scholarly journals Multisegment Scheme Applications to Modified Chebyshev Picard Iteration Method for Highly Elliptical Orbits

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Donghoon Kim ◽  
John L. Junkins ◽  
James D. Turner
Keyword(s):  
Iteration Method ◽  
Initial Boundary ◽  
Picard Iteration ◽  
True Anomaly ◽  
Multiple Time ◽  
Data Sampling ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
The Given ◽  
Dense Sampling

A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly.

Initial-boundary value problems for a time-fractional differential equation with involution perturbation

2019 ◽  
Vol 14 (3) ◽  
pp. 312 ◽  
Author(s):  
Nasser Al-Salti ◽  
Sebti Kerbal ◽  
Mokhtar Kirane
Keyword(s):  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
Formal Solutions ◽  
The Given

Direct and inverse initial-boundary value problems of a time-fractional heat equation with involution perturbation are considered using both local and nonlocal boundary conditions. Results on existence of formal solutions to these problems are presented. Solutions are expressed in a form of series expansions using appropriate orthogonal basis obtained by separation of variables. Convergence of series solutions are obtained by imposing certain conditions on the given data. Uniqueness of the obtained solutions are also discussed. The obtained general solutions are illustrated by an example using an appropriate choice of the given data.

scholarly journals Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Perturbation Method ◽  
Boundary Value Problems ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Initial Boundary Value

A variational homotopy perturbation method (VHPM) which is based on variational iteration method and homotopy perturbation method is applied to solve the approximate solution of the fractional initial boundary value problems. The nonlinear terms can be easily handled by the use of He's polynomials. It is observed that the variational iteration method is very efficient and easier to implements; illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

scholarly journals Comment on “Variational Iteration Method for Fractional Calculus Using He’s Polynomials”

2012 ◽  
Vol 2012 ◽  
pp. 1-2 ◽  
Author(s):  
Ji-Huan He
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Initial Boundary ◽  
Iteration Algorithm ◽  
Initial Boundary Value Problems ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Fractional Differential ◽  
Initial Boundary Value ◽  
Modified Variational Iteration Method

Recently Liu applied the variational homotopy perturbation method for fractional initial boundary value problems. This note concludes that the method is a modified variational iteration method using He’s polynomials. A standard variational iteration algorithm for fractional differential equations is suggested.

scholarly journals On the solutions of nonlinear initial-boundary value problems

2004 ◽  
Vol 2004 (5) ◽  
pp. 407-424 ◽  
Author(s):  
Vladimír Ďurikovič ◽  
Monika Ďurikovičová
Keyword(s):  
Evolution Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Qualitative Properties ◽  
Science Models ◽  
Initial Boundary Value ◽  
The Given

We deal with the general initial-boundary value problem for a second-order nonlinear nonstationary evolution equation. The associated operator equation is studied by the Fredholm and Nemitskii operatortheory. Under local Hölder conditions for the nonlinear member, we observe quantitative and qualitative properties of the set of solutions of the given problem. These results can be applied to different mechanical and natural science models.

Existence, Uniqueness and Asymptotic Behaviour for a Non-Linear Beam Equation

1995 ◽  
Author(s):  
G. Judith Boertjens ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Methods ◽  
Flexible Structures ◽  
Initial Boundary ◽  
Beam Equation ◽  
Multiple Time ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
The Right ◽  
Multiple Time Scale Method ◽  
Asymptotic Validity

Abstract The use of perturbation methods for fourth order PDE’s has not yet been examined extensively. Usually approximating power series are applied, which are truncated to one or two modes. Very little — or nothing — is said about the relation between this approximation and the exact solution. In this paper initial boundary value problems for the following equation will be discussed: w t t + w x x x x + ϵ ( u ( π , t ) − u ( 0 , t ) + ∫ 0 π w x 2 d x ) w x x = ϵ g ( x , t , w , w t ) . This equation can be regarded as a model describing wind-induced oscillations of flexible structures like elastic beams, where the small term on the right hand side of the equation represents the windforce acting on the structure. Existence and uniqueness for solutions of these problems will be discussed, as well as finding approximations using a multiple time-scale method. Finally the asymptotic validity of these approximations will be considered.

On high-order approximations for describing the lagging behavior of heat conduction

2018 ◽  
Vol 24 (6) ◽  
pp. 1648-1667 ◽  
Author(s):  
Stan Chiriţă
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Constitutive Equation ◽  
Heat Transport ◽  
Initial Boundary ◽  
High Order ◽  
Order Effects ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
The Given

This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) [Formula: see text]; (ii) [Formula: see text]; and (iii) [Formula: see text]. This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for [Formula: see text], while for [Formula: see text] some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

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