Chebyshev Solution of Ordinary Differential Equations

Fox, L.

doi:10.1016/b978-0-08-009660-5.50013-0

Cited by 101 publications

(63 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Known as Bratu's equation, this problem arises in gas dynamics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. It may be see 1,16] that for values of 2>0 sufficiently small, there are two solutions, y* <y~, and that as 2 increases, the two solutions come closer together until, at approximately 2c=3.5137, they coincide.…”

Section: Ii/~y* -Yjll O~ <5 (44)mentioning

confidence: 98%

“…y3 = 0, y(0) = 0 = y(1), yo(1/2) = 3.7. This example describes the motion of a mass suspended between two springs and actually has a countably infinite number of solutions [1][2][3][4][5][6][7][8][9][10][11][12][13]. There is, however, only one solution with y*>0 on (0, 1).…”

Section: Ii/~y* -Yjll O~ <5 (44)mentioning

confidence: 98%

“…[2,3]) or deferred corrections ( [2]) or iterated deferred corrections ( [4][5][6]). In general in these methods, occasional mesh refinements by halving are performed according to certain "event-driven" criteria.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems

Allgower

McCormick

1978

Numer. Math.

View full text Add to dashboard Cite

Summary. The object of this paper is the numerical solution of nonlinear two-point boundary value problems by Newton's method applied to the discretized problem on successively refined grids. The first part consists of a theoretical development of a phenomenon that often occurs in practice, namely, that the number of iterations for Newton's method to converge to within a fixed tolerance and for a fixed starting vector is essentially independent of the mesh size. The second part develops a process based on these results for determining an efficient mesh refinement strategy. Numerical results are also provided. This paper deals with the numerical solution of nonlinear two-point boundary value problems by Newton's method applied to the discretized problem on successively refined grids. One aim is to give a rigorous demonstration for a phenomenon which has frequently been observed in computation, viz. that the number of iterations for Newton's method to converge to within a fixed tolerance and for a fixed starting value is essentially independent of the mesh size. Several theorems having rather complex developments are established in Sections 2 and 3, culminating with the establishment of independence of the number of Newton iterations from the mesh size. In the last section, we use this result as a basis for an efficient mesh refinement strategy. The idea here is that if the discrete problem under consideration is to have ultimately a large number, N, of equally spaced grid points, then the process should begin with a coarse grid of, say, No points. After the Newton iterates have converged to within a prescribed tolerance, the grid is to be refined by a factor p, interpolating the last iterate to the new grid and using this as a new starting value for the Newton

show abstract

Section: Ii/~y* -Yjll O~ <5 (44)mentioning

confidence: 98%

Section: Ii/~y* -Yjll O~ <5 (44)mentioning

confidence: 98%

See 1 more Smart Citation

Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems

Allgower

McCormick

1978

Numer. Math.

View full text Add to dashboard Cite

show abstract

“…Both these systems are of Runge-Kutta type, the solution of which is well documented in the literature [6], and are easily solved on a high speed computer. Ther,e is a difficulty in the present problem owing to the fact that the vectors [Y] and [Y] are not completely known at the end points r = Ri or r = Ro.…”

Section: All Ranmentioning

confidence: 99%

Failure probability analysis of an elastic orthotropic brittle cylinder subjected to axisymmetric thermal and pressure loading

Stanley

Margetson

1977

Int J Fract

View full text Add to dashboard Cite

Procedures for the calculation of the principal stresses in a long hollow circular cylinder of a brittle, orthotropically anisotropic material, subjected to a uniform internal pressure and an axisymmetric radial temperature gradient are developed; the Runge-Kutta method is used in the solution, after an initial iterative stage. An equation for the failure probability of the cylinder, derived from the Weibull distribution, is presented, in which allowance is made for the anisotropy in mechanical strength.In order to illustrate their application, the equations are used to determine the stresses in and failure probability of an isotropic cylinder and a particular anisotropic cylinder, subjected to an internal pressure or a radial temperature gradient. The results are critically discussed.

show abstract

“…This higher order F-D approximation is similar to that of Greenspan [20] in the case of nonequal spacings and to that of Bickley [21] and others [22][23][24][25] 3. No negative coefficients; therefore, no subsequent problems with reducing round-off error [30].…”

Section: Literature Reviewmentioning

confidence: 51%

Formulation and analysis of higher order finite difference approximations to the neutron diffusion equation

Benghanem

View full text Add to dashboard Cite

INFORMATION TO USERSWhile the most advanced technology has been used to photograph and reproduce this manuscript, the quality of the reproduction is heavily dependent upon the quality of the material submitted. For example;• Manuscript pages may have indistinct print. In such cases, the best available copy has been filmed.• Manuscripts may not always be complete. In such cases, a note will indicate that it is not possible to obtain missing pages.• Copyrighted material may have been removed from the manuscript. In such cases, a note will indicate the deletion.Oversize materials (e.g., maps, drawings, and charts) are photographed by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each oversize page is also filmed as one exposure and is available, for an additional charge, as a standard 35mm slide or as a 17"x 23" black and white photographic print.

show abstract

Chebyshev Solution of Ordinary Differential Equations

Cited by 101 publications

References 0 publications

Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems

Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems

Failure probability analysis of an elastic orthotropic brittle cylinder subjected to axisymmetric thermal and pressure loading

Formulation and analysis of higher order finite difference approximations to the neutron diffusion equation

Contact Info

Product

Resources

About