High Accuracy Computational Method for Ordinary Differential Equation with Two Small Parameters

2011 ◽  
Vol 179-180 ◽  
pp. 64-69
Author(s):  
Xin Cai
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Asymptotic Properties ◽  
High Accuracy ◽  
Computational Method ◽  
Asymptotic Decomposition ◽  
Small Parameters ◽  
Error Estimations ◽  
Special Case

Ordinary differential equation with two small parameters was considered. Since the presence of two small parameters, the solution of the problem will change rapidly near both sides of the boundary layer. Firstly, the equation was decomposed into several equations in order to have fourth order asymptotic decomposition. The asymptotic properties of all these equations were discussed. Secondly, high order numerical methods were constructed for left side and right side singular component. Thirdly, a class of high numerical methods were presented when the special case. Finally, the error estimations for all these numerical methods were given.

Non-Equidistant Numerical Methods for Ordinary Differential Equation with Periodical Boundary Value

2011 ◽  
Vol 467-469 ◽  
pp. 383-388
Author(s):  
Xin Cai
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Boundary Value ◽  
Singular Component ◽  
Thin Region ◽  
Smooth Component ◽  
Error Estimations ◽  
Transition Points ◽  
Derivatives Of

Ordinary differential equation with periodical boundary value and small parameter multiplied in the highest derivative was considered. The solution of the problem has boundary layers, which is thin region in the neighborhood of the boundary of the domain. Firstly, the properties of boundary layer were discussed. The solution was decomposed into the smooth component and the singular component. The derivatives of the smooth component and the singular component were estimated. Secondly, mesh partition techniques were presented according to one transition point method and multi-transition points method. Thirdly numerical methods based on non-equidistant mesh partition were presented to solve the problem. Finally error estimations were given for both computational methods.

scholarly journals Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 155
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Stochastic Models ◽  
Maximum Likelihood Method ◽  
Diffusion Processes ◽  
Oscillatory Behavior ◽  
Likelihood Method ◽  
Systems Of Equations

Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.

Asymptotic solutions of the Orr—Sommerfeld equation

1975 ◽  
Vol 280 (1295) ◽  
pp. 271-316
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Properties ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Asymptotic Approximations ◽  
General Kind ◽  
Work Done ◽  
Sommerfeld Equation ◽  
Special Case

A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

scholarly journals General Solution Of Nth-Order Linear Ordinary Differential Equation

2021 ◽  
Author(s):  
Rajnish Kumar Jha
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Linear Ordinary Differential Equation ◽  
Order Linear ◽  
Special Case ◽  
Integrating Factors

In this paper we present a solution expression for the general Nth-order linear ordinary differential equation as our main result which involves the use of Integrating Factors where the Integrating Factors are determined using a set of equations such that when this set of equations can be solved, the solution of the concerned differential equation can be determined completely. In this regard we also present result for a special case corresponding to the main result where the solution of the general Nth-order linear ordinary differential equation can be determined completely when N-1 out of N complementary solutions are known.

scholarly journals A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

Open Physics ◽  
2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Eid Doha ◽  
Ali Bhrawy ◽  
Mohamed Abdelkawy ◽  
Ramy Hafez
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Spectral Methods ◽  
Numerical Approximation ◽  
Burgers Equation ◽  
Jacobi Polynomials ◽  
Initial Boundary ◽  
Nonlinear Ordinary Differential Equation ◽  
The Given

AbstractThis article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers’ equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers’ equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.

scholarly journals A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
I. B. Aiguobasimwin ◽  
R. I. Okuonghae
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Initial Value ◽  
Large Interval ◽  
New Class ◽  
Second Derivatives ◽  
Stiff Odes ◽  
Runge Kutta ◽  
Special Case

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

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