scholarly journals A decomposition method for a semilinear boundary value problem with a quadratic nonlinearity

2005 ◽  
Vol 2005 (6) ◽  
pp. 855-861 ◽  
Author(s):  
Michael S. Gordon
Keyword(s):  
Boundary Value Problem ◽  
Decomposition Method ◽  
Series Solution ◽  
Boundary Value ◽  
Quadratic Nonlinearity ◽  
One Dimensional ◽  
Semilinear Heat Equation ◽  
Convergence Results ◽  
Dimensional Boundary ◽  
Local And Global Convergence

The author adapts the decomposition method of Adomian to find a series solution of a one-dimensional boundary value problem for a semilinear heat equation with a quadratic nonlinearity. Local and global convergence results are obtained.

scholarly journals MÉTODO DE DIFERENCIAS FINITAS PARA UN PROBLEMA DE VALOR DE FRONTERA UNIDIMENSIONAL

2019 ◽  
pp. 5-8
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Sparse Matrices ◽  
Boundary Value ◽  
Bending Moment ◽  
One Dimensional ◽  
Dimensional Boundary ◽  
The One

MÉTODO DE DIFERENCIAS FINITAS PARA UN PROBLEMA DE VALOR DE FRONTERA UNIDIMENSIONAL THE FINITE- DIFERENCE METHOD FOR A ONE-DIMENSIONAL BOUNDARY-VALUE PROBLEM Luis Jaime Collantes Santisteban, Samuel Collantes Santisteban DOI: https://doi.org/10.33017/RevECIPeru2006.0011/ RESUMEN En este trabajo se considera el problema de valor de frontera unidimensional dado en (1). Se aproxima la solución del problema mediante el método de diferencias finitas suponiendo que la función c(x) es no negativa sobre 0,1, lo que permite establecer la convergencia del método de aproximación. El uso del método de diferencias finitas, a la vez, involucra la solución de sistemas de ecuaciones lineales con matrices muy ralas, cuyos ceros están posicionados de una manera remarcable. Dichas matrices son de tipo tridiagonal. Para la solución de dichos sistemas se ha utilizado el método de Thomas. Palabras clave: problema de valor de frontera unidimensional, diferencias finitas, matriz tridiagonal, método de Thomas, momento flexionante. ABSTRACT In this work the one-dimensional boundary-value problem given in (1) is considered. The solution of the problem by means of finite-difference method comes near supposing that the function c(x) is nonnegative on 0,1, which allows to establish the convergence of the considered method of approximation. The use of the finite-difference method, in turn, involves the solution of linear systems with very sparse‟ matrices, whose zeros are arranged in quite remarkable fashion. These matrices are of tridiagonal type. For the solution of these systems the Thomas‟ method has been used. Keywords: one-dimensional boundary-value problem, finite-difference, tridiagonal matrix, Thomas‟ method, bending moment.

scholarly journals Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils

2020 ◽  
Vol 4 (4) ◽  
pp. 59
Author(s):  
Vsevolod Bohaienko ◽  
Volodymyr Bulavatsky
Keyword(s):  
Boundary Value Problem ◽  
Fractal Structure ◽  
Boundary Value ◽  
Finite Thickness ◽  
Fractal Model ◽  
Consolidation Process ◽  
One Dimensional ◽  
Fractal Approach ◽  
Filtration Consolidation ◽  
Dimensional Boundary

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries.

scholarly journals Algorithm for reducing computational costs in problems of calculation of asymmetrically loaded shells of rotation

2020 ◽  
pp. 99-113
Author(s):  
Anatolii Dzyuba ◽  
Inha Safronova ◽  
Larysa Levitina
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Trigonometric Series ◽  
Strain State ◽  
Boundary Value ◽  
Stress Strain ◽  
One Dimensional ◽  
Dimensional Boundary

The problem of calculating the shells of rotation of a variable along the meridian of rigidity under asymmetric loading is reduced o a set of systems of one-dimensional boundary value problems with respect to the amplitudes of decomposition of the required functions into trigonometric Fourier series. A method for reducing the number of one-dimensional boundary value problems required to achieve a given accuracy in determining the stress-strain state of the shells of rotation with a variable along the meridian wall thickness under asymmetric load. The idea of the proposed approach is to apply periodic extrapolation (prediction) of the values of the decomposition coefficients of the required functions using the results of calculations of previous coefficients of the corresponding trigonometric series, thus replacing them with some prediction values calculated by simple formulas. To solve this problem, we propose the joint use of Aitken-Steffens extrapolation dependences and Adams method in the form of incremental component, which is quite effective in solving the Cauchy problem for systems of ordinary differential equations and is based on Lagrange and Newton extrapolation dependences. The validity of the proposed approach was verified b the results of a systematic numerical experiment by predicting the values of the expansion coefficients in the Fourier series of known functions of one variable. The approach is quite effective in the calculation of asymmetrically loaded shells of rotation with variable along the meridian thickness, when the coefficients of decomposition of the required functions into Fourier series are functions of the longitudina lcoordinate and are calculated by solving the corresponding boundary value problem. In this case, the approach allows solving solutions of differential equations for the amplitudes of decompositionin to trigonometric series only for individual "reference" harmonics, and the amplitudes for every third harmonic can be calculated by interpolating their values for all node integration points of the corresponding boundary value problem. This significantly reduces the computational cost of obtaining the solution as a whole. As an example, the results of the calculation of the stress-strain state of a steel annular plate under asymmetric transverse loading are given.

On the solution to a two-dimensional boundary value problem of heat conduction in a degenerating domain

2020 ◽  
Vol 98 (2) ◽  
pp. 100-109
Author(s):  
Minzilya T. Kosmakova ◽  
◽  
Valery G. Romanovski ◽  
Dana M. Akhmanova ◽  
Zhanar M. Tuleutaeva ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Two Dimensional ◽  
Dimensional Boundary

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

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