fundamental solutions for solving non-linear Berger equation of thin elastic plate

In this paper, we utilized the method of fundamental solutions, which is meshless and integral-free, to analyze the non-linear Berger equation for thin elastic plate. Based on the proposed numerical scheme, the deflection can be expressed as the linear combination of the homogeneous solution and the particular solutions. The particular solution, based on the polyharmonic splines, is derived and then the spatial-dependent loading term of the Berger equation can be approximated by the polyharmonic splines. After the particular solution is obtained, the homogeneous solution, which is governed by the homogeneous partial differential equations, can be solved by the method of fundamental solutions. Several numerical examples are adopted to demonstrate the flexibility and robustness of the proposed meshless scheme, especially the irregular plate with spatial-dependent loading function. Furthermore, we also performed the convergence test for various orders of the polyharmonic splines.

The local Kansa׳s method for solving Berger equation

In this paper, we present the local Kansa׳s method using radial basis functions (RBFs) to solve Berger equation which is a fourth order partial differential equation. To overcome the difficulty of solving higher order differential equations using localized RBF methods, we split the given equation into two second order partial differential equations. Furthermore, we use Matern function and normalized MQ as basis functions and make a comparison between the two radial basis functions in terms of accuracy and stability. LOOCV (Leave-One-Out-Cross-Validation) is used to find a good shape parameter of MQ and Matern function. To demonstrate the effectiveness of the local Kansa׳s method for solving Berger equation, three examples are given.

Application of the charge simulation method to analysis of large deflections of elastic plates

The non-linear Berger equation is used to obtain solutions for deformation of thin elastic plates, and is solved by applying the charge simulation method. The general solution for the deflection is first obtained by a combination of two kinds of series of Green's functions. Satisfying the boundary conditions at the collocation points, the unknown constants in the general solution are determined, and the deflection of the plate is calculated. Numerical results are presented in dimensionless graphical form for rectangular and isosceles triangular plates.

Simulation of medical electron linac bremsstrahlung beam transport in typical shielding materials

The Monte Carlo N-particle code MCNP version 4C3 was used to investigate the backscattering and transmission of high-energy photons in concrete, iron and lead at deep penetration. A typical bremsstrahlung beam from a 24 MV linac was used, and the transmission up to 15 mean-free paths was studied. Broad beam slab geometry was used. Estimates of the transmission in terms of absorbed dose to tissue ratio and air kerma ratio were performed for the primary and secondary components of the transmitted beam in the three materials. The tissue dose and air kerma buildup factors were calculated and fitted to Berger's equation. Finally, the differential dose albedo values for common reflected angles were determined.

An expansion of the solution of Dirichlet boundary value problem for Berger equation

The Dirichlet boundary value problem for Berger equation is reduced to the successive sequence of boundary value problems, which may be decomposed into a coupled systems of Poisson and Helmholtz equations. Convergence of a series in solutions of the systems of coupled equations to the solution of Berger boundary value problem with Dirichlet and the mixed boundary conditions is established. The bounds for the coupling function are found and explicit value of the upper bound is obtained for the biharmonic boundary value problem in a circular domain.

A meshless method for large deflection of plates

The nonlinear integro-differential Berger equation is used for description of large deflections of thin plates. An iterative solution of Berger equation by the local boundary integral equation method with meshless approximation of physical quantities is proposed. In each iterative step the Berger equation can be considered as a partial differential equation of the fourth order. The governing equation is decomposed into two coupled partial differential equations of the second order. One of them is Poisson's equation whereas the other one is Helmholtz's equation. The local boundary integral equation method is applied to both these equations. Numerical results for a square plate with simply supported and/or clamped edges as well as a circular clamped plate are presented to prove the efficiency of the proposed formulation.

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.

Historical Heresies and Modern Jewish Identity

n The Heretical Imperative, Peter Berger argued that modernity, characterized as it is by a plurality of choices, necessarily embraces heresy, for heresy, as he defines it (going back hereisis in Greek) means to choose. In the Hellenistic oikumene, the word most commonly referred a faction or party, from which, in the letters of Paul, for example,1 it eventually developed the pejorative meaning of factionalism or sectarian disagreement. With the gradual emergence of an orthodox Church, heresy came mean dissent from dogma, and, indeed, orthodoxy constructed itself precisely by naming and categorizing heresies. Berger contrasts the world of with modernity in the following terms: For premodern man, heresy is a possibility-usually a rather remote one; for modern man, heresy typically becomes a necessity. Or again, modernity creates a new situation in which picking and choosing becomes an imperative.2 Berger's equation of secularism with heresy is a singularly interesting attempt describe the dialectic of modernization in religious terms. Yet it seems me fail, at least in the Jewish case, one that Berger himself quotes repeatedly. Jewish tradition does not conform his definition of tradition, the static character of which probably applies no historical tradition. Consequently, the relationship of modern Jews heresy is not quite the diametric opposite of tradition. Rabbinic culture, which represents tradition for Berger, certainly developed a concept of heresy and even a lexicon of terms (min, kofer, epikoros) by which label the heretic. But the process of marking the boundaries

Large Deflection Analysis of Rectangular Plates on Two-parameter Elastic Foundation by the Boundary Strip Method.

