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.

scholarly journals On the Theory of Relaxation

1953 ◽  
Vol 1 (3) ◽  
pp. 101-110 ◽  
Author(s):  
A. R. Mitchell ◽  
D. E. Rutherford
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Difference Equations ◽  
Linear Equations ◽  
Finite Difference Equations ◽  
Linear Differential

§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

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.

III. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

1863 ◽  
Vol 12 ◽  
pp. 420-424
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamical Problem ◽  
General Character ◽  
Partial Differential ◽  
Central Function ◽  
First Order ◽  
Linear Partial Differential Equations

Jacobi in a posthumous memoir, which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the differential equations of dynamics which was established by Sir W. R. Hamilton in the 'Philosophical Transactions’ for 1834-35. The knowledge, indeed, that the solution of the equation of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results.

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.

Integration of Partial Differential Equation for Transient Radial Flow of Gas-Condensate Fluids in Porous Structures

1965 ◽  
Vol 5 (02) ◽  
pp. 141-152 ◽  
Author(s):  
C.K. Eilerts ◽  
E.F. Sumner ◽  
N.L. Potts
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Difference Equations ◽  
Radial Flow ◽  
Gas Condensate ◽  
Apparent Velocity ◽  
Partial Differential ◽  
Darcy Equation ◽  
Finite Difference Equations

Abstract The second-order, nonlinear, partial-differential equation representing the transient radial flow of gas-condensate fluids in reservoirs has been integrated by using finite-difference equations and electronic computers. Effect was given to pressure-dependent permeability, viscosity, and compressibility and to distance-dependent permeability. The influence of a second-degree velocity term in the Darcy equation was investigated. Implicit methods were used and practical, convergent solutions were obtained with material balance to less than 6 x 10 for recovery of one-half the reserve at constant flow rate. Integration results provide the productive period of a reservoir for a given constant rate and the fraction of the fluid initially in place that can be recovered in that period. The properties of a lean and a rich fluid are represented in a set of integrations designed to demonstrate the effect of different constant-recovery rates and significant variables emphasized one at a time. Introduction A method is needed for computing the availability or reserve of a gas in a formation, making use of all technical and engineering information that is pertinent. As a step in this direction, a program for computing transient linear flow was developed that utilizes principles of earlier finite-difference computing for a similar purpose and gives effect to significant pressure- and distance-dependent properties of reservoirs and their contents. The present paper pertains to transient, radial flow of gas-condensate fluids. In the partial-differential equation of flow, effect is given to the variables viscosity and compressibility, and also to permeability. Included in the study is the quadratic form of the Darcy equation of flow that has been the subject of field tests and that has been applied to a gas with constant properties in transient flow computing. Inclusion in the differential equation of variable coefficients to represent properties greatly complicates the higher derivatives of the equation. Because it was impractical to make a required improvement in certain of the finite-difference derivatives by Taylor-series expansion, five-term derivatives were used in the implicit computing. Related methods were developed that can improve general facility in manipulating finite-difference equations. BASIC EQUATIONS PARTIAL DIFFERENTIAL EQUATION The basic partial differential equation for transient radial flow in the direction of decreasing radius is (1) where r radius, Lv = apparent velocity along radius, L/tp = density, M/L3t = time, tphi = porosity, dimensionless. In this equation, only the porosity is regarded constant. The density is (2) where p = pressure, m/LtM = molecular weight of fluid, mT = constant temperature of fluid, Tz(p)= pressure- dependent compressibilityfactor, dimensionlessR = gas constant, mL2/t2T. SPEJ P. 141ˆ

On matrix methods for the solution of partial differential equations

1965 ◽  
Vol 61 (1) ◽  
pp. 129-132 ◽  
Author(s):  
M. Wadsworth ◽  
A. Wragg
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Laplace Transformation ◽  
Time Derivative ◽  
Partial Differential ◽  
Matrix Methods ◽  
Finite Difference Equations ◽  
Difference Form ◽  
Image Position

Bickley and McNamee (1) describe techniques for obtaining the solution of finite difference equations, arising from partial differential equations, making extensive use of matrix methods. In all cases solutions are obtained by solving algebraic equations as distinct from differential equations. For example, in order to solvethe second space derivative is replaced by finite differences and the time derivative is replaced either by substituting the backward finite difference form or by using the Laplace transformation.

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