The numerical solution of elliptic differential equations when the boundary conditions involve a derivative

1950 ◽  
Vol 242 (849) ◽  
pp. 345-378 ◽  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Elastic Plates ◽  
Boundary Values ◽  
Curved Boundaries ◽  
Elliptic Differential Equations ◽  
Fourth Order Equations ◽  
Second Order Equations ◽  
Approximate Equations ◽  
Laborious Method

The numerical solution of differential equations involves the replacement of derivatives by finitedifference equivalents. The idea of using approximate equations and subsequently correcting for the higher differences, already applied to second-order equations with specified boundary values, is here extended to the case where conditions at the boundary involve a derivative. The method is applied with examples to second- and fourth-order equations. The more difficult problems associated with curved boundaries are discussed, with particular reference to problems of stretching of flat elastic plates. An alternative but more laborious method of obtaining accurate solutions, the method of ‘the deferred approach to the lim it’, is illustrated by examples.

scholarly journals Wang-Ball Polynomials for the Numerical Solution of Singular Ordinary Differential Equations

2021 ◽  
pp. 941-949
Author(s):  
Ahmed Kherd ◽  
Azizan Saaban ◽  
Ibrahim Eskander Ibrahim Fadhel
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Operational Matrix ◽  
Second Order ◽  
Operational Matrices ◽  
Second Order Equations

This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions.

scholarly journals An efficient integrator that uses Gauss-Radau spacings

1985 ◽  
Vol 83 ◽  
pp. 185-202 ◽  
Author(s):  
Edgar Everhart
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Second Order ◽  
Double Precision ◽  
The Past ◽  
Fortran Code ◽  
Order Case ◽  
Speed And Accuracy ◽  
Second Order Equations

AbstractThis describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R. over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First- and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

POST ADAPTATION FOR A NUMERICAL SOLUTION OF THE SPHERICALLY-SYMMETRIC RIEMANN PROBLEM

1990 ◽  
Vol 01 (04) ◽  
pp. 285-298
Author(s):  
B. YUDANIN ◽  
M. LAX
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Riemann Problem ◽  
General Purpose ◽  
Spherically Symmetric ◽  
Lagrangian Coordinates ◽  
Point Boundary ◽  
Boundary Values

A"folding" transformation is used to reduce a two point boundary value problem to one point boundary values (with double the number of functions). A transformation to quasi-Lagrangian coordinates is used to transform discontinuities to rest. After these transformations the general purpose numerical package POST (partial and ordinary differential equations solver in time and one space coordinates) can be successfully applied to a system of hydrodynamical equations, whose solution exhibits jumps and cusp-type discontinuities. Numerical results are presented for the spherically-symmetric shock problem.

A Note on the Numerical Solution of Fourth Order Differential Equations

1954 ◽  
Vol 5 (4) ◽  
pp. 176-184 ◽  
Author(s):  
L. C. Woods
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Viscous Flow ◽  
Numerical Method ◽  
Fourth Order ◽  
Flow Equation ◽  
Higher Order ◽  
Relaxation Procedure ◽  
Fourth Order Equations ◽  
Correction Terms

SummaryAn old numerical method of solving fourth order differential equations is put in relaxation form. The higher order correction terms are included and the technique is illustrated by an example. The method has the advantage of being more rapidly convergent than the usual relaxation procedure for fourth order equations. Some comments are made on the numerical solution of the viscous flow equation.

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