A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

2015 ◽  
Vol 5 (2) ◽  
pp. 192-208 ◽  
Author(s):  
Ning Li ◽  
Bo Meng ◽  
Xinlong Feng ◽  
Dongwei Gui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Finite Difference ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Numerical Comparison ◽  
Fe Method

AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Riccardo Barsotti
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Nonlinear Ordinary Differential Equation ◽  
Experimental Results ◽  
Limit Case ◽  
Reasonable Approximation ◽  
Constituent Material ◽  
Actual Solution ◽  
Wrinkled Membrane

The axisymmetric inflation problem for a wrinkled membrane is solved by means of a simple nonlinear ordinary differential equation. The solution is illustrated in full details. Both the free and constrained cases are addressed, in the limit case where the membrane is fully wrinkled. In the constrained inflation problem, no slippage is allowed between the membrane and the constraining surfaces. It is shown that an actual membrane can in no way reach the fully wrinkled configuration during free inflation, regardless of the membrane's initial configuration and constituent material. The fully wrinkled solution is compared to some finite element results obtained by means of an expressly developed iterative–incremental procedure. When the values of the inflating pressure and length of the meridian lie within a suitable applicability range, the fully wrinkled solution may represent a reasonable approximation of the actual solution. A comparison with some numerical and experimental results available in the literature is illustrated.

Diffusion Approximation of State-Dependent G-Networks Under Heavy Traffic

2008 ◽  
Vol 45 (2) ◽  
pp. 347-362 ◽  
Author(s):  
Saul C. Leite ◽  
Marcelo D. Fragoso
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Approximation ◽  
Numerical Experiments ◽  
Heavy Traffic ◽  
Weak Sense ◽  
Feedforward Network ◽  
State Dependent ◽  
Number Of Customers

This paper is concerned with the characterization of weak-sense limits of state-dependent G-networks under heavy traffic. It is shown that, for a certain class of networks (which includes a two-layer feedforward network and two queues in tandem), it is possible to approximate the number of customers in the queue by a reflected stochastic differential equation. The benefits of such an approach are that it describes the transient evolution of these queues and allows the introduction of controls, inter alia. We illustrate the application of the results with numerical experiments.

On The Fluid-Solid Interaction in Reference to Thermoelastohydrodynamic Analysis of Journal Bearings

1991 ◽  
Vol 113 (2) ◽  
pp. 398-404 ◽  
Author(s):  
M. M. Khonsari ◽  
S. H. Wang
Keyword(s):  
Finite Element ◽  
Thermal Expansion ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Journal Bearings ◽  
Collective Effect ◽  
Thermoelastic Deformation ◽  
Simulation Results ◽  
Thermal Dilation ◽  
Fluid Solid Interaction

Thermohydrodynamic analysis of journal bearings is extended to include provisions for the shaft thermal dilation as well as the bush thermoelastic deformation. Numerical solutions using a combination of the finite difference and finite element methods are presented. Comparison of the simulation results with those obtained experimentally yielded satisfactory agreement. It was found that while the shaft and bush thermal expansion and the bush elastic deformation are individually important, the collective effect of these factors must be considered for meaningful end results.

Development of a Model for Analysing Radial Flow of Slightly Compressible Fluids

2009 ◽  
Vol 62-64 ◽  
pp. 629-636
Author(s):  
John A. Akpobi ◽  
E.D. Akpobi
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference ◽  
Radial Flow ◽  
Compressible Fluids ◽  
The Finite Element Method ◽  
Time Approximation ◽  
Slightly Compressible ◽  
Fluid Properties

In this work, we develop a finite element-finite difference method to solve the differential equation governing the radial flow of slightly compressible fluids. The finite element method is used to carry out spatial approximations so as to study the variation of fluid properties at the various nodes to which effect we divided the entire radial domain of the fluid into a mesh of four radial 1-D quadratic elements which exposes nine nodes to intense study. Time approximation is done with the aid of the Crank-Nicolson finite difference scheme.

On the theory and application of one-step numerical schemes for solving quantum stochastic differential equation (QSDE)

2014 ◽  
Vol 07 (03) ◽  
pp. 1450037
Author(s):  
T. O. Akinwumi ◽  
B. J. Adegboyegun
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Numerical Schemes ◽  
Quantum Stochastic Differential Equation ◽  
One Step ◽  
Definition Of ◽  
Convergent Sequences ◽  
Solution Techniques

This paper presents one-step numerical schemes for solving quantum stochastic differential equation (QSDE). The algorithms are developed based on the definition of QSDE and the solution techniques yield rapidly convergent sequences which are readily computable. As well as developing the schemes, we perform some numerical experiments and the solutions obtained compete favorably with exact solutions. The solution techniques presented in this work can handle all class of QSDEs most especially when the exact solution does not exist.

scholarly journals On the link between finite difference and derivative of polynomials

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Differential Calculus ◽  
Divided Difference ◽  
Applied Mathematics ◽  
Numerical Differentiation ◽  
Discrete Analog ◽  
Derivatives Of

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.MSC 2010: 46G05, 30G25, 39-XXarXiv:1608.00801Keywords: Finite difference, Derivative, Divided difference, Ordinary differential equation, Partial differential equation, Partial derivative, Differential calculus, Difference Equations, Numerical Differentiation, Finite difference coefficient, Polynomial, Power function, Monomial, Exponential function, Exponentiation, arXiv, Preprint, Calculus, Mathematics, Mathematical analysis, Numerical methods, Applied Mathematics

A Numerical Procedure for Obtaining the Static and Pull-In Deflection and Voltage of Capacitive Microcantilever Beams

2006 ◽  
Author(s):  
Seyed Babak Ghaemi Oskouei ◽  
Aria Alasty
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Numerical Procedure ◽  
Nonlinear Ordinary Differential Equation ◽  
Static Deflection ◽  
Element Analysis ◽  
Fringing Field ◽  
Good Agreement

A numerical procedure is proposed for obtaining the static deflection, pull-in (PI) deflection and PI voltage of electrostatically excited capacitive microcantilever beams. The method is not time and memory consuming as Finite Element Analysis (FEA). Nonlinear ordinary differential equation of the static deflection of the beam is derived, w/wo considering the fringing field effects. The nondimensional parameters upon which PI voltage is dependent are then found. Thereafter, using the parameters and the numerical method, three closed form equations for pull-in voltage are developed. The results are in good agreement with others in literature.

scholarly journals Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Josef Rebenda ◽  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Neutral System ◽  
Initial Value ◽  
Differential Transformation ◽  
Functional Differential

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

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