scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Sinc-Galerkin method for solving hyperbolic partial differential equations

2018 ◽  
Vol 8 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Hyperbolic Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Numerical Examples

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

a retrospective view of Richardson's book on weather prediction*

1967 ◽  
Vol 48 (8) ◽  
pp. 514-551 ◽  
Author(s):  
George W. Platzman
Keyword(s):  
Vertical Velocity ◽  
Stability Criterion ◽  
Weather Prediction ◽  
Pressure Change ◽  
Time Interval ◽  
Time Step ◽  
Initial Field ◽  
Static Approximation ◽  
Modern Development ◽  
The Stability

In 1922 Lewis F. Richardson published a comprehensive numerical method of weather prediction. He used height rather than pressure as vertical coordinate but recognized that a diagnostic equation for the vertical velocity is a necessary corollary to the quasi-static approximation. His vertical-velocity equation is the principal, substantive contribution of the book to dynamic meteorology. A comparison of Richardson's model with one now in operational use at the U. S. National Meteorological Center shows that, if only the essential attributes of these models are considered, there is virtually no fundamental difference between them. Even the vertical and horizontal resolutions of the models are similar. Richardson made a forecast at two grid points in central Europe and obtained catastrophic results, in particular a surface pressure change of 145 mb in 6 hours. This failure resulted partly, as Richardson believed, from inadequacies of upper wind data. Underlying this was a more fundamental difficulty which he did not seem to recognize clearly at the time he wrote his book: the impossibility of using observed winds to calculate pressure change from the pressure-tendency equation, a principle stated many years earlier by Margules. However, he did point in the direction in which a remedy was later found: suppression or smoothing of the initial field of horizontal velocity divergence. The 6-hr time interval used by Richardson violates the condition for computational stability, a constraint then unknown. It is sometimes said that this is one of the reasons his calculation failed, but that interpretation is misleading because the stability criterion becomes relevant only after several time steps have been made. Since Richardson did not go beyond a calculation of initial tendencies—in other words, he took only one time step—violation of the stability criterion had no effect on the result. Richardson's book surely must be recorded as a major scientific achievement. Nevertheless, it appears to have had little influence in the decades that followed, and indeed, the modern development of numerical weather prediction, which began about twenty-five years later, did not evolve primarily from Richardson's work. Shaw said it would be misleading to regard the book as “a soliloquy on the scientific stage,” but in fact that is what it proved to be. The intriguing problem of explaining this strange irony is one that leads beyond the obvious facts that when Richardson wrote, computers were nonexistent and upper-air data insufficient.

On the Buckling of Circular Cylindrical Shells Under Pure Bending

1961 ◽  
Vol 28 (1) ◽  
pp. 112-116 ◽  
Author(s):  
Paul Seide ◽  
V. I. Weingarten
Keyword(s):  
Galerkin Method ◽  
Compressive Stress ◽  
Cylindrical Shells ◽  
Bending Stress ◽  
Pure Bending ◽  
Circular Cylindrical Shells ◽  
The Stability ◽  
The Galerkin Method

The stability of circular cylindrical shells under pure bending is investigated by means of Batdorf’s modified Donnell’s equation and the Galerkin method. The results of this investigation have shown that, contrary to the commonly accepted value, the maximum critical bending stress is for all practical purposes equal to the critical compressive stress.

τ-D Decomposition to the One Dimensional Delay Differential Equation by Fixed the Coefficient b<0

2013 ◽  
Vol 785-786 ◽  
pp. 1418-1422
Author(s):  
Ai Gao
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Stability Domain ◽  
Transcendental Equation ◽  
One Dimension ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Delay Differential

In this paper, we provide a partition of the roots of a class of transcendental equation by using τ-D decomposition ,where τ>0,a>0,b<0 and the coefficient b is fixed.According to the partition, one can determine the stability domain of the equilibrium and get a Hopf bifurcation diagram that can provide the Hopf bifurcation curves in the-parameter space, for one dimension delay differential equation .

FRACTAL EVOLUTION IN DETERMINISTIC AND RANDOM MODELS

1996 ◽  
Vol 06 (11) ◽  
pp. 1977-1995
Author(s):  
SIEGFRIED FUSSY ◽  
GERHARD GRÖSSING ◽  
HERBERT SCHWABL
Keyword(s):  
Order Parameter ◽  
Specific Model ◽  
Coupled Map Lattices ◽  
Normalization Procedure ◽  
Quantum Cellular Automata ◽  
Time Step ◽  
One Dimensional ◽  
Random Models ◽  
Approximate Estimation ◽  
The Stability