A method is proposed for calculating the large deflection of a rectangular plate on an elastic foundation by the boundary integral equation method. The elastic foundation is assumed to be of the Pasternak type. The large deflection is formulated based on the nonlinear Berger equation. In the analysis, the governing nonlinear partial differential equation for a plate is transformed into an ordinary nonlinear differential equation using the Kantrovich method. The resultant fourth-order nonlinear differential equation was analyzed by the boundary element method. The nonlinear boundary integral equation is solved by using an iteration scheme. A fundamental solution required in the formulation is derived by solving the singular differential equation. It is found that the numerical results obtained by the present method agree well with the analytical solutions. To show the usefulness of the method, serveral numerical results based on the present formulation are given for various types of boundary and loading conditions.

A Simplified Approach to Analysis of Large Deflections for Rectangular Plates with a Cross-Stiffener.

In this study, the nonlinear Berger equation for large deflection problems of isotropic plates and the finite-difference method are used to obtain solutions for deformation in rectangular elastic plates with a cross-stiffener. Numerical solutions are presented in order to illustrate the influence of the stiffener on the center deflection in uniformly loaded rectangular plates, which are supported by two beams orthogonally intersecting at the center of the plate. The numerical procedure is simpler than the procedures based on the von Karman theory, and takes into account the continuity conditions related to particular loads more rigorously than tha variational methods, such as the Rayleigh-Ritz energy method and finite element method.

Analysis of large deflections for pressure loaded plates with rigid inclusions

In the present paper, the non-linear Berger equation for large deflection problems of isotropic plates and the point-matching method are utilized to obtain solutions for out-of-plane deformations in rectangular isotropic plates with a circular rigid body at the centre. Numerical solutions are presented in order to illustrate the influence of the rigid body size on the deflection distribution in rectangular plates subjected to uniform static load. The numerical procedure is easier than other procedures based on the von Kármán theory, and reasonable results are obtained.

Analysis of Large Deflection for Doubly-Connected Plates on an Elastic Foundation.

In the present paper, the nonlinear Berger equation for large deflection problems of plates and the finite difference method are utilized to obtain solutions for deformation in doubly-connected plates on an elastic foundation. The analysis is carried out for a rectangular isotropic plate with a rectangular rigid body at the center. Numerical solutions are presented in order to illustrate the influence of the rigid body size on the maximum deflection in uniformly loaded plates on an elastic foundation. The numerical procedure is easier than other procedures, such as the finite element method, and reasonable results are obtained.

Analysis of Large Deflection for Rectangular Plates with a Circular Rigid Body.

In the bending analysis of thin elastic plates subjected to lateral load, formulas based on the linear theory are quite satisfactory, provided that the deflections are small compared to the plate thickness. If deflections are of the same order of magnitude as the plate thickness, the linear theory may yield results which are erroneous and therefore a large deflection theory should be applied. The nonlinear Berger equation for large deflection problems of plates and the point-matching method are utilized to obtain solutions for deformation in rectangular isotropic plates with a circular rigid body at the center. Numerical solutions are presented in order to illustrate the influence of the rigid body size on the deflection distribution in plates subjected to concentrated load at the center of the rigid body. The numerical procedure is easier than other procedures, such as the boundary element method, and reasonable results are obtained.

Nonlinear bending analysis without domain-cell discretization

A new computational scheme for the analysis of the finite deflection, geometrically nonlinear plate bending is presented in this paper using the dual reciprocity boundary element method (DR/BEM). Integral equations, formulated by the combined use of the Berger equation governing approximate nonlinear bending and the conventional fundamental solution to the biharmonic differential operator, include domain integrals relating the nonlinear term of the original governing equation. Such domain integrals containing unknowns are transformed into equivalent boundary integrals according to the idea of the dual reciprocity process. Derived integral equations are formulated only on the boundary of the plate and do not require any domain discretization by internal cells. Availability of the present formulation and the solution scheme for the finite deflection problem is shown by some simple example calculations.

Boundary strip method for large deflection analysis of elastic rectangular plates.

A new iterative soltion of the nonlinear Berger equation for plate bending problems is proposed. The solution is derived by utilizing both the idea of the Kantrovich method and the boundary integral equation method. The present method, named the boundary strip metheod, could reduce the degrees of freedom of the problem to be solved and input data, although the plate configuration and the boundary conditions of the problems are limited to some extent. In order to show the validity of the proposed method, several examples under various boundary conditions are computed. The numerical results are in good agreement with the results obtained by other authors.