Lacunary Spline Solutions of Fourth Order Initial Value Problem

2010 ◽  
Vol 3 (3) ◽  
pp. 119-129
Author(s):  
K.H. Jwamer ◽  
R.G. Karem
Keyword(s):  
Fourth Order ◽  
Initial Value Problem ◽  
Initial Value

A New Efficient Ode Solver For Solving Initial Value Problem

1998 ◽  
pp. 47-56
Author(s):  
Nazeeruddin Yaacob ◽  
Bahrom Sanugi
Keyword(s):  
Stability Analysis ◽  
Explicit Formula ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Initial Value ◽  
Order Accuracy ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper we develop a new three-stage,fourth order explicit formula of Runge-Kutta type based on Arithmetic and Harmonic means.The error and stability analyses of this method indicate that the method is stable and efficient for nonstiff problems.Two examples are given which illustrate the fcurth order accuracy of the method. Keywords: Runge-Kutta method, Harmonic Mean, three-stage, fourth-order, covergence and stability analysis.

scholarly journals Adam-Bash forth-Multan Predictor Corrector Method for Solving First Order Initial value problem and Its Error Analysis

2021 ◽  
pp. 30-40
Author(s):  
Kedir Aliyi Koroche ◽  
◽  
Geleta Kinkino Mayu ◽  
Keyword(s):  
Fourth Order ◽  
Initial Value Problem ◽  
Present Method ◽  
Numerical Scheme ◽  
Mesh Size ◽  
Stability And Convergence ◽  
Initial Value ◽  
Finite Difference Approximations ◽  
The Stability ◽  
Predictor Corrector

This paper presents fourth order Adams predictor corrector numerical scheme for solving initial value problem. First, the solution domain is discretized. Then the derivatives in the given initial value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of difference equations is developed. The starting points are obtained by using fourth order Runge-Kutta method and then applying the present method to finding the solution of Initial value problem. To validate the applicability of the method, two model examples are solved for different values of mesh size. The stability and convergence of the present method have been investigated. The numerical results are presented by tables and graphs. The present method helps us to get good results of the solution for small value of mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method improves the findings of some existing numerical methods reported in the literature.

scholarly journals THE LARGE-TIME SOLUTION OF A NONLINEAR FOURTH-ORDER EQUATION INITIAL-VALUE PROBLEM I. INITIAL DATA WITH A DISCONTINUOUS EXPANSIVE STEP

2009 ◽  
Vol 51 (2) ◽  
pp. 178-190
Author(s):  
J. A. LEACH ◽  
ANDREW P. BASSOM
Keyword(s):  
Initial Data ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Large Time ◽  
Order Equation ◽  
Order Partial Differential Equation ◽  
Initial Value ◽  
Asymptotic Structure ◽  
Fourth Order Equation ◽  
Negative Constant

AbstractIn this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation ut+uux+γuxxxx=0, −∞<x<∞, t>0, where x and t represent dimensionless distance and time respectively and γ is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x,0)=u0(>0) for x≥0 and u(x,0)=0 for x<0. The method of matched asymptotic expansions is used to obtain the large-time asymptotic structure of the solution to this problem which exhibits the formation of an expansion wave. Whilst most physical applications of this type of equation have γ>0, our calculations show how it is possible to infer the large-time structure of a whole family of solutions for a range of related equations.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

2017 ◽  
Vol 147 (6) ◽  
pp. 1243-1277 ◽  
Author(s):  
Eiji Onodera
Keyword(s):  
Fourth Order ◽  
Energy Method ◽  
Initial Value Problem ◽  
Local Existence ◽  
Three Dimensional ◽  
Closed Curve ◽  
Initial Value ◽  
Curve Flow ◽  
Local Existence And Uniqueness ◽  
Loss Of Derivatives

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

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