On perturbation problems associated with finite boundaries

1948 ◽  
Vol 44 (2) ◽  
pp. 251-262 ◽  
Author(s):  
G. D. Wassermann
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Taylor Expansion ◽  
Displacement Vector ◽  
Surface Displacement ◽  
Equation Boundary ◽  
Subsequent Separation ◽  
Perturbation Problems ◽  
Good Agreement

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

scholarly journals Boundary conditions for the wave equation

1933 ◽  
Vol 141 (843) ◽  
pp. 216-217
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Existence Theorems ◽  
Theoretical Discussion ◽  
Wave Mechanics ◽  
Eigen Values ◽  
Eigen Solutions ◽  
Good Agreement

A single electron in the field of two fixed nuclei, constituting the idealized hydrogen molecular ion, provides the simplest case for the application of wave mechanics to molecular, as distinct from atomic, problems. The most extensive theoretical discussion of the corresponding wave equation has been given by A. H. Wilson in these 'Proceedings.’ He was led to conclude that this equation possesses no eigen-solutions satisfying the usual boundary conditions for an atomic problem. Subsequent investigators have succeeded, however, in obtaining by numerical methods eigen-values in good agreement with observed values of the energy. But, with the exception of Teller, they appear not to have taken account of Wilson’s result. It is therefore worth while to investigate the existence of their solutions and to clear up, if possible, any doubt as to the applicability of the familiar boundary conditions to this type of problem. The usual existence theorems for eigen-values apply only to boundary conditions at ordinary points of the differential equation. The difficulty in cases like Wilson’s equation is that the conditions are given at singular points.

scholarly journals Criticality dependence on data and parameters for a problem in combustion theory

1986 ◽  
Vol 27 (4) ◽  
pp. 416-441 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Boundary Data ◽  
Main Part ◽  
Original Problem ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Central Problem ◽  
Good Agreement

AbstractA central problem in the theory of combustion, consisting of a nonlinear parbolic equation together with initial and boundary conditions, is considered. The influence of the initial and boundary data examined. In the main part of the study, a two-step linearization is developed such that the interesting features of the original problem are given by the solution of a non-liner and ordinary differential equation. Approximate solutions are obtained and upper and lower solutions are used to assess the validity of the approximations. Whenever possible, results are compared with those obtained previously and there is good agreement in all cases.

Load-Supporting Fluid-Filled Cylindrical Membranes

1985 ◽  
Vol 52 (4) ◽  
pp. 913-918 ◽  
Author(s):  
V. Namias
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Entire Range ◽  
Exact Analytical Solution ◽  
Asymptotic Expressions ◽  
Flexible Membranes ◽  
Approximate Expressions

When long cylindrical flexible membranes are filled with a fluid and used to support external weights, the shape they assume and the relevant geometrical and dynamical quantities are governed by a nonlinear differential equation subject to particular boundary conditions. First, a complete and exact analytical solution is obtained for an unloaded membrane. Very accurate approximate expressions are derived directly from the exact solution for the entire range of applied pressures and fluid densities. Next, the nonlinear differential equation is solved exactly under boundary conditions corresponding to the loading of the membrane. Simple asymptotic expressions are also obtained in the limit of large loads.

An Analysis of Nonsimilar Problems on Spherical Blast Waves

1976 ◽  
Vol 31 (7) ◽  
pp. 723-727
Author(s):  
Fumio Higashino
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Blast Wave ◽  
Particle Velocity ◽  
Analytical Method ◽  
Blast Waves ◽  
Adiabatic Exponent ◽  
Von Neumann ◽  
Propagation Regime ◽  
Good Agreement

Abstract An approximate analytical method that is valid for the entire propagation regime of a blast wave is established for point explosions. The method is based on an application of the shock ex-pansion method to boundary conditions. The numerical calculations were carried out for spherical flows with constant adiabatic exponent. The results show that the value of the shock decay coefficient gives good agreement with the numerically exact solution by Goldstine and von Neumann. The profiles of the pressure, density, and particle velocity behind the shock were obtained. The solutions show the possible profiles as compared to other existing analyses.

scholarly journals The elastic stability of a thin twisted strip—II

1937 ◽  
Vol 161 (905) ◽  
pp. 197-220 ◽  
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Main Part ◽  
Fourier Expansion ◽  
Elastic Stability ◽  
Numerical Work ◽  
The Stability ◽  
Good Agreement

I—In a previous paper the present writer discussed both theoretically and experimentally the equilibrium and elastic stability of a thin twisted strip, and the results obtained by the theory were found to be in good agreement with observation. It has, however, been pointed out by Professor Southwell, F. R. S., that the solution of the stability equations which was given in that paper may only be regarded as an approximate solution for, although it satisfies exactly the differential equations and two boundary conditions along the edge of the strip, it only satisfies the two remaining boundary conditions approximately. The author has also noticed that the coefficients n a m in the Fourier expansion of θ 2 cos mθ which were used in A are incorrect when m = 0, and this has led to errors in the numerical work so that the values of ᴛb 2 / π 2 h which are given in Table I of A are wrong. In the present paper a solution of the stability equations is obtained which satisfies all the boundary conditions. This solution is very much more complicated than the approximate solution and much greater labour is required for the numerical work. The numerical work for the approximate solution of A has also been revised and the corrected results are given in 9, 10. It is found that the results for the approximate solution are in good agreement with those obtained from the exact solution and that both agree moderately well with the experimental results which are given in A. The main part of this paper is an extension of the previous work and is concerned with the stability of a thin twisted strip when it is subjected to a tension along its length. The theory has been compared with experiment and satisfactorily good agreement between them was found.

scholarly journals An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Mariusz Ciesielski ◽  
Tomasz Blaszczyk
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Constant Coefficients ◽  
Initial Boundary Value

We analysed the initial/boundary value problem for the second-order homogeneous differential equation with constant coefficients in this paper. The second-order differential equation with respect to the fractional/generalised boundary conditions is studied. We presented particular solutions to the considered problem. Finally, a few illustrative examples are shown.

scholarly journals Computation of Stray Losses in Transformer Bushing Regions Considering Harmonics in the Load Current

2020 ◽  
Vol 10 (10) ◽  
pp. 3527
Author(s):  
Sohail Khan ◽  
Serguei Maximov ◽  
Rafael Escarela-Perez ◽  
Juan Carlos Olivares-Galvan ◽  
Enrique Melgoza-Vazquez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Eddy Current ◽  
Special Functions ◽  
Separation Of Variables ◽  
Practical Implementation ◽  
Load Current ◽  
Good Agreement

The presence of harmonics in the load current considerably increases stray losses in electric transformers. In this research paper, a new model for computing the electromagnetic field (EMF) and eddy current (EC) losses in transformer tank covers is derived considering harmonics. Maxwell’s equations are solved with their corresponding boundary conditions. The differential equation thus obtained is solved using the method of separation of variables. The obtained expressions do not require the use of special functions, accommodating them for practical implementation in the industry. The obtained formulas are evaluated for different spectrum contents of the load current and losses. The results are in good agreement with simulations carried out using the Altair Flux finite element (FE) software.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

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