7,728 research outputs found
Modified Sumudu Transform Analytical Approximate Methods For Solving Boundary Value Problems
In this study, emphasis is placed on analytical approximate methods. These methods include the combination of the Sumudu transform (ST) with the homotopy perturbation method (HPM), namely the Sumudu transform homotopy perturbation method (STHPM), the combination of the ST with the variational iteration method (VIM), namely the Sumudu transform variational iteration method (STVIM) and finally, the combination of the ST with the homotopy analysis method (HAM), namely the Sumudu transform homotopy analysis method (STHAM). Although these standard methods have been successfully used in solving various types of differential equations, they still suffer from the weakness in choosing the initial guess. In addition, they require an infinite number of iterations which negatively affect the accuracy and convergence of the solutions. The main objective of this thesis is to modify, apply and analyze these methods to handle the difficulties and drawbacks and find the analytical approximate solutions for some cases of linear and nonlinear ordinary differential equations (ODEs). These cases include second-order two-point boundary value problems (BVPs), singular and systems of second-order two-point BVPs. For the proposed methods, the trial function was employed as an initial approximation to provide more accurate approximate solutions for the considered problems. In addition, for the STVIM method, a new algorithm has been proposed to solve various kinds of linear and nonlinear second-order two-point BVPs. In this algorithm, the convolution theorem has been used to find an optimal Lagrange multiplier
Modified Sumudu Transform Analytical Approximate Methods For Solving Boundary Value Problems
In this study, emphasis is placed on analytical approximate methods. These
methods include the combination of the Sumudu transform (ST) with the homotopy
perturbation method (HPM), namely the Sumudu transform homotopy perturbation
method (STHPM), the combination of the ST with the variational iteration method
(VIM), namely the Sumudu transform variational iteration method (STVIM) and finally,
the combination of the ST with the homotopy analysis method (HAM), namely
the Sumudu transform homotopy analysis method (STHAM). Although these standard
methods have been successfully used in solving various types of differential equations,
they still suffer from the weakness in choosing the initial guess. In addition, they
require an infinite number of iterations which negatively affect the accuracy and convergence
of the solutions. The main objective of this thesis is to modify, apply and
analyze these methods to handle the difficulties and drawbacks and find the analytical
approximate solutions for some cases of linear and nonlinear ordinary differential
equations (ODEs). These cases include second-order two-point boundary value
problems (BVPs), singular and systems of second-order two-point BVPs. For the proposed
methods, the trial function was employed as an initial approximation to provide
more accurate approximate solutions for the considered problems. In addition, for the
STVIM method, a new algorithm has been proposed to solve various kinds of linear
and nonlinear second-order two-point BVPs. In this algorithm, the convolution theorem
has been used to find an optimal Lagrange multiplier
An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method
In this paper we propose a collocation method for solving some well-known
classes of Lane-Emden type equations which are nonlinear ordinary differential
equations on the semi-infinite domain. They are categorized as singular initial
value problems. The proposed approach is based on a Hermite function
collocation (HFC) method. To illustrate the reliability of the method, some
special cases of the equations are solved as test examples. The new method
reduces the solution of a problem to the solution of a system of algebraic
equations. Hermite functions have prefect properties that make them useful to
achieve this goal. We compare the present work with some well-known results and
show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Variational methods for solving nonlinear boundary problems of statics of hyper-elastic membranes
A number of important results of studying large deformations of hyper-elastic
shells are obtained using discrete methods of mathematical physics. In the
present paper, using the variational method for solving nonlinear boundary
problems of statics of hyper-elastic membranes under the regular hydrostatic
load, we investigate peculiarities of deformation of a circular membrane whose
mechanical characteristics are described by the Bidermann-type elastic
potential. We develop an algorithm for solving a singular perturbation of
nonlinear problem for the case of membrane loaded by heavy liquid. This
algorithm enables us to obtain approximate solutions both in the presence of
boundary layer and without it. The class of admissible functions, on which the
variational method is realized, is chosen with account of the structure of
formal asymptotic expansion of solutions of the corresponding linearized
equations that have singularities in a small parameter at higher derivatives
and in the independent variable. We give examples of calculations that
illustrate possibilities of the method suggested for solving the problem under
consideration
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