scholarly journals LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 10 (24) ◽  
pp. 1795-1802 ◽  
Author(s):  
YURI SMIRNOV ◽  
ALEXANDER TURBINER
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Heisenberg Algebra ◽  
Difference Operators ◽  
Discrete Version ◽  
Classical Orthogonal Polynomials ◽  
Finite Difference Equations ◽  
Exactly Solvable

A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl 2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.

scholarly journals Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

2020 ◽  
Vol 7 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Bolat Duissenbekov ◽  
Abduhalyk Tokmuratov ◽  
Nurlan Zhangabay ◽  
Zhenis Orazbayev ◽  
Baisbay Yerimbetov ◽  
...  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Constant Load ◽  
Time Interval ◽  
Test Case ◽  
Algebraic Equations ◽  
Nonlinear Properties ◽  
Finite Difference Equations ◽  
Concrete Shells

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables, i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

Optimal Finite Analytic Methods

1982 ◽  
Vol 104 (3) ◽  
pp. 432-437 ◽  
Author(s):  
R. Manohar ◽  
J. W. Stephenson
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Linear Combination ◽  
Difference Equations ◽  
New Method ◽  
Linear Partial Differential Equations ◽  
Finite Difference Equations

A new method is proposed for obtaining finite difference equations for the solution of linear partial differential equations. The method is based on representing the approximate solution locally on a mesh element by polynomials which satisfy the differential equation. Then, by collocation, the value of the approximate solution, and its derivatives at the center of the mesh element may be expressed as a linear combination of neighbouring values of the solution.

On the operational solution of linear mixed difference differential equations

1934 ◽  
Vol 30 (4) ◽  
pp. 389-391 ◽  
Author(s):  
Jacob Neufeld
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Finite Difference Equations ◽  
Operational Solution

L. M. Milne-Thomson has recently described a method of solving linear finite difference equations by using processes some-what analogous to those employed by Heaviside.

On the Operational Solution of Linear Finite Difference Equations

1931 ◽  
Vol 27 (1) ◽  
pp. 26-36 ◽  
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Operational Method ◽  
Finite Difference Equations ◽  
Operational Solution ◽  
Linear Differential

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.

scholarly journals A study of resolvent set for a class of band operators with matrix elements

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Andrey Osipov
Keyword(s):  
Asymptotic Behavior ◽  
Finite Difference ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Jacobi Matrices ◽  
Matrix Elements ◽  
Polynomial Solutions ◽  
Resolvent Set ◽  
Finite Difference Equations

AbstractFor operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.

Divergence of solutions of polynomial finite-difference equations

2012 ◽  
Vol 142 (4) ◽  
pp. 787-804 ◽  
Author(s):  
Harry Gingold
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Difference Equations ◽  
Step Method ◽  
Quadratic Equations ◽  
Finite Difference Equations ◽  
Order Polynomial ◽  
Independent Parameters

A theorem is proven for kth-order polynomial finite-difference equations that guarantees the divergence of solutions. A ‘basin of divergence’ is characterized and an order of divergence is provided. The basin of divergence is shown to depend on k independent parameters. An unconventional compactification method is used. Applications include the multi-step method in the numerical integration of ordinary differential equations, quadratic equations and the Henon map.

scholarly journals On the Discrete Version of the Schwarzschild Problem

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 185
Author(s):  
Vladimir Khatsymovsky
Keyword(s):  
Finite Difference ◽  
Difference Equations ◽  
Planck Scale ◽  
Discrete Version ◽  
Length Scale ◽  
Slow Rotation ◽  
Finite Difference Equations ◽  
Elementary Length ◽  
Starting Point ◽  
Schwarzschild Problem

We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert–Einstein action. Using for the Schwarzschild problem a priori general non-spherically symmetrical ansatz, we get finite-difference equations for its discrete version. This defines a solution which at large distances is close to the continuum Schwarzschild geometry, and the metric and effective curvature at the center are cut off at the elementary length scale. Slow rotation can also be taken into account (Lense–Thirring-like metric). Thus, we get a general approach to the classical background in the quantum framework in zero order: it is an optimal starting point for the perturbative expansion of the theory, finite-difference equations are classical, and the elementary length scale has quantum origin. Singularities, if any, are resolved.

Application of discrete differential operators of periodic functions to solve 1D boundary-value problems

2020 ◽  
Vol 39 (4) ◽  
pp. 885-897 ◽  
Author(s):  
Tadeusz Sobczyk ◽  
Marcin Jaraczewski
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Differential Operators ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Periodic Functions ◽  
Content Type ◽  
Finite Difference Equations

Purpose Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations. Design/methodology/approach The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images. Findings An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems. Research limitations/implications The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes. Practical implications Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established. Originality/value New finite-difference expressions for the first- and second-order derivatives have been applied.

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