One-dimensional coupled map lattices or quantum cellular automata with any additionally implemented temporal feedback operations (involving some memory of the system’s states) and a normalization procedure after each time step exhibit a universal dynamic property called fractal evolution [Fussy & Grössing, 1994]. It is characterized by a power-law behavior of a system’s order parameter with regard to a resolution-like parameter which controls the deviation from an undisturbed (i.e., feedback-less) system’s evolution and provides a dynamically invariant measure for the emerging spatiotemporal patterns. By comparison with another, simpler model without memory, where the patterns are generated randomly, the underlying principles of fractal evolution are studied. It is shown that our system evolving entirely deterministically, exhibits properties occurring usually only in random models, where the global measures, up to a certain degree, are calculable. Other properties like the fractal evolution exponent remain in general computationally irreducible due to the self-referential feedback dynamics. A specific model with an approximate estimation of the fractal evolution exponent is discussed. The stability of fractal evolution with respect to the dependence of pattern formation on the systems variables is also analyzed.

Stability of Human Balance With Reflex Delays Using Galerkin Approximations

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Zaid Ahsan ◽  
Thomas K. Uchida ◽  
Akash Subudhi ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Galerkin Approximation ◽  
The Elderly ◽  
Double Pendulum ◽  
Neutral Delay ◽  
Stability Margins ◽  
Human Balance ◽  
The Stability ◽  
The Galerkin Method ◽  
Delay Differential

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

scholarly journals On The Correlation between Properties of One-Dimensional Mappings of Control Functions and Chaos in a Special Type Delay Differential Equation

2017 ◽  
Vol 12 (2) ◽  
pp. 385-397 ◽  
Author(s):  
В.А. Лихошвай ◽  
V.A. Likhoshvai
Keyword(s):  
Differential Equation ◽  
Chaotic Dynamics ◽  
Molecular Genetic ◽  
Numerical Examples ◽  
One Dimensional ◽  
Control Functions ◽  
Delayed Argument ◽  
The One ◽  
Delay Differential ◽  
Dimensional Mapping

A differential equation of a special form, which contains two control functions f and g and one delayed argument, is analyzed. This equation has a wide application in biology for the description of dynamic processes in population, physiological, metabolic, molecular-genetic, and other applications. Specific numerical examples show the correlation between the properties of the one-dimensional mapping, which is generated by the ratio f /g, and the presence of chaotic dynamics for such equation. An empirical criterion is formulated that allows one to predict the presence of a chaotic potential for a given equation by the properties of the one-dimensional mapping f /g.

Steady-state analysis of DC converter using Galerkin’s method

2019 ◽  
Vol 38 (6) ◽  
pp. 2057-2069
Author(s):  
Igor Korotyeyev
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Galerkin Method ◽  
State Space ◽  
Time Interval ◽  
Time Varying ◽  
Content Type ◽  
State Process ◽  
The Galerkin Method ◽  
Steady State Process

Purpose The purpose of this paper is to present the Galerkin method for analysis of steady-state processes in periodically time-varying circuits. Design/methodology/approach A converter circuit working on a time-varying load is often controlled by different signals. In the case of incommensurable frequencies, one can find a steady-state process only via calculation of a transient process. As the obtained results will not be periodical, one must repeat this procedure to calculate the steady-state process on a different time interval. The proposed methodology is based on the expansion of ordinary differential equations with one time variable into a domain of two independent variables of time. In this case, the steady-state process will be periodical. This process is calculated by the use of the Galerkin method with bases and weight functions in the form of the double Fourier series. Findings Expansion of differential equations and use of the Galerkin method enable discovery of the steady-state processes in converter circuits. Steady-state processes in the circuits of buck and boost converters are calculated and results are compared with numerical and generalized state-space averaging methods. Originality/value The Galerkin method is used to find a steady-state process in a converter circuit with a time-varying load. Processes in such a load depend on two incommensurable signals. The state-space averaging method is generalized for extended differential equations. A balance of active power for extended equations is shown.

The Galerkin method applied to convective instability problems

1968 ◽  
Vol 33 (1) ◽  
pp. 201-208 ◽  
Author(s):  
Bruce A. Finlayson
Keyword(s):  
Exact Solution ◽  
Time Dependence ◽  
Galerkin Method ◽  
Convective Instability ◽  
Fluid Layer ◽  
Oscillatory Instability ◽  
Rotating Fluid ◽  
Exponential Time ◽  
The Stability ◽  
The Galerkin Method

The Galerkin method is applied in a new way to problems of stationary and oscillatory convective instability. By retaining the time derivatives in the equations rather than assuming an exponential time-dependence, the exact solution is approximated by the solution to a set of ordinary differential equations in time. Computations are simplified because the stability of this set of equations can be determined without finding the detailed solution. Furthermore, both stationary and oscillatory instability can be studied by means of the same trial functions. Previous studies which have treated only stationary instability by the Galerkin method can now be extended easily to include oscillatory instability. The method is illustrated for convective instability of a rotating fluid layer transferring heat.

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